Chap5 - ADSP 21K Manual

ADSP-21K Optimized DSP Library User’s Manual

Wideband Computers, Inc. 5-55

CHAPTER 5 Function Descriptions For The ADSP-21K Optimized DSP Library

Each function described in the following pages includes the following topics in order to better understand its use:

• Name

• Description of the function's operation

• The algorithm as applicable

• Synopsis of function prototype

• Domain valid for arguments

• Accuracy of the returned value(s)

• Execution time in machine cycles

• Notes applicable to this function


5-56 Wideband Computers, Inc.

acort ( a, c, m, n )

NAME Auto-correlation (Time Domain)

DESCRIPTION Computes the time domain auto-correlation of the real elements stored in input vectora[ ]. Values m and n define the number of auto-correlation values to compute. Theresulting auto-correlation values are stored in output vector c[ ].

ALGORITHM

SYNOPSIS void acort ( a, c, m, n )

float *a ; /* Pointer to input vector a[ ] */

float *c ; /* Pointer to output vector c[ ] */

int m ; /* Lag count m */

int n ; /* Number of elements in vector a[ ] */

DOMAIN -3.4E+38 to 3.4E+38

ACCURACY 7.75 decimal digits

EXECUTION TIME 31 + 9*M + (M+1) (2*N-M)

NOTES The file tacort.c included in the distribution tape provides an example of this func-tion’s use.

Note that the lag count m must be less than or equal to the number of floating-point elements (i.e. ).

Ci Ai j+ Aj• i 0 1 2 …m 1–, , ,{ }=j 0=

n i– 1–

∑=

m n≤



acos_wci ( x )

NAME Arc Cosine

DESCRIPTION This function computes the arc cosine of a floating-point number, x. The computed value returned from this function is in the range [0 to π ] radians. A domain error is returned if x is not in the range [-1 to +1].

ALGORITHM

SYNOPSIS float acos_wci ( float x )

DOMAIN -1.0 < x < +1.0


EXECUTION TIME If A <= 0.5 then 55 cycles, Else if A >0.5 then 75 cycles

NOTES The file tacos.c included in the distribution tape provides an example of this function's use.

acosh_wci ( x )

NAME Inverse Hyperbolic Cosine

DESCRIPTION This function computes the inverse hyperbolic cosine of a floating-point number, x.

ALGORITHM

SYNOPSIS float acosh_wci ( float x )

DOMAIN 1.0 to 3.4E+38


EXECUTION TIME 72 cycles

NOTES The file tacosh.c included in the distribution tape provides an example of this func-tion's use.

return x( )1–cos=

return x( )1–cosh=



alawc ( a, i, c, k, n )

NAME a-Law Compression

DESCRIPTION This routine performs an a-law compression on the elements in input vector a and out-puts the compressed results to output vector c.

ALGORITHM

SYNOPSIS void alawc ( a, i, c, k, n )

int *a ; /* Pointer to input vector a */

int i ; /* Element stride for vector i */

int *c ; /* Pointer to output vector c */

int k ; /* Element stride for vector c */

int n ; /* Number of floating-point elements */

DOMAIN 0 to 255


EXECUTION TIME 49 + 12 * ( N-1 )

NOTES The file talawc.c included in the distribution tape provides an example of this func-tion’s use.

The alawc() routine takes a linear 13-bit signed speech sample and compresses it according to CCITT (now ITU) recommendation G.711. The 8-bit compressed sample is output to vector c.

This function is found on the serial port hardware for the ADSP-2106x DSP proces-sors.

Cmk alaw compression of Ami=

m 0 1 2 …n 1–, , ,{ }=



alawe ( a, i, c, k, n )

NAME a-Law Expansion

DESCRIPTION This routine performs an a-law expansion on the elements in input vector a and out-puts the expanded results to output vector c.

ALGORITHM

SYNOPSIS void alawe ( a, i, c, k, n )






DOMAIN 0 to 255



NOTES The file talawe.c included in the distribution tape provides an example of this func-tion’s use.

The alawe() routine takes an 8-bit compressed speech sample and expands it accord-ing to CCITT (now ITU) recommendation G.711. The 13-bit signed sample is output to vector c.

This function is found on the serial port hardware for the ADSP-2106x DSP proces-sors.

Cmk alaw expansion of Ami=

m 0 1 2 …n 1–, , ,{ }=



alpha ( df, a, &al, &n )

NAME Kaiser-Bessel Window Shape Parameter

DESCRIPTION Computes a Kaiser-Bessel window shape parameter for later use by the kaiser( ) win-dow mutiply library function. The computation is based on the input attenutation specified in input scalar a and the transition width specified in real input scalar df. From this, a count of floating-point elements (output scalar n) and an output window shape parameter (output scalar al) is computed.

ALGORITHM

SYNOPSIS void alpha ( df, a, &al, &n )

float dm *df ; /* Input transition width in fs units */

float dm a ; /* Input ripple attenutation in dB */

float dm &al ; /* Output alpha window shape parameter */

int &n ; /* Output floating-point element count */

-3.4E+38 to 3.4E+38


If A 21 then al 0=≤Else If

A 50 then al 0.5842 A 21–( )0.40.07886 A 21–( )•+•=<

Else Ifal 0.1102 A 8.7–( )•=

Number of Elements n is computed as follows:

If A 21 then d A 7.95–( )

14.36------------------------- else d 0.922==>

n 1 ceiling d df⁄( )+=

n n 1 remainder n 2⁄( )–+=



EXECUTION TIME If a >= 50 then 143 Cycles

If 21 < a < 50 then 221 Cycles

If A <= 21 then 124 Cycles

NOTES The file talpha.c included in the distribution tape provides an example of this func-tion’s use.

asin_wci ( x )

NAME Arc Sine

DESCRIPTION This function computes the arc sine of a floating-point number, x. The computed value returned from this function is in the range [-π/2 to π/2] radians. A domain error is returned if x is not in the range [-1 to +1].

ALGORITHM

alpha ( df, a, &al, &n )

df ∆f fs⁄ A ripple attentuation in dB, δ=,= 10 A 20⁄–=

return x( )1–sin=



SYNOPSIS float asin_wci ( float x )

DOMAIN - 1.0 < x < +1.0


EXECUTION TIME If A <= 0.5 then 55 cycles, Else if A >0.5 then 73 cycles

NOTES The file tasin.c included in the distribution tape provides an example of this function's use.

asinh_wci ( x )

NAME Inverse Hyperbolic Sine

DESCRIPTION This function computes the inverse hyperbolic sine of a floating-point number, x.

ALGORITHM

SYNOPSIS float asinh_wci ( float x )

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tasinh.c included in the distribution tape provides an example of this func-tion's use.

asin_wci ( x )

return x( )1–sinh=



aspec ( a, c, n )

NAME Accumulating Auto-spectrum

DESCRIPTION Computes the auto-spectrum of complex input vector a by multiplying vector a by its complex conjugate and adding the resulting real number to the current value of vector c. Vector c must be initialized prior to invoking a series of accumulating auto-spec-trum calls.

ALGORITHM

SYNOPSIS void aspec ( a, c, n )

complex *a ; /* Pointer to input vector a */

float *c ; /* Pointer to output vector c */

int n ; /* Element count for vector c */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 28 + 6*N cycles

NOTES The file taspec.c included in the distribution tape provides an example of this func-tion’s use.

The stride of vectors a and c must always be 1.

If you wish to clear the auto-spectrum results before they are added to output vector cuse the vclr( ) function. If the results are not cleared using vclr( ), autospectrum resultsare added to output vector c, thus computing an accumulating autospectrum.

Note that input vector a is of type complex, and data arguments supplied to this routinewill be treated as interleaved real and imaginary data.

Cm Cm Re2Am Im2Am+ +⇐

m 0 1 2 …n 1–, , ,{ }=



atan_wci ( x )

NAME Arc Tangent

DESCRIPTION This function computes the arc tangent of a floating-point number x. The computed value returned from this function is in the range [-π/2 to +π/2] radians.

ALGORITHM

SYNOPSIS float atan_wci ( float x )

DOMAIN - 4.2E+37 < x < +4.2E+37



NOTES The file tatan.c included in the distribution tape provides an example of this function's use.

return x( )1–tan=



atan2_wci ( y, x )

NAME Arc Tangent 2 Arguments

DESCRIPTION This function computes the arc tangent of a floating-point number x. The computed

value returned from this function is in the range [-π to +π] radians.

ALGORITHM

SYNOPSIS float atan2_wci ( y, x )

float dm y ; /* Input value y */

float dm x ; /* Input value x */

DOMAIN - 4.2E+37 < y/x < +4.2E+37, except x = 0.0



NOTES The file tatan2.c included in the distribution tape provides an example of this func-tion's use.

returnyx---

1–

tan=



atanh_wci ( x )

NAME Inverse Hyperbolic Tangent

DESCRIPTION This function computes the inverse hyperbolic tangent of a floating-point number, x.

ALGORITHM

SYNOPSIS float atanh_wci ( float x )

DOMAIN -1.0 to +1.0



NOTES The file tatanh.c included in the distribution tape provides an example of this func-tion's use.

return x( )1–tanh=



bartlett ( a, i, c, k, n )

NAME Bartlett Window

DESCRIPTION This function generates a Bartlett window multiply on the elements of input vector a and places the results in output vector c.

ALGORITHM

SYNOPSIS void bartlett ( a, i, c, k, n )

float *a ; /* Pointer to input vector a */

int i ; /* Address stride in words for input vector a */


int k ; /* Address stride in words for output vector c */

int n ; /* Element count */

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 44 + 17 * ( N-1 ) cycles

NOTES The file tbartlett.c included in the distribution diskette provides an example of this function’s use.

The Bartlett window is also known as a triangular window.

Cmk Ami 1m

12---n–

12---n

----------------–

•=

m 0 1 2 …n, 1 }–, ,{=



biquad ( x, d, c, y, n )

NAME Bi-Quad IIR Filter

DESCRIPTION Using a bi-quad implementation, this function computes an IIR ( Infinite Impulse Response ) filter using coefficients stored in input vector c, delay node points stored in input buffer d, and applied to the elements of input vector x. The results are stored in output vector y.

ALGORITHM

where

SYNOPSIS void biquad ( x, d, c, y, n )

float *x ; /* Pointer to input buffer vector x of length n */

float *d ; /* Pointer to input delay node buff vector d of length 2 */

float *c ; /* Pointer to input coeff buffer vector c of length 5 */

float *y ; /* Pointer to output buffer vector y of length n */

int n ; /* Number of input/output samples to compute */

DOMAIN -3.4E+38 to 3.4E+38


H z( )B0 B1z

1–B2z

2–+ +

1 A1z1–

– A2z2–

–------------------------------------------------=

Dm A2 Dm 2– A1 Dm 1– xm+•+•=

Ym B2 Dm 2– B1 Dm 1– Dm+•+•=

m 0 1 2 … n 1–, , , ,{ }=



EXECUTION TIME 65 + 13*N

NOTES This is a single bi-quad form of an infinite impulse response filter (IIR), defined by the first equation shown above. It is implemented using a delay node buffer d shown in the second and third equation shown above. The coefficients a[ ] and b[ ] are passed in a single array c[ ] given by the following:

Prior to executing the filter loop, the two “oldest” delay node values are loaded from buffer d[ ]. When the filter loop has completed (n samples have been processed) the two “newest” delay node values are written to d[ ]. In this way the filter delay node states are retained between calls, allowing filtering on blocks of contiguous samples. The user is responsible for allocating the delay node array and for initializing its ele-ments to zero prior to the first call to biquad( ).

Defining

Then

The coefficient buffer length is defined symbolically in the file dsppac.h as DSP_BIQUAD_NCOEFF. The delay node buffer length is defined symbolically in the file dsppac.h as DSP_BIQUAD_NDELAY.

The number of input samples n must be greater than or equal to 5.

The file tbiquad.c included in the distribution tape provides an example of this func-tion’s use.

biquad ( x, d, c, y, n )

c 0[ ] A2 c 1[ ] B2 c 2[ ] A1 c 3[ ] B1 c 4[ ] B0 =====

d0 Dm d1 Dm 1 – d2 Dm 2 –===

d0 c0 d2 c2 d1 xm+•+•=

ym c1 d2 c3 d1 c4 d0•+•+•=

d2 d1=

d1 d0=

m 0 1 2 … n 1–, , , ,{ }=



blkman ( a, i, c, k, w, h, n )

NAME Blackman Window Multiply

DESCRIPTION Multiplies the input vector a[ ] by a Blackman window and stores the result to vector c[ ].

ALGORITHM

SYNOPSIS void blkman ( a, i, c, k, w, h, n )

float dm *a ; /* Pointer to input vector a */

int i ; /* Element stride for vector a */

float dm *c ; /* Pointer to output vector c */


float pm *w ; /* Pointer to cosine weights array */

int h ; /* Element stride for weights array */


DOMAIN -3.4E+38 to 3.4E+38


Cmk Ami 0.42 0.50 2πmiN

------------- 0.08 4πmiN

-------------cos•+cos•–•=

m 0 1 2 … n, 1–, , ,{ }=



EXECUTION TIME 41 + 4*(N-1) cycles

NOTES The file tblkman.c included in the distribution tape provides an example of this func-tion’s use.

For real-time applications, the Blackman window can be computed once, and a simplemultiply used to window data as shown in the variable . The Blackman Win-dow is computed using the winwts( ) function found in the DSP Pac library. The win-wts( ) function computes the weights array using the sin and cosine functions. Thisarray is pointed to by variable w listed in the synopsis section above.

The blkman( ) function is a vector function. You may therefore use the stride argu-ments i, k and h to decimate both the input and output for data congruence. For exam-ple, suppose you use winwts( ) to compute the FFT weights for a 16K FFT. This wouldresult in an fftwts array whose length would be 16,384 points. If you were to laterdecide to compute an FFT of length 1,024 and run a Blackman Window on the results,you would not need to rerun the winwts( ) function to generate new weights. Simplyuse the old weights and stride by 16 (16,384/1024 = 16) on stride element h to obtainthe correct Blackman window FFT weights . In this manner you need only computewinwts( ) once and later us them for varying length FFTs and windowing functions.

The cosine arguments are held in input vector w[ ] and can be computed from the win-wts( ) function. Note that larger vector sizes of w[ ] can be used by changing the stridefor w[ ]. For example, if w[ ] were computed for a window of size 2,048, but a Black-man Window of 1,024 was needed, use a stride of 2,048/1,024 = 2.

Note that the Blackman window has a passband ripple of 0.0017 dB, a maximum stop-band attenuation of 74 dB, and a 57 dB main lobe relative to side lobe.

blkman ( a, i, c, k, w, h, n )

Wml



blkmanh ( a, i, c, k, w, h, n )

NAME Blackman-Harris Window Multiply

DESCRIPTION Multiplies the input vector a[ ] by a Blackman-Harris window and stores the result to output vector c[ ].

ALGORITHM

SYNOPSIS void blkmanh ( a, i, c, k, w, h, n )








DOMAIN -3.4E+38 to 3.4E+38


Cmk Ami 0.35875 0.48829 2πmiN

------------- 0.14128 4πmiN

-------------cos•+cos•– 0.01168 6πmiN

-------------cos•–•=

m 0 1 2 … n, 1–, , ,{ }=




NOTES The file tblkmanh.c included in the distribution tape provides an example of thisfunction’s use.

For real time applications, the Blackman-Harris window can be computed once, and asimple multiply used to window data, as shown in the variable . The Blackman-Harris Window is computed using the winwts( ) function found in the DSP Pac library.The winwts( ) function computes the weights array using the sin and cosine functions.This array is pointed to by variable w listed in the synopisis section above.

The blkmanh function is a vector function. You may therefore use the stride argu-ments i, k and h to decimate both the input and output for data congruence. For exam-ple, suppose you use winwts( ) to compute the FFT weights for a 16K point FFT. Thiswould result in an fftwts array whose length would be 16,384 points. If you were tolater decide to compute an FFT of length 1,024 and run a Blackman-Harris Window onthe results, you would not need to rerun the winwts( ) function to generate newweights. Simply use the old weights and stride by 16 (16,384/1024 = 16) on stride ele-ment h to obtain the correct window FFT weights . In this manner you need only com-pute winwts( ) once and later us them for varying length FFTs and windowingfunctions.

The cosine arguments are held in input vector w[ ] and can be computed from the win-wts( ) function. Note that larger vector sizes of w[ ] can be used by changing the stridefor w[ ]. For example, if w[ ] were computed for a window of size 2,048, but a Black-man Window of 1,024 was needed, use a stride of 2,048/1,024 = 2.

Note that the Blackman-Harris window has a passband ripple of 0.0017 dB, a maxi-mum stopband attenuation of 74 dB, and a 57 dB main lobe relative to side lobe.

blkmanh ( a, i, c, k, w, h, n )

Wml



cacort ( a, c, m, n )

NAME Complex Auto-Correlation (Time Domain)

DESCRIPTION Computes the time domain auto-correlation of the complex elements stored in inputvector a[ ]. Values m and n define the number of auto-correlation values to compute.The resulting auto-correlation values are stored in output complex vector c[ ].

ALGORITHM

SYNOPSIS void cacort ( a, c, m, n )

complex dm *a ; /* Pointer to input vector a[ ] */

complex dm *c ; /* Pointer to output vector c[ ] */


int n ; /* Number of elements in vector a[ ] */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 39 + ( 9 + 5 * n ) * n

NOTES The file tacort.c included in the distribution tape provides an example of this func-tion’s use.

Note that the lag count m must be less than or equal to the number of floating-point elements (i.e. ).

The strides of vectors a[ ] and c [ ] must be 1.

Ci Ai j+ Aj• i 0 1 2 …m 1–, , ,{ }=

j 0=

n i– 1–

∑=

m n≤



ccdotpr ( a, i, b, j, c, k, n )

NAME Complex Dot Product Multiply by Conjugate

DESCRIPTION This function computes the complex dot product of complex input vector a by the complex conjugate of input vector b and stores the results in complex output vector c. This can be alternatively expressed as C=AB*.

ALGORITHM

SYNOPSIS void ccdotpr ( a, i, b, j, c, k, n )

complex *a ; /* Pointer to complex input vector a */


complex *b ; /* Pointer to complex input vector b */

int j ; /* Address stride in words for input vector b */

complex *c ; /* Pointer to complex output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tccdotpr.c included in the distribution diskette provides an example of this function’s use.

Re C{ } Re Ami{ } Re Bmj{ } Im Ami{ } Im Bmj{ }•+•

m 0=

n 1–

∑=

Im C{ } R– e Ami{ } Im Bmj{ } Im Ami{ } Re Bmj{ }•+•

m 0=

n 1–

∑=

m 0 1 2…n 1–, ,{ }=



ccmmul ( a, b, x, y, b, z, c )

NAME Complex Matrix Multiply By Congugate of Complex Matrix

DESCRIPTION This function computes the multiplication of the conjugate of complex input matrix a [ ] [ ] times the elements of complex input matrix b[ ] [ ]. The dimensions of com-plex input matrix a[ ] [ ] are x and y, while the dimensions of complex input matrix b[ ] [ ] are defined by input scalars y and z. The results are stored in complex output matrix c[ ] [ ], which is of dimensions x and z.

ALGORITHM

SYNOPSIS void ccmmul( a, x, y, b, z, c )

complex dm *a ; /* Pointer to complex input matrix a[ ][ ] */

int x ; /* Number of rows in complex matrix a[ ][ ] */

int y ; /* Number of columns in matrix a[ ][ ] And */

/* Number of rows in complex matrix b[ ][ ] */

complex dm *b ; /* Pointer to complex input matrix b[ ][ ] */

int z ; /* Number of columns in matrix b[ ][ ] */

complex dm *c ; /* Pointer to complex output matrix c[ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


Re Cij( ) Re( )Aik Re( )Bkj Im( )Aik Im( )Bkj•+•[ ]

k 1=

y

∑=

Im Cij( ) Re( )Cik Im( )Bkj Re( )Bkj Im( )Aik•–•[ ]

k 1=

y

∑=

for i 0 1 …x, ,{ }=for j 0 1 …z, ,{ }=



EXECUTION TIME 62 + ( 6 + ( 12 + 7 * Y ) * Z ) * X cycles

NOTES The file tccmmul.c included in the distribution diskette provides an example of this function’s use.

a[x][y] =

b[y][z] =

x = 3, y = 4, z = 3 ;

ccmmul ( a, x, y, b, z, c ) ;

The resulting values in output matrix c [ ] [ ] would be as follows:

c[x][y] =

The storage methodology for matrices is by rows. Matrices can be thought of as one long array (vector) where the beginning of each row is offset by the number of col-umns.

ccmmul ( a, b, x, y, b, z, c )

1 1, 2 2, 3 3, 4 4,5 5, 6 6, 7 7, 8 8,9 9, 10 10, 11 11, 12 12,

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,

270 10, 310 10, 350 10,606 26, 610 26, 814 26,942 42, 1110 42, 1278 42,



ccmsmul ( a, x, y, b, c )

NAME Complex Scalar-Complex Congugate Matrix Multiplication

DESCRIPTION This function computes the multiplication of the conjugate of the complex input matrix a[ ] [ ] times complex input scalar b. The dimensions of complex input matrix a[ ] [ ] are x and y. The results are stored in complex output matrix c[ ] [ ], which is of dimensions x and y.

ALGORITHM

SYNOPSIS void ccmsmul( a, x, y, b, c )

complex dm *a ; /* Pointer to complex input matrix a[ ][ ] */


int y ; /* Number of columns in matrix a[ ][ ] */

complex dm *b ; /* Pointer to complex input scalar b */

complex dm *c ; /* Pointer to complex output matrix c[ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


Cxy B Axy•=



EXECUTION TIME 46 + 2 * X * Y cycles

NOTES The file tccmsmul.c included in the distribution diskette provides an example of this function’s use.

a[x][y] =

b = {8,2}

x = 8, y = 7 ;

ccmsmul ( a, x, y, b, c ) ;

The resulting values in output matrix c [ ] [ ] would be as follows:

c[x][y] =


ccmsmul ( a, x, y, b, c )

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,25 26, 27 28, 29 30,

12 14–, 32 26–, 52 38–,72 50–, 92 62–, 112 74–,

132 86–, 152 98–, 172 110–,192 122–, 212 134–, 232 146–,252 158–, 272 170–, 292 182–,



cccort ( a, b, c, m, n )

NAME Complex Cross-Correlation (Time Domain)

DESCRIPTION Computes the time domain (real) cross-correlation of the time domain (real) elements stored in complex input vectors a[ ] and b[ ]. The result is stored in complex output vector c [ ]. Values m and n define the number of cross-correlation values to compute. The implementation uses a time domain technique.

ALGORITHM

SYNOPSIS void cccort ( a, b, c, m, n )

complex dm *a ; /* Pointer to input vector a[ ] */

complex dm *b ; /* Pointer to input vector b[ ] */

complex dm *c ; /* Pointer to output vector c[ ] */


int n ; /* Number of elements in vector c[ ] */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 41 + ( 9 + 5 * n ) * m

NOTES The file tcccort.c included in the distribution tape provides an example of this func-tion’s use.

Note that the lag count must be less than or equal to the number of floating-point ele-ments (i.e. ).

The strides of vectors a[ ], b[ ], and c[ ] must always be 1.

Ci Ai j+ Bj i 0 1 2 … m, 1–, , ,{ }=•

j 0=

n i– 1–

∑=

m n≤



ccort ( a, b, c, m, n )

NAME Cross-Correlation (Time Domain)

DESCRIPTION Computes the time domain (real) cross-correlation of the time domain (real) elements stored in input vectors a[ ] and b[ ]. The result is stored in output real vector c [ ]. Val-ues m and n define the number of cross-correlation values to compute. The implemen-tation uses a time domain technique.

ALGORITHM

SYNOPSIS void ccort ( a, b, c, m, n )

float *a ; /* Pointer to input vector a[ ] */

float *b ; /* Pointer to input vector b[ ] */

float *c ; /* Pointer to output vector c[ ] */


int n ; /* Number of elements in vector c[ ] */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 32 + 9 * M + (M+1)(2*N-M)

NOTES The file tccort.c included in the distribution tape provides an example of this func-tion’s use.

Note that the lag count must be less than or equal to the number of floating-point ele-ments (i.e. ).

The strides of vectors a, b, and c must always be 1.

Cm Ai j+ Bj i 0 1 2 … m, 1–, , ,{ }=•

j 0=

n i– 1–

∑=

m n≤



cdesamp ( data, coeff, output, d, n, p )

NAME Complex Decimating Finite Impulse Response (FIR) Filter

DESCRIPTION The function computes the convolution of complex vectors data [ ] and coeff [ ] plac-ing the results in complex vector output [ ]. The number of output samples n and the number of coefficients p may be dissimilar. n elements will be written to output [ ].

Complex vector data [ ] represents the real and imaginary (I and Q) components of the input data respectively. Likewise, complex vector coeff [ ] represents the real and imaginary ( I and Q) components of the coefficient data. A complex multiply and add is performed to compute the convolutional output. The decimation factor d is used to stride the next starting point in data [ ].

ALGORITHM

SYNOPSIS void cdesamp ( data, coeff, output, d, n, p )

complex dm *data ; /* Complex input data ( len n+p-1 ) */

complex pm *coeff ; /* Complex coefficients ( len p ) */

complex dm *output ; /* Complex output data ( len n ) */

int d ; /* Decimation factor */

int n ; /* Number of output samples */

int p ; /* Number of coefficients */

DOMAIN -3.4E+38 to 3.4E+38


Output i[ ] data i d j+•[ ] coeff p j– 1–[ ]•

j 0=

p 1–

∑=

i 0 1 2…n 1–, ,{ }=



EXECUTION TIME 36 + ( 7 + 5 * p ) * n cycles

NOTES The file tcdesamp.c included in the distribution tape provides an example of this func-tion’s use.

The number of filter output samples to generate can be obtained as follows:

where ndata is the number of elements in data[ ].

A complex correlation can be performed by reversing the order of the coefficients vec-tor.

cdesamp ( data, coeff, output, d, n, p )

n ndata p–( ) d⁄ 1+=



cdotpr ( a, i, b, j, c, k, n )

NAME Complex Dot Product

DESCRIPTION This function computes the complex dot product of complex input vector a and com-plex input vector b and stores the results in complex output vector c. This can altena-tively thought of as .

ALGORITHM

SYNOPSIS void cdotpr ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcdotpr.c included in the distribution diskette provides an example of this function’s use.

C A B•=

Re C{ } Re Ami{ } Re Bmj{ } Im Ami{ } Im Bmj{ }•–•

m 0=

n 1–

∑=

Im C{ } Re Ami{ } Im Bmj{ } Im Ami{ } Re Bmj{ }•+•

m 0=

n 1–

∑=

m 0 1 2…n 1–, ,{ }=



ceil_wci ( x )

NAME Round Up to Nearest Integer

DESCRIPTION This function computes the smallest integral value greater than or equal to the float-ing-point number x. A floating-point representation of this integer value is returned.

ALGORITHM

SYNOPSIS float ceil_wci ( float x )

DOMAIN -3.4E+38 to 3.40E+38



NOTES The file tceil.c included in the distribution tape provides an example of this function's use.

return smallest int x≥=



cfft ( xr, xi, wr, wi, wstr, yr, yi, n )

NAME Fast Fourier Transform Of Complex Input Data

DESCRIPTION Computes the Fast Fourier Transform of the complex input elements stored in com-plex input vector a. The results are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cfft ( xr, xi, wr, wi, wstr, yr, yi, n )

float dm *xr ; /* Pointer to real input data */

float dm *xi ; /* Pointer to imaginary input data */

float pm *wr ; /* Pointer to cosine table */

float dm *wi ; /* Pointer to sine table */

int wstr ; /* Cosine/sine table stride */

float dm *yr ; /* Pointer to real output data */

float pm *yi ; /* Pointer to imaginary output data */

int n ; /* FFT Size (In Complex Elements) */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME See Attached Table Below

Cm Akei2πmk– n⁄

m 0 1 2 … n, 1–, , ,{ }=

k 0=

n 1–

∑=



NOTES This is a radix-2 Fast Fourier Transform using parallel data memory/program memorydata accesses to maximize the throughput on the 21020/60/62 processor. The complexinput data is separated into real and imaginary parts, xr and xi. These vectors must bealigned on an address which is an integer multiple of the FFT size, as required for21K bit-reverse addressing. The input vectors are both in data memory; the imagi-nary data is bit-reversed into program memory at the beginning of the routine. Thenumber of elements n supplied to the algorithm must be an integral power of two anda minimum of 32.

The complex output is separated into real and imaginary parts, yr and yi. These vec-tors may have arbitrary address alignment; however yr is in the data memory and yi isin the program memory. Vectors xr and xi must be in data memory and each must bealigned to an integral multiple of n.

Vectors wr and wi are in program memory and data memory respectively and aregiven the values:

The weight stride wst allows cfft() to be called with varying sizes n from a single setof weights.These weights are generated using the fftwts() function.

This precomputed FFT weight approach was implemented in order to ensure accurateresults and boost the available cfft() dynamic range to approximately 130 dB forlonger length (>16K) FFTs. This is accomplished by using an implementation thatdoes not rely on a recursive call to a sin/cosine approximation routine, as found inother implementations. Rather, the FFT weights are precomputed accurately using thefftwts() function. This is sufficient for A/D converters with bit lengths up to 22 bits.

The number of elements n must be an integral power of two and a minimum of 32.Vector yr is in data memory and has a minimum size of n. Vector yi is in programmemory and has a minimum size of n.

The file tcfft.c included in the distribution tape provides an example of this function’suse.


wr k[ ] 2πk wst*n⁄[ ] k 0 1 … wstn 2 1–⁄, , ,( )program memory=cos=

wi k[ ] 2πk wst*n⁄[ ]sin k 0 1 … wstn 2 1–⁄, , ,( )data memory==



SPECIAL NOTES Previous users have sometimes reported problems associated with implementing inter-rupt service routines (ISRs), when used in conjunction with the FFT routines ( cfft( ),cffti( ), rfft( ), rffti( ) ). Observations related to the Wideband technical staff typicallyinclude a description of the Wideband routine executing perfectly, but unable to returnto an exact state after being interrupted by the ISR ( what is described as a “tumbleinto the weeds.” )

The Wideband Fast Fourier transforms, both complex and real, forward and inverse,use the built-in bit reversing and circular addressing capabilites of the SHARC archi-tecture. Also, other routines such as some of the FIR filters use the SHARC’s internalcircular addressing capabilities.

End users are usually cognizant that their ISR calling routine is responsible for savingand restoring the registers of the Wideband routines. However, end users sometimesforget to save and restore ( push and pop ) the mode 1 regiser, which is associated withbir reversing and the B ( base ) and L ( length ) registers associated with circularaddressing. In such circumstances where they are not saved and restored by the ISRthey are unable to return the proper length parameter ( L Register ) used for circularaddressing or the proper mode ( Mode 1 Register ) used in Bit Reversing. This resultsin the strange manefestations users sometimes report.

To properly save and restore the above mentioned registers in an ISR, refer to page 4-21, section 4.3 of the Analog Devices ADSP-21000 Family C Tools Manual (#31-000005-08, dated August 95) which references examples of in line assembly codewithin C code to save and restore registers.

For a detailed review of the relationships between the various FFT functions and how to use them with one another, see the final section of Chapter 4.




Performance Issues

The inital timing shown below for the 32 point to 4,096 point FFTs were timed using theAnalog Devices simulator.

Performance Timings For Complex FFTs

Number of Points Processor Cycles

8 See cfft8( ) function

16 See cfft16( ) function

32 771 Cycles

64 1,274 Cycles

128 2,368 Cycles

256 4,724 Cycles

512 10,060 Cycles

1,024 21,618 Cycles

2,048 46,744 Cycles

4,096 101,054 Cycles

8,192 217,828 Cycles

16,384 467,722 Cycles

32,768 1,000,240 Cycles

65,536 2,130,774 Cycles



cfft2d ( xr, xi, wr, wi, wstr, tmpdm, tmppm, n )

NAME Complex 2-Dimensional Fast Fourier Transform

DESCRIPTION Computes a 2-Dimensional Fast Fourier Transform of the complex input elements stored in vector a[ ]. The results are stored in complex output vector c[ ].

ALGORITHM

SYNOPSIS void cfft2d ( xr, xi, wr, wi, wstr, tmpdm, tmppm, n )

float dm *xr ; /* Pointer to real input/output data */

float dm *xi ; /* Pointer to imaginary input/output data */



int wstr ; /* Consine/sine Table table */

float dm *tmpdm ; /* Pointer to real output data */

float pm *tmppm ; /* Pointer to imag output data */

int n ; /* CFFT2D Size (Complex Elements n x n) */

DOMAIN -3.4E+38 to 3.4E+38


Cr c, Ak

c 0=

n 1–

∑ e2– πj r R c C⋅+⋅( ) n⁄( )

r 0=

n 1–

∑=

R 0 1 …n 1–, ,{ }=

C 0 1 …n 1–, ,{ }=



EXECUTION TIME 32 x 32 Pts. 44,532 cycles

64 x 64 Pts. 165,364 cycles

128 x 128 Pts. 659,572 cycles




NOTES The input data is an nxn complex matric x separated into real and imaginary parts xrand xi stored as follows:

Variables r and c are the row and column numbers.

The DFT output replaces the input, and is stored as follows:

A radix-2 Fast Fourier Transform (FFT) algorithm is used to compute the individualrow and column DFTs.

The number of elements n must be an integral power of two and a minimum of 32.

Vectors xr and xi must be in data memory and are adress-aligned to an integral multi-ple of n.

Vectors wr and wi must be in program memory and data memory respectively and arepre-computed to be:

Vector tmpdm must be in data memory, having a minimum size of n, and be address-aligned to an integral multiple of n.

Vector tmppm must be in program memory and have a minimum size of n,and beaddress-aligned to an integral multiple of n.

The file tcfft2d.c included in the distribution tape provides an example of this func-tion’s use.


Re xr c,( ) xr r n c+•[ ]=

r 0 1 … n 1–, , ,{ } c 0 1 … n 1–, , ,{ }==

Im xr c,( ) xi r n c+•[ ]=

r 0 1 … n 1–, , ,{ } c 0 1 … n 1–, , ,{ }==

Re FR C,( ) xr R n C+•[ ]=

R 0 1 … n 1–, , ,{ } C 0 1 … n 1–, , ,{ }==

Im FR C,( ) xi R n C+•[ ]=

R 0 1 … n 1–, , ,{ } C 0 1 … n 1–, , ,{ }==

wr k[ ] 2πk wst*n⁄[ ] k 0 1 … wstn 2 1–⁄, , ,( )=cos=

wi k[ ] 2πk wst*n⁄[ ]sin k 0 1 … wstn 2 1–⁄, , ,( )==



cfft8 ( xr, xi, yr, yi )

NAME 8-Point Complex Fast Fourier Transform (Inline)

DESCRIPTION Computes the Fast Fourier Transform of the complex input elements stored in input vector xr and xi. The results are stored in output vector yr and yi.

ALGORITHM

SYNOPSIS void cfft8 ( xr, xi, yr, yi )





DOMAIN -3.4E+38 to 3.4E+38


Ym Xke2πj m k 8⁄•( )–

m 0 1 2 … 7,, , ,{ }=

k 0=

7

∑=



EXECUTION TIME 184 Cycles

NOTES This is an 8-point radix-2 Fast Fourier Transform using parallel data memory/programmemory data accesses to maximize the throughput on the 21020/60/62 processor.

The complex input data is separated into real and imaginary parts, xr and xi. Thesevectors must be aligned on an address which is an integer multiple of the FFTsize, as required for 21K bit-reverse addressing. The input vectors are both in datamemory; the imaginary data is bit-reversed into program memory at the beginning ofthe routine.

This algorithm utilizies a decimation in time approach. As the cffti( ) function requiresa minimum of 32-points as input, there is no corresponding inverse algorithm for thisroutine. The complex output is separated into real and imaginary parts, yr and yi.These vectors may have arbitrary address alignment; however yr is in the data mem-ory and yi is in the program memory.

•Vectors xr and xi are defined in cfft8dta.asm using the dm_align segment to ensure address alignment.


The file tcfft8.c included in the distribution tape provides an example of this func-tion’s use.





NAME 16-Point Complex Fast Fourier Transform (Inline)

DESCRIPTION Computes the Fast Fourier Transform of the complex input elements stored in input vector xr and xi. The results are stored in output vector yr and yi.

ALGORITHM

SYNOPSIS void cfft16 ( xr, xi, yr, yi )





DOMAIN -3.4E+38 to 3.4E+38


Ym Xke2πj16–

m 0 1 2 … 15,, , ,{ }=

k 0=

15

∑=




NOTES This is an 16-point radix-2 Fast Fourier Transform using parallel data memory/pro-gram memory data accesses to maximize the throughput on the 21020/60/62 proces-sor.

The complex input data is separated into real and imaginary parts, xr and xi. Thesevectors must be aligned on an address which is an integer multiple of the FFTsize, as required for 21K bit-reverse addressing. The input vectors are both in datamemory; the imaginary data is bit-reversed into program memory at the beginning ofthe routine.

This algorithm utilizies a decimation in time approach. As the cffti( ) function requiresa minimum of 32-points as input, there is no corresponding inverse algorithm for thisroutine. The complex output is separated into real and imaginary parts, yr and yi.These vectors may have arbitrary address alignment; however yr is in the data mem-ory and yi is in the program memory.

•Vectors xr and xi are defined in cfft16dt.asm using the dm_align segment to ensure address alignment.


The file tcfft16.c included in the distribution tape provides an example of this func-tion’s use.




cffti ( xr, xi, wr, wi, wstr, yr, yi, n )

NAME Inverse Complex FFT

DESCRIPTION Computes the Inverse Fast Fourier Transform of the input elements stored in vectors xr and xi. The results are stored in complex output vector c. Note the Inverse FFT is the same as the Forward FFT except that the sign of the imaginary components of the twiddle factors is negated. The Inverse FFT swaps the real and imaginary input data, perform the Forward FFT with the same weights table, and swaps the real and imagi-nary ouptut data. Scaling by 1/N is then performed.

ALGORITHM

SYNOPSIS void cffti ( xr, xi, wr, wi, wstr, yr, yi, n )








int n ; /* FFT Size (In Complex Elements) */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 22,650 Cycles @ 1,024 Points - Data and Program In On-Board Cache

Cm1n--- Ake

i2πmk n⁄ m 0 1 2 … n, 1–, , ,{ }=

k 0=

n 1–

∑=



NOTES This is a radix-2 inverse Fast Fourier Transform using parallel DM/PM data accessesto maximize the throughput on the 21020 processor.The complex input data is sepa-rated into real and imaginary parts, xr and xi. These vectors must be aligned on anaddress which is an integer multiple of the FFT size, as required for 21K bit-reverse addressing. The input vectors are both in DM; the imaginary data is bit-reversed into PM at the beginning of the routine.The number of elements n must be anintegral power of two and a minimum of 32.

The complex output is separated into real and imaginary parts, yr and yi. These vec-tors may have arbitrary address alignment; however yr is in the DM and yi is in thePM. Vectors xr and xi mus be in data memory and each must be aligned to an integralmultiple of n.


The weight stride, wst, allows for calling cfft() with varying sizes n from a single setof weights. These weights are generated using the fftwts( ) function.

Vector yr is in data memory and has a minimum size of n.Vector yi is in programmemory and has a minimum size of n.

The file tcfft.c included in the distribution tape provides an example of this function’suse.


wr k[ ] 2πk wst*n⁄[ ] k 0 1 … wstn 2 1–⁄, , ,( )=cos=

wi k[ ] 2πk wst*n⁄[ ]sin k 0 1 … wstn 2 1–⁄, , ,( )==



SPECIAL NOTES Previous users have sometimes reported problems associated with implementing inter-rupt service routines (ISRs), when used in conjunction with the FFT routines (cfft ( ),cffti ( ), rfft ( ), rffti ( ) ). Observations related to the Wideband technical staff typi-cally include a description of the Wideband routine executing perfectly, but unable toreturn to an exact state after being interrupted by the ISR ( a “tumble into the weeds.” )




For a detailed review of the relationships between the various FFT functions and howto use them with one another, see the final section of Chapter 4.




TABLE 8 Table of Inverse Complex FFT Timing

Number of Points

Processor Cycles

32 868 Cycles

64 1,435 Cycles

128 2,657 Cycles

256 5,319 Cycles

512 11,117 Cycles

1,024 23,699 Cycles

2,048 50,873 Cycles

4,096 109,281 Cycles

8,192 234,244 Cycles

16,384 500,525 Cycles

32,768 1.072,560 Cycles

65,536 2,288,128 Cycles



cfir ( ii, qq, ci, cq, oi, oq, d, n, p )

NAME Complex Finite Impulse Response Filter

DESCRIPTION The function cfir( ) computes the convolution of vectors ii[ ], iq[ ], ci[ ], and cq[ ] placing the results in oi[ ] and oq[ ] respectively. The number of output samples n and the number of coefficients p may be dissimilar. n elements will be writtento oi[ ] and oq[].

The vectors ii[ ] and iq[ ] represent the real and imaginary (I and Q) components of the input data respectively. Likewise,the vectors ci[ ] and cq[ ] represent the real and imaginary (I and Q) components of the coefficient data. A complex multiply and add is performed to compute the convolutional output. The decimation factor d is used to stride the next starting ii[ ] and iq[ ] data.

ALGORITHM

SYNOPSIS void cfir ( ii, qq, ci, cq, oi, oq, d, n, p )

*/ float dm *ii ; Input samples for I data ( len n+p-1 ) */

*/ float dm *iq ; Input samples for Q data ( len n+p-1 ) */

*/ float pm *ci ; Coefficients for I data ( len p ) */

*/ float pm *cq ; Coefficients for Q data ( len p ) */

*/ float dm *oi ; Output samples for I data ( len n ) */

*/ float dm *oq ; Output samples for Q data ( len n ) */

*/ int d ; Decimation factor */

*/ int n ; Number of output samples */

*/ int p ; Number of coefficients */

C i[ ] a a d j+•[ ] b p j– 1–[ ]•

j 0=

p 1=

∑=

m 0 1 2 … n 1–,, , ,{ }=

where

a [ ] compromises complex components ii [ ] and iq [ ]

b [ ] compromises complex components ci [ ] and cq [ ]

c [ ] compromises complex components oi [ ] and oq [ ]



DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 59 + ( 9 + 5 * p ) * n cycles

NOTES The file tfir.c included in the distribution tape provides an example of this function’s use.

The number of filter output samples to generate can be obtainted as follows:

where ndata is the number of elements in ii[ ] and iq[ ].

A correlation can be performed by reversing the order of the coefficients vector.

cfir ( ii, qq, ci, cq, oi, oq, d, n, p )

n ndata p–( )d 1+

----------------------------=



chksum ( a, i, type, n )

NAME Perform Checksum

DESCRIPTION This function performs a checksum on a memory block. The memory block is defined by the start address a offset by n. The type flag determines whether dm or pm memory is tested ( 1 = dm, 0 = pm).

ALGORITHM

SYNOPSIS void chksum ( a, i, type, n )

int a ; /* Start address of memory */

int i ; /* Memory Stride */

int type ; /* Type of memory to test ( dm or pm ) */

int n ; /* Length of block to be checked */

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 17 + 2 * N cycles

NOTES The file tchksum.c included in the distribution diskette provides an example of this function’s use.

chksum( ) performs a two’s complement on the sum of the elements within the mem-ory block. The check sum value is returned.

Return Checksum⇐



cmadd ( a, b, x, y, c )

NAME Complex Matrix Addition

DESCRIPTION This function computes the addition of complex input matrix a with complex input matrix b and stores the results to complex output matrix c.

ALGORITHM

SYNOPSIS void cmadd ( a, b, x, y, c )

complex dm *a ; /* Pointer to input matrix a [ ][ ] */

complex dm *b ; /* Pointer to input matrix b [ ][ ] */

int x ; /* Number of rows in matrix a[ ][ ] & b[ ][ ] */

int y ; /* Number of columns in matrix a[ ][ ]& b[ ][ ]*/

complex dm *c ; /* Pointer to output matrix c [ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


Cri11 Cri12 Cri13Cri21 Cri22 Cri23

Ari11 Ari12 Ari13Ari21 Ari22 Ari23

Bri11 Bri12 Bri13Bri21 Bri22 Bri23

+=

where ri indicates a real and imaginary component



EXECUTION TIME 32 + 3*X*Y cycles

NOTES The file tcmadd.c included in the distribution diskette provides an example of this function’s use.

The addition of a complex matrix is mathematically expressed as follows:

An example of the additon of one complex matrix to another is as follows:

cmadd ( a, b, x, y, c )

Real C x[ ] y[ ] A x[ ] y[ ]Real B x[ ] y[ ]Real+=

Imaginary C x[ ] y[ ] A x[ ] y[ ]Imaginary B x[ ] y[ ]Imaginary+=

A x[ ] y[ ]

1 2, 3.8 1.7, 8.8 5.5, 9.9 14,7.1 5, 9.3 1.6, 0.4 1, 51 3.3,0.9 1, 8 5, 2.1 6, 3.1 1–,–9.3 1, 2.5 1.5, 6.9 9, 10 22.1,

1.3 1.4, 0.2 4.5, 0.9 51.4, 1.5 4.4,9.2 4, 7.8 1.7, 61 3.4, 14.3 1.4,

=

B [x] [y]

3.2 1, 8.8 2, 9.9 3, 44.3 13.3,8.1 4, 6.5 5, 3.2 6, 2.3 9.9–,–8.9 7, 2.8 8, 1.7 9, 8.1 2.2–,–

6.4 10, 11 1.3, 12 4.5, 22.9 5.4–,6.5 7, 2.1 8, 2.2 9, 32 9.8,1.1 4, 7.7 5, 4.4 6, 2.1 0.3–,–

=

x=6, y=4

C [x] [y]

4.2 3, 12.6 3.7, 18.7 8.5, 54.2 27.3,15.2 9, 15.8 6.6, 3.6 7, 48.7 6.6–,9.8 8, 10.8 13, 3.8 15, 11.2 3.2–,–

15.7 11, 13.5 2.8, 18.9 13.5, 32.9 16.7,7.8 8.4, 2.3 12.5, 3.1 60.4, 33.5 14.2,10.3 8, 15.5 6.7, 65.4 9.4, 12.2 1.1,

=



cmmov ( a, x, y, b )

NAME Complex Matrix Move

DESCRIPTION This function moves a source complex input matrix a to a destination complex output matrix b.

ALGORITHM

SYNOPSIS void cmmov ( a, x, y, b )


int x ; /* Number of rows in matrix a[ ][ ] */


complex dm *b ; /* Pointer to output matrix b [ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 13 + ( 2 * X * Y ) cycles

NOTES The file tcmmov.c included in the distribution diskette provides an example of this function’s use.

The storage methodology for matrices is by rows. Matricies can be thought of as one long array (vector) where the beginning of each row is offset by the length of the col-umn.



⇐




cmmul ( a, x, y, b, z, c )

NAME Complex Matrix Multiplication

DESCRIPTION This function computes the multiplication of complex input matrix a times complex input matrix b and stores the results to complex output matrix c. The dimension of complex matrix a [ ] [ ] is x and y and the dimension of complex input matrix b [ ] [ ] is y and z. The resulting complex output matrix c [ ] [ ] is of dimension x and z.

ALGORITHM

SYNOPSIS void cmmul ( a, x, y, b, z, c )




/* Number of rows in matrix b[ ][ ] */




DOMAIN -3.4 x 1038 to +3.4 x 1038


Cri11 Cri12

Cri21 Cri22

Ari11 Ari12 Ari13

Ari21 Ari22 Ari23

Bri11 Bri12

Bri21 Bri22

Bri31 Bri32

•=




EXECUTION TIME 45 + (4 + ( 10 + 5 * Y) * Z) * X cycles

NOTES The file tcmmul.c included in the distribution diskette provides an example of this function’s use.

The multiplication of a complex matrix is as follows:

The storage methodology for matrices is by rows. Matrices can be thought of as one long array (vector) where the beginning of each row is offset by the length of the col-umn.

The first row of a [ ] [ ] times the first column of b [ ] [ ] is the first element of c [ ] [ ] (row 1, column 1). The first row of a [ ] [ ] times the second row of b [ ] [ ] is the sec-ond element of c [ ] [ ] (row 1, column 2 ) ... etc.

This algorithm follows the general law of matrix multiplication whereby the number of columns of input matrix a must equal the number of rows of input matrix b.

ri indicates that each component of the matrix is composed of a complex number which has both a real and imaginary component.

An example of the multipication of one complex matrices by another is as follows:

cmmul ( a, x, y, b, z, c )

C x[ ] y[ ] Real Sum + Imaginary Sum( ) where

k 1=

y

∑=

Real Sum AReal BReal• AImaginary BImagainary•–=

Imaginary Sum AReal BImaginary• BReal AImaginary•+=

A x[ ] y[ ]1 1, 2 2, 3 3, 4 4,5 5, 6 6, 7 7, 8 8,9 9, 10 10, 11 11, 12 12,

B [y] [z]

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,

==

x=3, y=4, z=3

C [ ] 10 270,– 10 310,– 10– 350,26 606,– 26 710,– 26 814,–42 942,– 42 1110,– 42 1278,–

=



cmmul_dpd ( a, x, y, b, z, c )

NAME Complex Matrix Multiplication (Data Memory x Program Memory to Data Memory)

DESCRIPTION This function computes the multiplication of complex input matrix a[ ] (in data mem-ory) times complex input matrix b[ ] (in program memory) and stores the results to complex output matrix c[ ] (in data memory). The dimension of complex matrix a [ ] [ ] is x and y and the dimension of complex input matrix b [ ] [ ] is y and z. The resulting complex output matrix c [ ] [ ] is of dimension x and z.

ALGORITHM

SYNOPSIS void cmmul_dpd ( a, x, y, b, z, c )





complex pm *b ; /* Pointer to input matrix b [ ][ ] */



DOMAIN -3.4 x 1038 to +3.4 x 1038


Cri11 Cri12

Cri21 Cri22

Ari11 Ari12 Ari13

Ari21 Ari22 Ari23

Bri11 Bri12

Bri21 Bri22

Bri31 Bri32

•=





NOTES The file tcmmuldpd.c included in the distribution diskette provides an example of this function’s use.







cmmul_dpd ( a, x, y, b, z, c )


k 1=

y

∑=



A x[ ] y[ ]1 1, 2 2, 3 3, 4 4,5 5, 6 6, 7 7, 8 8,9 9, 10 10, 11 11, 12 12,

B [y] [z]

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,

==

x=3, y=4, z=3

C [ ] 10 270,– 10 310,– 10– 350,26 606,– 26 710,– 26 814,–42 942,– 42 1110,– 42 1278,–

=



cmmul_dpp ( a, x, y, b, z, c )

NAME Complex Matrix Multiplication (Data Memory x Program Memory to Program Mem-ory)

DESCRIPTION This function computes the multiplication of complex input matrix a[ ] (in data mem-ory) times complex input matrix b[ ] (in program memory) and stores the results to complex output matrix c[ ] (in program memory). The dimension of complex matrix a [ ] [ ] is x and y and the dimension of complex input matrix b [ ] [ ] is y and z. The resulting complex output matrix c [ ] [ ] is of dimension x and z.

ALGORITHM

SYNOPSIS void cmmul_dpp ( a, x, y, b, z, c )





complex pm *b ; /* Pointer to input matrix b [ ][ ] */


complex pm *c ; /* Pointer to output matrix c [ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


Cri11 Cri12

Cri21 Cri22

Ari11 Ari12 Ari13

Ari21 Ari22 Ari23

Bri11 Bri12

Bri21 Bri22

Bri31 Bri32

•=





NOTES The file tcmmuldpp.c included in the distribution diskette provides an example of this function’s use.







cmmul_dpp ( a, x, y, b, z, c )


k 1=

y

∑=



A x[ ] y[ ]1 1, 2 2, 3 3, 4 4,5 5, 6 6, 7 7, 8 8,9 9, 10 10, 11 11, 12 12,

B [y] [z]

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,

==

x=3, y=4, z=3

C [ ] 10 270,– 10 310,– 10– 350,26 606,– 26 710,– 26 814,–42 942,– 42 1110,– 42 1278,–

=



cmmul_pdd ( a, x, y, b, z, c )

NAME Complex Matrix Multiplication (Program Memory x Data Memory to Data Memory)

DESCRIPTION This function computes the multiplication of complex input matrix a[ ] (in program memory) times complex input matrix b[ ] (in data memory) and stores the results to complex output matrix c[ ] (in data memory). The dimension of complex matrix a [ ] [ ] is x and y and the dimension of complex input matrix b [ ] [ ] is y and z. The resulting complex output matrix c [ ] [ ] is of dimension x and z.

ALGORITHM

SYNOPSIS void cmmul_pdd ( a, x, y, b, z, c )

complex pm *a ; /* Pointer to input matrix a [ ][ ] */







DOMAIN -3.4 x 1038 to +3.4 x 1038


Cri11 Cri12

Cri21 Cri22

Ari11 Ari12 Ari13

Ari21 Ari22 Ari23

Bri11 Bri12

Bri21 Bri22

Bri31 Bri32

•=





NOTES The file tcmmulpdd.c included in the distribution diskette provides an example of this function’s use.







cmmul_pdd ( a, x, y, b, z, c )


k 1=

y

∑=



A x[ ] y[ ]1 1, 2 2, 3 3, 4 4,5 5, 6 6, 7 7, 8 8,9 9, 10 10, 11 11, 12 12,

B [y] [z]

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,

==

x=3, y=4, z=3

C [ ] 10 270,– 10 310,– 10– 350,26 606,– 26 710,– 26 814,–42 942,– 42 1110,– 42 1278,–

=



cmmul_pdp ( a, x, y, b, z, c )

NAME Complex Matrix Multiplication (Program Memory x Data Memory to Program Mem-ory)

DESCRIPTION This function computes the multiplication of complex input matrix a[ ] (in program memory) times complex input matrix b[ ] (in data memory) and stores the results to complex output matrix c[ ] (in program memory). The dimension of complex matrix a [ ] [ ] is x and y and the dimension of complex input matrix b [ ] [ ] is y and z. The resulting complex output matrix c [ ] [ ] is of dimension x and z.

ALGORITHM

SYNOPSIS void cmmul_pdp ( a, x, y, b, z, c )

complex pm *a ; /* Pointer to input matrix a [ ][ ] */






complex pm *c ; /* Pointer to output matrix c [ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


Cri11 Cri12

Cri21 Cri22

Ari11 Ari12 Ari13

Ari21 Ari22 Ari23

Bri11 Bri12

Bri21 Bri22

Bri31 Bri32

•=





NOTES The file tcmmulpdp.c included in the distribution diskette provides an example of this function’s use.







cmmul_pdp ( a, x, y, b, z, c )


k 1=

y

∑=



A x[ ] y[ ]1 1, 2 2, 3 3, 4 4,5 5, 6 6, 7 7, 8 8,9 9, 10 10, 11 11, 12 12,

B [y] [z]

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,

==

x=3, y=4, z=3

C [ ] 10 270,– 10 310,– 10– 350,26 606,– 26 710,– 26 814,–42 942,– 42 1110,– 42 1278,–

=



cmsmul ( a, x, y, b, c )

NAME Complex Scalar-Matrix Multiplication

DESCRIPTION This function computes the multiplication of complex input matrix a[ ] [ ] times com-plex input scalar b and stores the results to complex output matrix c. The dimensionsof complex input matrix a[ ] [ ] and complex output matrix c[ ] [ ] are of dimensions xand y.

ALGORITHM

SYNOPSIS void cmsmul ( a, x, y, b, c )





complex dm *b ; /* Pointer to input scalar b [ ][ ] */


DOMAIN -3.4 x 1038 to +3.4 x 1038


Cri11 Cri12 Cri13

Cri21 Cri22 Cri23

Ari11 Ari12 Cri13

Ari21 Ari22 Ari23

Bri•=





NOTES The file tcmsmul.c included in the distribution diskette provides an example of this function’s use.


An example of a complex scalar-matrix multiplication is as follows:

cmsmul ( a, x, y, b, c )

C x[ ] y[ ]

4.00 18.0, 16.0 38.0, 28.0 58.0,40.0 78.0, 52.0 98.0, 64.0 118,76.0 138, 88.0 158, 100 178,112 198, 124 218, 136 238,148 258, 160 287, 172 298,

=

A x[ ] y[ ]

1.0 2.0, 3.0 4.0, 5.0 6.0,7.0 8.0, 9.0 10, 11 12,13 14, 15 16, 17 18,19 20, 21 22, 23 24,25 26, 27 28, 29 30,

B 8 2,[ ] x=8, y=7 ==



cmsub ( a, b, x, y, c )

NAME Complex Matrix Subtraction

DESCRIPTION This function computes the subtraction of complex input matrix b[ ] [ ] from complex input matrix a[ ] [ ] and stores the results to complex output matrix c[ ] [ ].

ALGORITHM

SYNOPSIS void cmsub ( a, b, x, y, c )



int x ; /* Number of rows in matrix a[ ][ ] & b[ ][ ] */

int y ; /* Number of columns in matrix a[ ][ ]& b[ ][ ]*/


DOMAIN -3.4 x 1038 to +3.4 x 1038




Bri11 Bri12 Bri13Bri21 Bri22 Bri23

–=




EXECUTION TIME 32 + ( 3 * X * Y ) cycles

NOTES The file tcmsub.c included in the distribution diskette provides an example of this function’s use.

The subtraction of a complex matrices is mathematically expressed as follows:

An example of the subtraction of one complex matrix from another is as follows:

cmsub ( a, b, x, y, c )

Real C x[ ] y[ ] A x[ ] y[ ]Real B x[ ] y[ ]Real–=

Imaginary C x[ ] y[ ] A x[ ] y[ ]Imaginary B– x[ ] y[ ]Imaginary=

A x[ ] y[ ]1.0 2.0, 3.8 1.7, 8.8 5.5,7.1 5.0, 9.3 1.6, 0.4 1.0,0.9 1.0, 8.0 5.0, 2.1 6.0,

=

B [x] [y] 3.2 1.0, 8.8 2.0, 9.9 3.0,8.1 4.0, 6.5 5.0, 3.2 6.0,

8.9 7.0·, 2.8 8.0, 1.7 9.0,

·

=

x=3, y=3

C [x] [y] 2.2 1.0,– 5.0 0.3–,– 1.1 2.5,–1.0 1.0,– 2.8 3.4–, 2.8 5.0–,–

8.0 6.0–,– 5.2 3.0–, 0.4 3.0–,

=



cmtrans ( a, b, x, y )

NAME Complex Matrix Transpose

DESCRIPTION This function transposes the contents of complex input matrix a[ ] [ ] and places the results in complex output matrix b[ ] [ ]. The dimensions of matrix a [ ] [ ] are x and y and the dimensions of output matrix b [ ] [ ] are y and x.

ALGORITHM

SYNOPSIS void cmtrans ( a, b, x, y )

complex dm *a ; /* Pointer to complex input matrix a [ ][ ] */

complex dm *b ; /* Pointer to complex output matrix b [ ][ ] */


/* Number of columns in matrix b[ ][ ] */



DOMAIN -3.4 x 1038 to +3.4 x 1038


If A

Ari11 Ari12

Ari21 A22

Ari31 Ari32

then AT Ari11 Ari21 Ari31

Ari12 Ari22 Ari32

==

Where ri indicates a real and imaginary pair of numbers



EXECUTION TIME 34 + ( 3+4*Y ) * X cycles

NOTES The file tcmtrans.c included in the distribution diskette provides an example of this function’s use.


Transposing a matrix is performed as follows:

An example of a complex matrix transpose is as follows:

cmtrans ( a, b, x, y )

B Realnm A Realmn=

B Imaginarynm A Imaginarymn=

for m=0, x and n=0, y

x 4 y=3,=

B y[ ] x[ ]1 2, 7 8, 13 14, 19 20,3 4, 9 10, 15 16, 21 22,5 6, 11 12, 17 18, 23 24,

=

A x[ ] y[ ]

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,

=



cmvmul ( a, x, y, b, c )

NAME Complex Matrix-Vector Multiply

DESCRIPTION This function multiplies the elements of complex input matrix a[ ] [ ] times the ele-ments of complex input vector b. The length of complex input vector b is defined by input scalar y, while the dimensions of complex input matrix a is defined by input sca-lars x and y. The results are stored in complex output vector c, which is of length x.

ALGORITHM

SYNOPSIS void cmvmul( a, x, y, b, c )

complex dm *a ; /* Pointer to complex input matrix a [ ][ ] */


int y ; /* Number of columns in matrix vector a[ ][ ] */

/* Number of elements in vector b[ ] */

complex dm *b ; /* Pointer to complex input vector b [ ] */

complex dm *c ; /* Pointer to complex output vector c */

DOMAIN -3.4 x 1038 to +3.4 x 1038


Cx Axk Bk•

k 1=

y

∑=



EXECUTION TIME 41 + ( 14 + 7 * Y ) * X cycles

NOTES The file tcmvmul.c included in the distribution diskette provides an example of this function’s use.

a[x] [y]=

b[x] = { 1, 1, 2, 2 3, 3 };

x = 4, y = 3 ;

cmvmul ( a, 4, 3, b, c ) ;

The resulting values in output vector [c] would be as follows:

c[3] = { -6, 50, -6, 122, -6, 194, -6, 266 }

The storage methodology for matricies is by rows. Matricies can be thought of as one long array (vector) where the beginning of each row is offset by the number of col-umns.

The first row of matrix a [ ] [ ] times vector b is the first element of vector c. The sec-ond row of matrix a [ ] [ ] times vector b is the second element of vector c, etc.

cmvmul ( a, x, y, b, c )

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,



coher ( a, b, c, d, n )

NAME Coherence Function

DESCRIPTION Computes the coherence function of two signals. The real input vectors a and b are the signal’s auto-spectra. Their cross-spectrum is in complex input vector c. The results are stored in real output vector d.

ALGORITHM

SYNOPSIS void coher ( a, b, c, d, n )


int *b ; /* Pointer to input vector b */

float *c ; /* Pointer to complex input vector c */

int *d ; /* Pointer to output vector d */

int n ; /* Number of elements in vector c */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 32 + 11*N Cycles

NOTES The file tcoher.c included in the distribution tape provides an example of this func-tion’s use.

The strides of vectors a, b, c, and d must always be 1.

Should real input vectors a or b should equal zero, this would result in an attempt to divide by zero, which is an illegal operation.

Dml

Re Cm{ }2Im Cm{ }2

+

Am Bm•------------------------------------------------------- m 0 1 2 … m, 1–, , ,{ }==



comb ( a, b )

NAME Combination

DESCRIPTION This function computes the combination of input scalar a by input scalar b, and returns the result in comb. The number of combinations refers to the different ways of choos-ing b objects from a total of a, without regard for their order.

ALGORITHM

SYNOPSIS void comb ( a, b )

int dm a ; /* Input value a */

int dm b ; /* Input value b */

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 33 + ( A - B ) * 4 cycles

NOTES The file tcomb.c included in the distribution diskette provides an example of this func-tion’s use.

The value of input scalar a must be greater than or equal to 0. If not, a value of -99 is returned.

The value of input scalar b must be less then the value of input scalar a. If not, a value of -99 is returned.

comb a!b! a b–( )• !-----------------------------=



conv ( a, i, b, j, c, k, na, nb )

NAME Convolution Function

DESCRIPTION Computes the convolution of input vectors a with vector b, placing the results in out-put vector c. na + nb-1 elements will be written to output vector c as a result of this operation. na and nb may be dissimilar.

ALGORITHM

SYNOPSIS void conv ( a, i, b, j, c, k, na, nb )


int i ; /* Address stride in words for vector a */

float pm *b ; /* Pointer to input vector b */

int j ; /* Address stride in words for vector b */


int k ; /* Address stride in words for vector c */

int na ; /* Number of element to compute */

int nb ; /* Element count in vector b */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 42 + (7+NC)*N cycles

NOTES The file tconv.c included in the distribution tape provides an example of this func-tion’s use.

na elements will be written to output vector c as a result of this operation, where:

Cm Am q+ Bnb q 1 ––•q 0=

nb 1–

∑=

m 0 1 2 … na 1–,, , ,{ }=

na sizeof a( ) nb– 1+≤



conv2d ( a, nr, nc, b, c, k )

NAME 2-Dimensional Convolution, KxK Kernel

DESCRIPTION Computes the 2-dimensional convolution of input matrix a[ ] [ ] with input kernel matrix b[ ] [ ] , placing the results in output matrix c[ ] [ ].

ALGORITHM

SYNOPSIS void conv2d ( a, nr, nc, b, c, k )

float dm *a ; /* Pointer to input matrix a [ ] [ ] */

int nr ; /* Number of rows in matrix a[ ][ ] to convolve */

int nc ; /* Number of columns in matrix a to convolve */

float dm *b ; /* Pointer to input kernel matrix b [ ] [ ] */

float dm *c ; /* Pointer to output matrix c */

int k ; /* Kernel size; */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 45 + ( 3 + ( 6 + ( 6 + 2k ) k ) (nc - k + 1 )) (nr - k + 1) cycles

NOTES The file tconv2d.c included in the distribution tape provides an example of this func-tion’s use.

C r c,( ) b k 1– i–[ ] k 1– j–[ ] a r i+[ ] c j+[ ]•

j 0=

k 1–

∑i 0=

k 1–

∑=

r 0 1 2 … nr k– 1+,, , ,{ }=

c 0 1 2 … nc k– 1+,, , ,{ }=



conv3x3 ( a, x, y, b, c )

NAME 2-Dimensional Convolution of a Matrix and a 3x3 Kernel

DESCRIPTION Computes the 2-D convolution of input matrices a and b, and places the results in out-put matrix c. The image in matrix a is filtered with a 3-by-3 kernel in matrix b.

ALGORITHM

SYNOPSIS void conv3x3 ( a, x, y, b, c )

float *a ; /* Pointer to input matrix a */

int x ; /* Number of rows in matrix a */

int y ; /* Number of columns in matrix a */

float *b ; /* Pointer to coeff matrix b */

float *c ; /* Pointer to output matrix c */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 54 + (30*(C-2)(R-2) cycles where C = # of columns and r = # of rows

NOTES The file tconv3x3.c included in the distribution tape provides an example of this func-tion’s use.

Matrix b must be 3x3.

The strides of matrices a, b, and c must be 1.

Cmn Am p n q+,+ Bpq•q 0=

2

∑p 0=

2

∑=

m 0 1 2 … x, 2–, , ,{ }=

n 0 1 2 … y, 2–, , ,{ }=



conv3x3p ( a, x, y, b, c )

NAME 2-Dimensional Convolution of a Matrix and a 3x3 Program Memory Kernel

DESCRIPTION Computes the 2-D convolution of input matrices a[ ] [ ] and b[ ] [ ], and places the results in output matrix c[ ] [ ]. The image in matrix a[ ] [ ] is filtered with a 3-by-3 kernel in matrix b[ ] [ ]. Note that the 3x3 kernel is in pm (program memory).

ALGORITHM

SYNOPSIS void conv3x3p ( a, x, y, b, c )


int x ; /* Number of rows in vector a */

int y ; /* Number of columns in vector a */

float pm *b ; /* Pointer to coeff vector b */


DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 54 + ( 30 + 27 (Y-2))(X-2)/2 cycles

NOTES The file tconv3x3p.c included in the distribution tape provides an example of this function’s use.



Cmn Am p n q+,+ Bpq•q 0=

2

∑p 0=

2

∑=

m 0 1 2 … x, 2–, , ,{ }=

n 0 1 2 … y, 2–, , ,{ }=





DESCRIPTION Computes the 2-Dimensional convolution of input matrices a[ ] [ ] and b[ ] [ ], and places the results in output matrix c[ ] [ ]. The image in matrix a[ ] [ ] is filtered with a 2-Dimensional Convolution with a 5-by-5 kernel in matrix b[ ] [ ].

ALGORITHM







DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 58 + 76*(X-4)(Y-4)cycles

NOTES The file tconv5x5.c included in the distribution tape provides an example of this func-tion’s use



Cmn Am p n q+,+ Bpq•

q 0=

4

∑p 0=

4

∑=

m 0 1 2 … x, 4–, , ,{ }=

n 0 1 2 … y, 4–, , ,{ }=



conv5x5p ( a, x, y, b, c )

NAME 2-Dimensional Convolution of a Matrix and a 5x5 Kernel in Program Memory

DESCRIPTION Computes the 2-D convolution of input matrices a[ ] [ ] and b[ ] [ ], and places the results in output matrix c[ ] [ ]. The image in matrix a[ ] [ ] is filtered with a 5-by-5 kernel in matrix b[ ] [ ]. Note that the 5 x 5 kernel is in pm (program memory).

ALGORITHM

SYNOPSIS void conv5x5p ( a, x, y, b, c )


int x ; /* Number of rows in vector a */

int y ; /* Number of columns in vector a */

float pm *b ; /* Pointer to coeff vector b */


DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 95 + (74+75(Y-4))(X-4)/2cycles

NOTES The file tconv5x5p.c included in the distribution tape provides an example of this function’s use



Cmn Am p n q+,+ Bpq•

q 0=

4

∑p 0=

4

∑=

m 0 1 2 … x, 4–, , ,{ }=

n 0 1 2 … y, 4–, , ,{ }=





DESCRIPTION Computes the 2-Dimensional convolution of input matrices a[ ] [ ] and b[ ] [ ], and places the results in output matrix c[ ] [ ]. The image in matrix a[ ] [ ] is filtered with a 2-Dimensional Convolution with a 7-by-7 kernel in matrix b[ ] [ ].

ALGORITHM







DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 33 + (3+235(Y-6))(X-6) cycles

NOTES The file tconv7x7.c included in the distribution tape provides an example of this func-tion’s use



Cm Am p+ n q+, Bpq•q 0=

6

∑p 0=

6

∑=

m 0 1 2 … n, 1–, , ,{ }=



coord2 ( a, b, c, d, n )

NAME 2-Dimensional Coordinate Transformation

DESCRIPTION Translates and rotates input matrix b[ ] [ ] according to the rotation matrix a[ ] [ ] and translation vector c. The results are stored in matrix d[ ] [ ].

ALGORITHM

SYNOPSIS void coord2 ( a, b, c, d, n )

float *a ; /* Pointer to rotation matrix a */

float *b ; /* Pointer to input vector b */

float *c ; /* Pointer to translation vector */

float *d ; /* Pointer to output vector c */

int n ; /* Number coordinate pairs */

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tcoord2.c included in the distribution tape provides an example of this func-tion’s use

Dm 0, Bm 0, A0 0,• Bm 1, A1 0,• C0+ +=

Dm 1, Bm 0, A0 1,• Bm 1, A1 1,• C1+ +=

m 0 1 2 … n, 1–, , ,{ }=



coord3 ( a, b, c, d, n )

NAME 3-Dimensional Coordinate Transformation

DESCRIPTION Translates and rotates input matrix b[ ] [ ] according to the rotation matrix a[ ] [ ] and translation vector c[ ] [ ]. The results are stored in matrix d[ ] [ ].

ALGORITHM

SYNOPSIS void coord3 ( a, b, c, d, n )

float *a ; /* Pointer to rotation matrix a */


float *c ; /* Pointer to translation vector */


int n ; /* Number coordinate pairs */

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tcoord3.c included in the distribution tape provides an example of this func-tion’s use.

Dm 0, Bm 0, A0 0,• Bm 1, A1 0,• Bm 2, A2 0,• C0+ + +=

Dm 1, Bm 0, A0 1,• Bm 1, A1 1,• Bm 2, A2 1,• C1+ + +=

Dm 2, Bm 0, A0 2,• Bm 1, A1 2,• Bm 2, A2 2,• C2+ + +=

m 0 1 2 … n, 1–, , ,{ }=



corr ( a, i, b, j, c, k, n, nb )

NAME Correlation Function

DESCRIPTION Computes the correlation of input vectors a with vector b, placing the results in output vector c.

ALGORITHM

SYNOPSIS void corr ( a, i, b, j, c, k, n, nb )


int i ; /* Address stride in words for vector a */

float pm *b ; /* Pointer to input vector b */

int j ; /* Address stride in words for vector b */


int k ; /* Address stride in words for vector c */

int n ; /* Number of elements to compute */

int nb ; /* Element count in vector b */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 52 + N*(11+(NB-1)*2) cycles

NOTES The file tcorr.c included in the distribution tape provides an example of this function’s use.

Note that the elements of input vector b must be placed in program memory (pm).

Cm Am p+ Bp•p 0=

nb 1–

∑=

m 0 1 2 … n 1–,, , ,{ }=



corr2d ( a, nr, nc, b, c, k )

NAME 2-Dimensional Correlation, K x K Kernel

DESCRIPTION Computes the 2-dimensional correlation of input matrix a[ ] [ ] with input kernel matrix b[ ] [ ], placing the results in output matrix c[ ] [ ].

ALGORITHM

SYNOPSIS void corr2d ( a, nr, nc, b, c, k )

float dm *a ; /* Pointer to input matrix a[ ][ ] */

int nr ; /* Number of rows in matrix a[ ][ ] */

int nc ; /* Number of columns in matrix a[ ][ ] */

float dm *b ; /* Pointer to input kernel matrix b[ ][ ] */

float dm *c ; /* Pointer to output matrix c[ ][ ] */

int k ; /* Kernel Size ( k x k ) of input matrix b */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 35 + ( 4 + ( 5 + ( 5 + 4k ) k )(nc - k + 1))(nr - k + 1) cycles

NOTES The file tcorr2d.c included in the distribution tape provides an example of this func-tion’s use.

C r c,( ) A r u c v+,+( ) B u v,( )•

v 0=

k 1–

∑u 0=

k 1–

∑=

r 0 1 2 … nr 1– 1+,, , ,{ }=

c 0 1 2 … nc k– 1+,, , ,{ }=



cos_wci ( x )

NAME Cosine

DESCRIPTION This function computes the cosine of a floating-point number, x ( x is measured in radians ). The computed value is returned from this function.

ALGORITHM

SYNOPSIS float cos_wci ( float x )

DOMAIN - 205,884 < x < 205,884 radians



NOTES The file tcos.c included in the distribution tape provides an example of this function's use.

cosh_wci ( x )

NAME Hyperbolic Cosine

DESCRIPTION This function computes the hyperbolic cosine of a floating-point number, x ( x is mea-sured in radians ). The computed value is returned from this function.

ALGORITHM

SYNOPSIS float cosh_wci ( float x )

DOMAIN -88 to +88



NOTES The file tcosh.c included in the distribution tape provides an example of this function's use.

return x( )cos=

return x( )cosh=



cspec ( a, b, c, n)

NAME Accumulating Cross-Spectrum

DESCRIPTION Computes the cross-spectrum of complex input vectors a and b by multiplying the conjugate of vector a by vector b and adding the result to the current value of vector c. Input/Output vector c must be initialized prior to invoking a series of accumulating cross-spectrum calls.

ALGORITHM

SYNOPSIS void cspec ( a, b, c, n )



complex *c ; /* Pointer to complex input/output vector c */


DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 37 + 8*N Cycles

NOTES The file tcspec.c included in the distribution tape provides an example of this func-tion’s use.

Cm Cm Am* Bm• m 0 1 2 … n, 1–, , ,{ }=+=



cvabs ( a, i, c, k, n )

NAME Complex Vector Absolute Value

DESCRIPTION This function computes the complex absolute value of complex input vector a and stores the results in real output vector c.

ALGORITHM

SYNOPSIS void cvabs ( a, i, c, k, n )



float *c ; /* Pointer to real output vector c */



DOMAIN -1.7 x 1019 < abs (a.im) +1.7 x 1019



NOTES The file tcvabs.c included in the distribution diskette provides an example of this function’s use.

Cmk Re2

Ami{ } Im2

Ami{ }+ m 0 1 2 …n 1–, , ,{ }==



cvadd ( a, i, b, j, c, k, n )

NAME Complex Vector Add

DESCRIPTION This function adds the elements of complex input vector a to the elements of complex input vector b. The results in are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvadd ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvadd.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami Bmj m 0 1 2 …n 1–, , ,{ }=+=



cvadd_pd ( a, i, b, j, c, k, n )



ALGORITHM

SYNOPSIS void cvadd_pd ( a, i, b, j, c, k, n )

complex dm *a ; /* Pointer to complex input vector a */


complex pm *b ; /* Pointer to complex input vector b */





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvaddpd.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami Bmj m 0 1 2 …n 1–, , ,{ }=+=



cvadd_pp ( a, i, b, j, c, k, n )



ALGORITHM

SYNOPSIS void cvadd_pp ( a, i, b, j, c, k, n )



complex pm *b ; /* Pointer to complex input vector b */


complex pm *c ; /* Pointer to complex output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 33 + 4 * (N-1) cycles

NOTES The file tcvaddpp.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami Bmj m 0 1 2 …n 1–, , ,{ }=+=



cvcma ( a, i, b, j, c, k, d, l, n )

NAME Complex Vector Conjugate, Multiply and Add

DESCRIPTION This function multiplies the elements of complex input vector a by the congugate ele-ments of complex input vector b, and then adds complex input vector c to the product. The results in are stored in complex output vector d.

ALGORITHM

SYNOPSIS void cvcma ( a, i, b, j, c, k, d, l, n )





complex *c ; /* Pointer to complex input vector c */

int k ; /* Address stride in words for input vector c */

complex *d ; /* Pointer to complex output vector d */

int l ; /* Address stride in words for output vector d */


DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 68 +8*(N-1) cycles

NOTES The file tcvcma.c included in the distribution diskette provides an example of this function’s use.

The * sign signifies the computation of the complex conjugate.

Dml Ami B*mj•( ) Cmk m 0 1 2 …n 1–, , ,{ }=+=



cvcmpx ( a, i, b, j, c, k, n )

NAME Complex Vector Create From Reals

DESCRIPTION This function creates a complex output vector c by combining the elements of real input vector a and real input vector b.

ALGORITHM

SYNOPSIS void cvcmpx ( a, i, b, j, c, k, n )

float *a ; /* Pointer to real input vector a */


float *b ; /* Pointer to real input vector b */





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvmpx.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Ami m 0 1 2 …n 1–, , ,{ }==

Im Cmk{ } Bmj=



cvcmpx_dpp ( a, i, b, j, c, k, n )

NAME Complex Vector Create From Reals, Data Memory and Program Memory to Program Memory

DESCRIPTION This function creates a complex output vector c by combining the elements of real input vector a[ ] and real input vector b[ ] and the results are placed in complex output vector c[ ]. Vector a[ ] is in data memory, while vectors b[ ] and c[ ] are in program memory.

ALGORITHM

SYNOPSIS void cvcmpx ( a, i, b, j, c, k, n )

float dm *a ; /* Pointer to real input vector a */


float pm *b ; /* Pointer to real input vector b */





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvmpx.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Ami m 0 1 2 …n 1–, , ,{ }==

Im Cmk{ } Bmj=



cvcmpxi ( a, i, b, j, c, k, n )

NAME Complex Vector Create From Integers

DESCRIPTION This function creates a complex output vector c by combining the elements of integer input vector a and integer input vector b.

ALGORITHM

SYNOPSIS void cvcmpxi ( a, i, b, j, c, k, n )

int *a ; /* Pointer to integer input vector a */


int *b ; /* Pointer to integer input vector b */


complexi*c ; /* Pointer to complex output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvmpxi.c included in the distribution diskette provides an example of this function’s use.

complexi is a new type known as complex integer.

Re Cmk{ } Ami m 0 1 2 …n 1–, , ,{ }==

Im Cmk{ } Bmj=



cvcmul ( a, i, b, j, c, k, n )

NAME Complex Vector Conjugate and Multiply

DESCRIPTION This function multiplies the conjugate of the elements of complex input vector a times the elements of complex input vector b. The results are stored in complex output vec-tor c.

ALGORITHM

SYNOPSIS void cvcmul( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvcmul.c included in the distribution diskette provides an example of this function’s use.

Cmk A*mi Bmj• m 0 1 2 …n 1–, , ,{ }==



cvconj ( a, i, c, k, n )

NAME Complex Vector Conjugate

DESCRIPTION This function conjugates the elements of complex input vector a and stores the results in complex output vector c.

ALGORITHM

SYNOPSIS void cvconj ( a, i, c, k, n )






DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tcvconj.c included in the distribution diskette provides an example of this function’s use.

Cmk A*mi m 0 1 2 …n 1–, , ,{ }==



cvdiv ( a, i, b, j, c, k, n )

NAME Complex Vector Division

DESCRIPTION This function divides the elements of complex input vector a bythe elements of com-plex input vector b. The results are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvdiv ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvdiv.c included in the distribution diskette provides an example of this func-tion’s use.

CmkAmi

Bmi------------ m 0 1 2 …n 1–, , ,{ } ==



cvexp ( a, i, c, k, n )

NAME Complex Vector Exponential

DESCRIPTION This function computes the complex exponential of real input vector a and stores the result in complex output vector c.

ALGORITHM

SYNOPSIS void cvexp ( a, i, c, k, n )






DOMAIN -109 < a[i] < +109



NOTES The file tcvexp.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Ami mcos= 0 1 2 …n 1–, , ,{ }=

Im Cmk{ } Ami sin =



cvexpm ( a, i, b, j, c, k, n )

NAME Complex Vector Exponential and Multiply

DESCRIPTION This function computes the complex exponential of real input vector a, multiplies it by a complex vector b, and stores the result in complex output vector c.

ALGORITHM

SYNOPSIS void cvexpm ( a, i, b, j, c, k, n )



complex dm *b ; /* Pointer to complex input vector b */





DOMAIN -109 < a[i] < +109



NOTES The file tcvexpm.c included in the distribution diskette provides an example of this function’s use.

Re Cmk( ) ami( ) Re bmj( )• ami( )sin Im bmj( )•–cos=

Im Cmk( ) ami( ) Re bmj( )• ami( )cos Im bmj( )•+sin=

m= 0 1 2 …n 1–, , ,{ }



cvfill ( a, c, k, n )

NAME Complex Vector Fill

DESCRIPTION This function copies the complex input scalar a to the complex output vector c.

ALGORITHM

SYNOPSIS void cvfill ( a, c, k, n )

complex *a ; /* Pointer to complex input scalar a */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvfill.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk A m 0 1 2 …n 1–, , ,{ }==



cvma ( a, i, b, j, c, k, d, l, n )

NAME Complex Vector Multiply and Add

DESCRIPTION This function multiplies complex input vector a times complex input vector b, and then adds complex input vector c to the product. The results are stored in complex out-put vector d.

ALGORITHM

SYNOPSIS void cvma ( a, i, b, j, c, k, d, l, n )



complex *b ; /* Pointer to complex intput vector b */


complex *c ; /* Pointer to complex intput vector c */


complex *d ; /* Pointer to complex outtput vector d */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvma.c included in the distribution diskette provides an example of this func-tion’s use.

Dml Ami Bmj• Cmk m 0 1 2…n 1–, ,{ }=+=



cvmags ( a, i, c, k, n )

NAME Complex Vector Magnitude Squared

DESCRIPTION This function computes the complex magnitude squared of complex input vector a and stores the result in real output vector c.

ALGORITHM

SYNOPSIS void cvmags ( a, i, c, k, n )






DOMAIN -1018 < abs(a[i]) < +1018



NOTES The file tcvmags.c included in the distribution diskette provides an example of this function’s use.

Cmk Re2 Ami{ } Im2 Ami{ } m 0 1 2 …n 1–, , ,{ }=+=



cvmagsa ( a, i, b, j, c, k, n )

NAME Complex Vector Magnitude Squared and Add

DESCRIPTION This function computes the complex magnitude squared of complex input vector a, adds real input vector b to the sum, and stores the result in real output vector c.

ALGORITHM

SYNOPSIS void cvmagsa ( a, i, b, j, c, k, n )








DOMAIN -10 x 1018 < a[i] < +10 x 1018



NOTES The file tcvmagsa.c included in the distribution diskette provides an example of this function’s use.

Cmk Re2 Ami{ } Im2 Ami{ }+[ ] Bmj+=

m 0 1 2 …n 1–, , ,{ }=



cvmmul ( a, b, x, y, c )

NAME Complex Vector-Matrix Multiply

DESCRIPTION This function multiplies the elements of complex input vector a times the elements of complex input matrix b[ ] [ ]. The length of complex input vector a is defined by input scalar x, while the dimensions of complex input matrix b[ ] [ ] is defined by input sca-lars x and y. The results are stored in complex output vector c, which is of length y.

ALGORITHM

SYNOPSIS void cvmmul( a, b, x, y, c )

complex *a ; /* Pointer to complex input vector a [ ] */

complex *b ; /* Pointer to complex input matrix b [ ][ ] */

int x ; /* Number of rows in cmplx matrix vector b[ ][ ] */

/* Number of elements in vector a[ ] */

int y ; /* Number of columns in matrix vector b[ ][ ] */


DOMAIN -3.4 x 1038 to +3.4 x 1038


Cj Ak Bk j,•

k 0=

x 1–

∑=

j 0 1 2…y, ,{ }=



EXECUTION TIME 44 + ( 12 + 7 * (X-1) ) * Y cycles

NOTES The file tcvmmul.c included in the distribution diskette provides an example of this function’s use.

a[x] = { 1, 1, 2, 2 3, 3 4, 4 };

b[x][y] =

x = 4, y = 3 ;

cmmul ( a, 4, 3, b, c ) ;

The resulting values in output vector c would be as follows:

c[3] = { -10, 270, -10, 310, -10, 350 }


Vector a time the first row of matrix b[ ] [ ] is the second element of vector c, etc.

cvmmul ( a, b, x, y, c )

1 2, 3 4, 5 6,7 8, 9 10, 11 12,

13 14, 15 16, 17 18,19 20, 21 22, 23 24,



cvmov ( a, i, c, k, n )

NAME Complex Vector Move

DESCRIPTION This function moves the complex input vector a to complex output vector c.

ALGORITHM

SYNOPSIS void cvmov ( a, i, c, k, n )

complex dm *a ; /* Pointer to input vector a */

int i ; /* Address stride for input vector a */

complex pm *c ; /* Pointer to output vector c */

int k ; /* Address stride for output vector c */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvmov.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami= m 0 1 2 …n 1–, , ,{ }=



cvmov_dp ( a, i, c, k, n )

NAME Complex Vector Move From Data Memory to Program Memory

DESCRIPTION This function moves the complex input vector a to complex output vector c. Complex vector a is in data memory while complex vector c is in program memory.

ALGORITHM

SYNOPSIS void cvmov_dp ( a, i, c, k, n )

complex dm *a ; /* Pointer to input vector a */


complex pm *c ; /* Pointer to output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvmovdp.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami= m 0 1 2 …n 1–, , ,{ }=



cvmov_pd ( a, i, c, k, n )

NAME Complex Vector Move From Program Memory to Data Memory

DESCRIPTION This function moves the complex input vector a to complex output vector c. Complex vector a is in program memory while complex vector c is in data memory.

ALGORITHM

SYNOPSIS void cvmov_pd ( a, i, c, k, n )

complex pm *a ; /* Pointer to input vector a */


complex dm *c ; /* Pointer to output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvmovpd.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami= m 0 1 2 …n 1–, , ,{ }=



cvmul ( a, i, b, j, c, k, n )

NAME Complex Vector Multiply

DESCRIPTION This function multiplies complex input vector a times complex input vector b, and stores the result in complex output vector c.

ALGORITHM

SYNOPSIS void cvmul ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvmul.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami Bmj• m 0 1 2 …n 1–, , ,{ }==



cvneg ( a, i, c, k, n )

NAME Complex Vector Negate

DESCRIPTION This function negates the elements of complex input vector a and stores them in com-plex output vector c.

ALGORITHM

SYNOPSIS void cvneg ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvneg.c included in the distribution diskette provides an example of this function’s use.

Cmk A– mj m 0 1 2 …n 1–, , ,{ }==



cvphase ( a, i, c, k, n )

NAME Complex Vector Phase

DESCRIPTION This function computes the phase of the elements of complex input vector a, and stores the result in real output vector c.

ALGORITHM

SYNOPSIS void cvphase ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038, Re{Amj} cannot equal zero or a divide by zero error will occur.



NOTES The file tcvphase.c included in the distribution diskette provides an example of this function’s use.

Cmk

Im Ami{ }

Re Ami{ }-------------------------atan m 0 1 2 …n 1–, , ,{ }==



cvradd ( a, i, b, j, c, k, n )

NAME Complex Vector And Real Vector Add

DESCRIPTION This function adds real input vector b to the elements of complex input vector a and stores the results in complex output vector c.

ALGORITHM

SYNOPSIS void cvradd ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvradd.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Re Ami{ } Bmj+=

Im Cmk{ } Im Ami{ }=

m 0 1 2 …n 1–, , ,{ }=



cvrdiv ( a, i, b, j, c, k, n )

NAME Complex Vector And Real Vector Divide

DESCRIPTION This function divides the elements of complex input vector a by the elements of real input vector b. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvrdiv ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038, [b] cannot be zero or a divide by zero error will occur.



NOTES The file tcvrdiv.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ }Re Ami{ }

Bmj-------------------------=

Im Cmk{ }Im Ami{ }

Bmj-------------------------=

m 0 1 2…n 1–, ,{ }=



cvrecp ( a, i, c, k, n )

NAME Complex Vector Reciprocal

DESCRIPTION This function computes the complex reciprocal of the elements of complex input vec-tor a. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvrecp ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038, the absolute value of a[real] must not equal the absolute value of a[imaginary] or a division by zero error will occur.



NOTES The file tcvrecp.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ }Re Ami{ }

Re2 Ami{ } Im2 Ami{ }+--------------------------------------------------------------- =

Im Cmk{ }Im– Ami{ }

Re2 Ami{ } Im2 Ami{ }+--------------------------------------------------------------- =

m 0 1 2 …n 1–, , ,{ }=



cvrmul ( a, i, b, j, c, k, n )

NAME Complex Vector Multiply By Real Vector

DESCRIPTION This function multiplies the elements of complex input vector a times the elements of real input vector b. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvrmul ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvrmul.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Re Ami{ } Bmj{ }•=

Im Cmk{ } Im Ami{ } Bmj{ }•=

m 0 1 2 …n 1–, , ,{ }=



cvrmul_dpd ( a, i, b, j, c, k, n )



ALGORITHM

SYNOPSIS void cvrmul_dpd ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 32+4*(N-1) cycles

NOTES The file tcvrmdpd.c included in the distribution diskette provides an example of this function’s use.



m 0 1 2 …n 1–, , ,{ }=



cvrmul_dpp ( a, i, b, j, c, k, n )



ALGORITHM

SYNOPSIS void cvrmul_dpp ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 32+4*(N-1) cycles

NOTES The file tcvrmdpp.c included in the distribution diskette provides an example of this function’s use.



m 0 1 2 …n 1–, , ,{ }=



cvrmul_pdd ( a, i, b, j, c, k, n )



ALGORITHM

SYNOPSIS void cvrmul_pdd ( a, i, b, j, c, k, n )

complex pm *a ; /* Pointer to complex input vector a */


float dm *b ; /* Pointer to real input vector b */





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvrmpdd.c included in the distribution diskette provides an example of this function’s use.



m 0 1 2 …n 1–, , ,{ }=



cvrmul_pdp ( a, i, b, j, c, k, n )



ALGORITHM

SYNOPSIS void cvrmul_pdp ( a, i, b, j, c, k, n )



float dm *b ; /* Pointer to real input vector b */





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvrmpdp.c included in the distribution diskette provides an example of this function’s use.



m 0 1 2 …n 1–, , ,{ }=



cvrsub ( a, i, b, j, c, k, n )

NAME Complex Vector Subtract Real Vector

DESCRIPTION This function subtracts the elements of real input vector b from the elements of com-plex input vector a. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvrsub ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvrsub.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Re Ami{ } Bmj–=

Im Cmk{ } Im Ami{ }=

m 0 1 2 …n 1–, , ,{ }=



cvsadd ( a, i, b, c, k, n )

NAME Complex Vector Add Complex Scalar

DESCRIPTION This function adds a complex input scalar b to the elements of complex input vector a. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvsadd ( a, i, b, c, k, n )



complex *b ; /* Pointer to complex input scalar b */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvsadd.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Re Ami{ } Re B{ } +=

Im Cmk{ } Im Ami{ } Im B{ }+=

m 0 1 2 …n 1–, , ,{ }=



cvsdiv ( a, i, b, c, k, n )

NAME Complex Vector Divide By Complex Scalar

DESCRIPTION This function divides the elements of complex input vector a by complex input scalar b. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvsdiv ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038, Re{Bmj} and Im{Bmj} must not both equal zero or a divide by zero error will occur.



NOTES The file tcvsdiv.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ }Re Ami{ } Re B{ } Im Ami{ } Im B{ }•+•

Re2

B{ } Im2

B{ }+---------------------------------------------------------------------------------------------------- =

Im Cmk{ }Im Ami{ } Re B{ } Re Ami{ } Im B{ }•–•

Re2

B{ } Im2

B{ }+---------------------------------------------------------------------------------------------------- =

m 0 1 2 …n 1–, , ,{ }=



cvsma ( a, i, b, c, k, d, l, n )

NAME Complex Vector Multiply By Complex Scalar & Add Complex Vector

DESCRIPTION This function multiplies the elements of complex input vector a by a complex input scalar b time, then the elements of complex input vector c to the product. The result is stored in complex output vector d.

ALGORITHM

SYNOPSIS void cvsma ( a, i, b, c, k, d, l, n )




complex *c ; /* Pointer to complex input vector c */


complex *d ; /* Pointer to complex output vector d */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvsma.c included in the distribution diskette provides an example of this function’s use.

An alternative way of expressing the algorithm is as follows:

Re Dml{ } Re Ami{ } Re B{ } I– m Ami{ } Im B{ }••[ ] Re Cmk{ }+=

Im Dml{ } Re Ami{ } Im B{ } Im Ami{ } Re B{ }•+•[ ] Im Cmk{ }+=

m 0 1 2 …n 1–, , ,{ }=

D Ab C+( )=



cvsmul ( a, i, b, c, k, n )

NAME Complex Vector Multiply By Complex Scalar

DESCRIPTION This function multiplies the elements of complex input vector a by a complex input scalar b time. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvsmul ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvsmul.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Re Ami{ } Re B{ } I– m Ami{ } Im B{ }••[ ]=

Im Cmk{ } Re Ami{ } Im B{ } Im Ami{ } Re B{ }•+•[ ]=

m 0 1 2 …n 1–, , ,{ }=



cvsqrt ( a, i, c, k, n )

NAME Complex Vector Square Root

DESCRIPTION This function computes the square root of the elements of complex input vector a and stores the results in complex output vector c.

ALGORITHM

SYNOPSIS void cvsqrt ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvsqrt.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } α θ cos=

Im Cmk{ } α θ where: sin=

α Re2

Ami{ } Im2

Ami{ }+

12---

=

θ

Im Ami{ }

Re Ami{ }-----------------------atan

2---------------------------------------=

m 0 1 2 …n 1–, , ,{ }=



cvssub ( a, i, b, c, k, n )

NAME Complex Vector Subtract Complex Scalar

DESCRIPTION This function subtracts a complex input scalar b from the elements of complex input vector a. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvssub ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvssub.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Re Ami{ } R– e B{ }=

Im Cmk{ } Im Ami{ } I– m B{ }=

m 0 1 2 …n 1–, , ,{ }=



cvsub ( a, i, b, j, c, k, n )

NAME Complex Vector Subtract

DESCRIPTION This function subtracts the elements of complex input vector b from the elements of complex input vector a. The result are stored in complex output vector c.

ALGORITHM

SYNOPSIS void cvsub ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tcvsub.c included in the distribution diskette provides an example of this function’s use.

Re Cmk{ } Re Ami{ } R– e Bmj{ }=

Im Cmk{ } Im Ami{ } I– m Bmj{ }=

m 0 1 2 …n 1–, , ,{ }=



dct8x8 ( im, dir )

NAME 8 x 8 Two-Dimensional Discrete Cosine Transform

DESCRIPTION This function calculates an 8x8 two dimensional discrete cosine transform (DCT) of the elements in input matrix im[ ] [ ]. The dir direction flag determines if a forward or inverse 2 dimensional discrete cosine transform is performed ( 1 = forward, 0 = inverse). The transform is returned in matrix im[ ] [ ].

ALGORITHM

SYNOPSIS void dct8x8 ( im, dir )

int *im ; /* Pointer to input/output matrix im [ ] [ ] */

int dir ; /* Decision flag ( forward = 1 , inverse = 0 ) */

DOMAIN -3.4 x 1038 to +3.4 x 1038


imkl2n---akal immn

πk 2m 1+( )2n

---------------------------- πl 2n 1+( )2n

--------------------------coscos

n 0=

N 1–

∑m 0=

N 1–

∑=

k 0 to N-1, l 0 to N-1==

a 0[ ]1

2------- a j[ ], 1 for j 0 n,≠ 8= = =



EXECUTION TIME 2,464 cycles forward

2,578 cycles inverse

NOTES The file t8x8dct.c included in the distribution diskette provides an example of this function’s use. The forward 8x8 2D-DCT is performed in two summation steps.

For the first summation step, the input matrix im[ ] [ ] is scaled by 2/N. The resulting matrix im[ ] [ ] is multiplied times the cosine matrix cm[ ] [ ] by rows. In otherwords, row 0 of im[ ] [ ] times row 0 of cm[ ] [ ] is row 0, col 0 of the result matrix[ ] [ ], row 0 of im[ ] [ ] times row 1 of cm[ ] [ ] is row 0, col 1 of the result matrix[ ] [ ], etc.

im * cm = Result matrix

| m1 m2 m3 |------->| c1 c2 c3 | | mr0*cr0 mr0*cr1 mr0*cr2 |

| m4 m5 m6 | \----->| c4 c5 c6 | = | mr1*cr0 mr1*cr1 mr1*cr2 |

| m7 m8 m9 | \--->| c7 c8 c9 | | mr2*cr0 mr2*cr1 mr2*cr2 |

where: mr0 = row 0 of im

mr1 = row 1 of im

mr2 = row 2 of im

cr0 = row 0 of cm

cr1 = row 1 of cm

cr2 = row 2 of cm

For the second summation step, the cosine matrix cm[ ] [ ] is multiplied by the result matrix[ ] [ ] using normal matrix multiplication (ie. row times column).

cm[ ][ ] * Result matrix = DCT matrix

| c1 c2 c3 | | mr0*cr0 mr0*cr1 mr0*cr2 |

| c4 c5 c6 | * | mr1*cr0 mr1*cr1 mr1*cr2 | = DCT matrix

| c7 c8 c9 | | mr2*cr0 mr2*cr1 mr2*cr2 |

As a final step, the first row and first column (row 0 and col 0) of the DCT matrix is scaled by the A0 constant ( 1/sqrt(2) ).

The inverse 8x8 2D-DCT is also performed in two summation steps. Before the first step, input matrix im[ ][ ] is scaled by the A0 constant ( 1/sqrt(2) ) for the first row and first column (row0 and col 0). Then the resulting matrix im[ ][ ] is scaled by2/N.

dct8x8( ) is an in-place computation (ie. the original data of matrix im[ ][ ] is destroyed).

dct8x8 ( im, dir )



desamp ( a, b, c, nc, p, d )

NAME Direct Form Decimating Finite Impulse Response (FIR) Filter

DESCRIPTION The function computes a FIR ( Finite Impulse Response) filter using coefficients stored in vector b applied to the elements of input vector a. The results are stored in vector c and computed according to a decimation index d. The number of desired out-put filter samples to generate is specified by the parameter nc, while the size of vector b (taps) is specified by p.

ALGORITHM

SYNOPSIS void desamp ( a, b, c, nc, p, d )


float *b ; /* Pointer to coefficients vector b */


int nc ; /* Number of samples to generate */

int p ; /* Number of coefficients (taps)in vector b */

int d ; /* Decimation index */

DOMAIN -3.4E+38 to 3.4E+38


Cm Am d• j+ Bp j– 1– m 0 1 2 … nc, 1–, , ,{ }=•j 0=

p 1–

∑=



EXECUTION TIME 25 + ( 7 + P-1 ) * NC cycles

NOTES The file tdesamp.c included in the distribution tape provides an example of this func-tion’s use.

The argument nc is the number of filter output samples to generate. It is obtained as follows:

where na is the number of elements in a.

The number of a samples consumed is

This function differs from the fir_wci( ) function in that it performs a direct convolu-tion of the input data. Since there is no delay memory, the convolution value for the first sample uses the first nc number of points (number of taps) of the input data.

desamp ( a, b, c, nc, p, d )

nc na p–( ) d⁄=

nc d•



desampd ( a, b, p, m, c, n, nc, d, fill )

NAME Compute Finite Impulse Response (FIR) Filter of Two Vectors with Decimation and Delay Memory

DESCRIPTION This function computes a FIR ( Finite Impulse Response) filter with decimation using input samples in vector a[ ] and coefficients in vector b[ ]. The results are returned in output vector pointed to by the double pointer c.

The double pointers m and c and the pointer to fill are defined such that they are updated by desampd( ). This allows processing of a continious sample stream across multiple function calls.

The address pointed to by m contains the current position within the delay line p. The address pointed to by c contains the current position within the output vector. The pointer fill contains the contents of the fill value.

ALGORITHM

For Example, Set the Decimation Factor d = 3, and fill = 1

First Set Second Set

a[ ] = 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

p [ ] = 0 0 0 0 1b [ ] = 5 4 3 2 1=1

0 1 2 3 45 4 3 2 1=2

3 4 5 6 75 4 3 2 1=65

6 7 8 9 05 4 3 2 1=100

9 0 1 2 35 4 3 2 1=55

2 3 4 5 65 4 3 2 1=50

5 6 7 8 95 4 3 2 1=95



SYNOPSIS void desampd ( a, b, p, m, c, n, nc, d, fill )

float dm *a ; /* Pointer to input sample vector a (length n) */

float dm *b ; /* Pointer to input coefficient vector b

(length nc) */

float pm *p ; /* Base address of delay line vector (length nc)*/

float dm **m ; /* Current Pointer in delay */

float dm **c ; /* Current Pointer in output vector c */

int n ; /* Count of floating-point elements in a [ ] */

int nc ; /* Count of floating-point elements in b [ ] */

int d ; /* Decimation factor */

int *fill ; /* Pointer to fill value for delay line */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 86 + fill + ( 6 + nc + d ) * w cycles

where: w = ( int ) ( ( n - fill ) / d ) + 1 Note: fill can vary call to call

NOTES The file tdesamp.c included in the distribution tape provides an example of this func-tion’s use.

Reverse the order of the coefficients (taps) to compute the convolution. Example- Compute the convolution for:

data = { x1, x2, x3, x4, ....} taps = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

To compute the convolution order the taps as follows:

taps = { 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 } and call the desampd( ) function.

Pipelining of the sample loop requires that there be at least two data samples (n>=2) when calling this function.

The decimation algorithm requires that the number of samples is greater than the deci-mation factor and that the decimation factor is greater than or equal to two.

desampd ( a, b, p, m, c, n, nc, d, fill )



doppler ( a, i, c, k, n )

NAME Generate Doppler Signal

DESCRIPTION The function generates a doppler signal using the input data signal contained within input real vector a and places the doppler signal in output real vector c.

ALGORITHM

SYNOPSIS void doppler ( a, i, c, k, n )



float dm *c ; /* Pointer to real output vector c */



DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tdoppler.c included in the distribution tape provides an example of this func-tion’s use.

Cmk Ami 1 Ami–( )2π

Ami 0.05+--------------------------

sin•=

0 Ami 1≤ ≤

m 0 1 2 … n, 1–, , ,{ }=



dotpr ( a, i, b, j, c, n )

NAME Dot Product

DESCRIPTION This function computes the dot product of vector a and vector b. The scalar result is stored in c.

ALGORITHM

SYNOPSIS void dotpr ( a, i, b, j, c, n )



float *b ; /* Pointer to complex input vector b */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tdotpr.c included in the distribution diskette provides an example of this func-tion’s use.

C Ami Bmj•

m 0=

n 1–

∑=

m 0 1 2 … n, 1–, , ,{ }=



dotpr_dp ( a, i, b, j, c, n )

NAME Dot Product

DESCRIPTION This function computes the dot product of vector a and vector b. The scalar result is stored in c.

ALGORITHM

SYNOPSIS void dotpr_dp ( a, i, b, j, c, n )



float pm *b ; /* Pointer to complex input vector b */




DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 23 + N cycles

NOTES The file tdotprdp.c included in the distribution diskette provides an example of this function’s use.

C Ami Bmj•

m 0=

n 1–

∑=

m 0 1 2 … n, 1–, , ,{ }=



endian ( a, i, c, k, n )

NAME Endian Order

DESCRIPTION This function computes reverses the byte order for input vector a and places the result in output vector c.

ALGORITHM

SYNOPSIS void endian ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tendian.c included in the distribution diskette provides an example of this function’s use.

The endian ( ) function reverses the byte order for input vector a [ ] and places the result in output vector c [ ].

For example:

int a [3] = { 0x04030201, 0x08070605, 0x12111009 } ;

endian ( a, 1, c, 1, 3 ) ;

int c [3] = { 0x01020304, 0x05060708, 0x09101112 } ;

Ck Ai in reverse byte order=



exp_wci ( x )

NAME Exponential Base e

DESCRIPTION This function computes the exponential of floating-point number, x

ALGORITHM

SYNOPSIS float exp_wci ( float x )

DOMAIN -88.0 to 88.0



NOTES The file texp.c included in the distribution tape provides an example of this function's use.

exp10_wci ( x )

NAME Exponential Base 10

DESCRIPTION This function computes the exponential base 10 of floating-point number, x.

ALGORITHM

SYNOPSIS float exp10_wci ( float x )

DOMAIN -38 to +38



NOTES The file texp10.c included in the distribution tape provides an example of this func-tion's use.

return ex=

return 10x=



fabs_wci ( x )

NAME Absolute Value

DESCRIPTION This function computes the absolute values of a floating-point number, x. The com-puted value is returned from this function.

ALGORITHM

SYNOPSIS float fabs_wci ( float x )

DOMAIN -3.40E38 to 3.40E38



NOTES The file tfabs.c included in the distribution tape provides an example of this function's use.

return x=



fact ( a )

NAME Factorial

DESCRIPTION This function computes the factorial based on the element contained within input sca-lar a.

ALGORITHM

SYNOPSIS float fact ( a )

int *a ; /* Input value */

DOMAIN 0 to 34



NOTES The file tfact.c included in the distribution diskette provides an example of this func-tion’s use.

The factorial is computed as follows:

c = 1 * 2 * 3 * 4 * 5 * ... a

Input scalar a must be >= 0. If not, a value of -99 is returned.

return a!=



fct ( x, y, wr, wi, tmpdm, tmppm, n )

NAME Fast Cosine Transform

DESCRIPTION Computes the Fast CosineTransform of complex input vectors x and y. The imaginary and real weights array are stored in complex output vectors wr and wi. Tempory stor-age space used to compute the Fast CosineTransform are designated by tmpdm for temporary data memory and tmppm for temporary program memory.

ALGORITHM

SYNOPSIS void fct ( x, y, wr, wi, tmpdm, tmppm, n )

float *x ; /* Pointer to real input vector x */

float *y ; /* Pointer to real output vector y */

float *wr ; /* Pointer to real (cosine) weights array wr */

float *wi ; /* Pointer to imaginary (sin) weights array wi */

float *tmpdm ; /* Pointer to temporary dm storage space */

float *tmppm ; /* Pointer to temporary pm storage space */

int n ; /* Element count for floating point input */

DOMAIN -3.4E+38 to 3.4E+38


Fπ k i 1 2⁄+( )••

2----------------------------------------cos

k i, 0=

n 1–

∑=

m 0 1 2 … n 1–,, , ,{ }=



EXECUTION TIME 16,465 cycles ( 128 Pts ) - Currently C Code Only

NOTES The file tfct.c included in the distribution tape provides an example of this function’s use.


Note that these are identical to the cosine/sin vectors used for a Complex Fast Fourier Transform (cfft( )) of size 4*n.

Vector x has a minimum size of n

Vectors tmppm and tmpdm are in program and data memory respectively and have mimimum sizes of .

n must be an integer power of 2 and greater than 32 points.

fct ( x, y, wr, wi, tmpdm, tmppm, n )

wr k[ ] 2πk wst*n⁄[ ] k 0 1 … wstn 2 1–⁄, , ,( )program memory=cos=

wi k[ ] 2πk wst*n⁄[ ]sin k 0 1 … wstn 2 1–⁄, , ,( )data memory==

n 2⁄



fftwts ( wr, wi, n )

NAME FFT Weights Array

DESCRIPTION The fftwts( ) function computers the weights (or twiddles) array for an FFT Computa-tion. Only a single weights array for the maximum length FFT is required. Typically a weights array is computed for half the FFT size.

ALGORITHM

SYNOPSIS void fftwts ( wr, wi, n )

float pm *wr ; /* Pointer to output real weight table */

float dm *wi ; /* Pointer to output imaginary weight table */

int n ; /* Number of weights to generate */

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tfftwts.c included in the distribution tape provides an example of this func-tion’s use.

Note that a new function called winwts( ) was introduced to correct a discrepancy with the hamm( ) window function, whose weights were originally calculated using the formula . This was incorrect as this required the FFT function to allo-cate twice the amount of memory to contain the weights. To correct the problem, a new function was introduced called the winwts( ) which uses the formula

to perform the weights calculation while the fftwts( ) function was mod-ified to use the formula for the fft weights calculation.

wrm πmn---- m 0 1 2 … n, 1–, , ,{ }=cos=

wim πmn---- m 0 1 2 … n, 1–, , ,{ }=sin=

2 π m•• n⁄

2 π m•• n⁄π m• n⁄



fht ( x, wr, wi, tmpdm, n )

NAME Fast Hartley Transform

DESCRIPTION Computes the Fast Hartley Transform of complex input/output vector x. The real and imaginary weights array are stored in wr and wi. Temporary storage space used to compute the fast cosine transform is designated by tmpdm for temporary dm memory.

ALGORITHM

SYNOPSIS void fht ( x, wr, wi, tmpdm, n )

float *x ; /* Pointer to real input/output vector x */

float *wr ; /* Pointer to real cosine weights array wr */

float *wi ; /* Pointer to imaginary weights array wi */

float *tmpdm ; /* Pointer to temporary dm storage space */

int n ; /* Element count for floating point input */

DOMAIN -3.4E+38 to 3.4E+38


H k( ) He k( ) Ho k( ) 2πk N⁄( ) Ho N 2⁄( ) k–( ) 2πk N⁄( )sin+cos[ ]+=

H k N 2⁄+( ) He k( ) Ho k( ) 2πk N⁄( ) Ho N 2⁄( ) k–( ) 2πk N⁄( )sin+cos[ ]–=

m 0 1 2 … n, 1–, , ,{ }=



EXECUTION TIME 82,173 cycles ( 128 Pts ) - ( Currently Only C Code Version Available )

NOTES The file tfht.c included in the distribution tape provides an example of this function’suse.

This implementation is a recursive algorithm that does not employ explicit bit-reversed addressing. Therefore, no address alignment restrictions are imposed.


Note that these are twice the length of the cosine/sin vectors used for a Complex FastFourier Transform of the same size.

Vector tmpdm, which is in data memory, and has a mimimum sizes of .

n must be an integer power of 2.

fht ( x, wr, wi, tmpdm, n )

wr k[ ] 2πk n⁄[ ] k 0 1 … n 1–, , ,( )program memory=cos=

wr k[ ] 2πk n⁄[ ]sin k 0 1 … n 1–, , ,( ) data memory==

4 n•



fir_wci ( x, h, y, b, d, n, p )

NAME Direct Form FIR Filter

DESCRIPTION Computes a FIR ( Finite Impulse Response Filter) using the coefficients stored in vec-tor h applied to the elements of vector x. The results are stored in vector y.

ALGORITHM

SYNOPSIS void fir_wci ( x, h, y, b, d, n, p )

float *x ; /* Pointer to input samples x */

float pm *h ; /* Pointer to coefficient in vector h */

float *y ; /* Pointer to output vector y */

float *b ; /* Pointer to start of delay memory */

float **d ; /* Double Pointer to delay memory d */

int n ; /* Number of input samples n */

int p ; /* Number of coefficients p */

DOMAIN -3.4E+38 to 3.4E+38


Yk Hp 1 i–– Xk i– k 0 1 2 … n 1–,, , ,{ }=•

i 0=

p 1–

∑=



EXECUTION TIME 50 + P + (N-1) * (4+P) cycles

NOTES The file tfir.c included in the distribution tape provides an example of this function’s use.

Negative indices of x in the above equation are handled using a delay memory buffer d. ‘d’ represents the initial filter state, and should be set to zero prior to the first calling of fir_wci( ). On subsequent calls d is updated by fir_wci( ), allowing processing of a continuous input sample stream across multiple function calls.

This routine uses pipelined coding techniques to execute in an optimal manner. There-fore, the number of data points (n) supplied to the routine must also be greater than or equal to 2.

float *x, the pointer to input samples x, is of size n

float pm *h, the pointer to coefficient in vector h, is of size p

float *y, the pointer to output vector y, is of size n

Reverse the order of the coefficients (taps) to compute the convolution. Example- To compute the convolution for:

data = { x1, x2, x3, x4, ....} taps = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}


taps = { 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 } and call the fir_wci function.

fir_wci ( x, h, y, b, d, n, p )



fir5 ( x, h, y, d, n )

NAME 5-Tap Finite Impulse Response (FIR) Filter

DESCRIPTION Computes a 5-Tap FIR ( Finite Impulse Response Filter) using the coefficients stored in vector h applied to the elements of vector x. The results are stored in vector y.

ALGORITHM

SYNOPSIS void fir5 ( x, h, y, d, n )

float *x ; /* Pointer to input samples x */

float pm *h ; /* Pointer to coefficient in vector h */

float *y ; /* Pointer to output vector y */

float **d ; /* Double Pointer to delay memory d */

int n ; /* Number of input samples n */

DOMAIN -3.4E+38 to 3.4E+38


Yk H4 i– Xk i– k 0 1 2 … n 1–,, , ,{ }=•

i 0=

4

∑=



EXECUTION TIME 55 + (N-1)*7 cycles

NOTES The file tfir5.c included in the distribution tape provides an example of this function’s use.

Negative indices of x in the above equation are handled using a delay memory buffer d. ‘d’ represents the initial filter state, and should be set to zero prior to the first calling of fir5( ). On subsequent calls d is updated by fir5( ), allowing processing of a contin-uous input sample stream across multiple function calls.

Reverse the order of the coefficients (taps) to compute the convolution. Example - Compute the convolution for:

data = { x1, x2, x3, x4, ....} taps = { 1, 2, 3, 4, 5 }


taps = { 5, 4, 3, 2, 1 } and call the fir5 function.

fir5 ( x, h, y, d, n )



firlms ( x, h, y, d, n, p, b )

NAME Adaptive FIR Filter

DESCRIPTION Computes an adaptive FIR (Finite Impulse Response) filter using coefficients stored in vector h applied to the elements of vector x. The results are stored in vector y.

ALGORITHM

SYNOPSIS Void firlms ( x, h, y, d, n, p, b)

float *x; /* Pointer to input samples vector xof length n */

float *h ; /* Pointer to coefficients vector h of length p */

float *y ; /* Pointer to output sampoles vector y of length n/p */

float *d ; /* Pointer to delay memory of length p */

int n ; /* Number of input samples */

int p ; /* Number of coefficients */

float b ; /* Adaption coefficient */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME TBS cycles

NOTES The file tfirlms.c included in the distribution tape provides an example of this func-tion’s use.

Negative indicies of x in the above equation are handled using a delay memory buffer d. d represents the initial filter state, and should be set to zero prior to firlms calling firlms( ). On subsequent calls d is updated by firlms( ), allowing processing of contin-ious input sample stream across multiple function calls.

The number of input samples n must be greater than or equal to the number of coeffi-cients p.

yk hk i, xk i– k 0 1 2 … n 1–,, , ,{ }=

i 0=

p 1–

∑=

hk h b+ k 1– xk 1–=



floor_wci ( x )

NAME Round Down to Nearest Integer

DESCRIPTION This function computes the largest integral value less than or equal to the floating-point number x. A floating-point representation of this integral value is returned.

ALGORITHM

SYNOPSIS float floor_wci ( float x )

DOMAIN -3.4E+38 to 3.40E+38



NOTES The file tfloor.c included in the distribution tape provides an example of this function's use.

fmod_wci ( x, y )

NAME Floating-Point Remainder

DESCRIPTION This function computes the floating-point remainder of x/y. The computed value is returned from this function.

ALGORITHM

SYNOPSIS float fmod_wci ( float x, float y )

DOMAIN -2.14E9 to 2.14E9



NOTES The file tfmod.c included in the distribution tape provides an example of this func-tion's use.

return largest int x≤=

return xy--- fix x

y---– sign x( )•=



frexp_wci ( x, n )

NAME Get Mantissa and Exponent

DESCRIPTION This function computes the fractional portion and the exponent of a floating-point number, x. The fractional value is returned from this function and the integer value is stored at the pointer, n.

ALGORITHM ,

SYNOPSIS float frexp_wci ( float x, int *n )

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tfrexp.c included in the distribution tape provides an example of this func-tion's use.

x f 2N• return f=,= n N=



hamm ( a, i, c, k, w, h, n )

NAME Hamming Window Multiply

DESCRIPTION Computes a Hamming window on the values held in real input vector a using the cosine weights array values held in real input vector w. The results are stored in real output vector c.

ALGORITHM

SYNOPSIS void hamm ( a, i, c, k, w, h, n )








DOMAIN -3.4E+38 to 3.4E+38


Cmk Ami 0.54 0.46Arg

n---------

cos•– •=

Arg 2 π i••=

i 0 1 2 … n, 1–, , ,{ }=




NOTES The file thamm.c included in the distribution tape provides an example of this func-tion’s use.

The Hamming Window is computed using the winwts( ) function. The winwts( ) func-tion computes the weights array using the cosine function. This array is pointed to byvector w listed in the synopsis section shown above.

The input vector w contains the pre-computed cosine value cos ( Arg / n ) which maybe computed from the winwts( ) function.

hamm( ) is a vector function. You may therefore use the stride arguments i, k and h todecimate both input and output for data congruence. For instance, suppose you use thewinwts( ) function to compute the FFT weights for a 16K point FFT. This would resultin an array whose length would be 16,384 points. If you were to later decide to com-pute an FFT of length 1,024 and run a Hamming Window on the results, you would notneed to rerun the winwts( ) function to generate new weights. Simply use the oldweights previously calculated and then stride by 16 (16,384/1024 = 16) on stride ele-ment h to obtain the correct window FFT weights. In this manner you may computewinwts( ) once, and later us them for varying length FFTs and windowing functions.

The Hamming window has a passband ripple of 0.0194 dB, a maximum stopbandattenuation of 53 dB, and a 41 dB main lobe relative to side lobe.

hamm ( a, i, c, k, w, h, n )



hann ( a, i, c, k, n, f, w, h )

NAME Hanning Window Multiply

DESCRIPTION Computes a Hanning window on the values held in real input vector a using the cosine weights array values held in real input vector w. The results are stored in real output vector c.

ALGORITHM

SYNOPSIS void hann ( a, i, c, k, n, f, w, h )



float dm *c ; /* Pointer to resultant output vector c */

int k ; /* Element stride for output vector c */

int n ; /* Element count for ooutput vector c */

int f ; /* Normalization flag */

float pm *w ; /* Pointer to weights array */


DOMAIN -3.4E+38 to 3.4E+38


Cm Ami α• 1Arg

n---------

cos–• =

if F 0 then α 0.5 (Un-normalized)==

Arg 2 π i••=

i 0 1 2 … n, 1–, , ,{ }=

if F 1 then α 0.8165 (Normalized)==




NOTES The file thann.c included in the distribution tape provides an example of this func-tion’s use.

This routine utilizes a normalization flag specified by the variable F. If the flag is setto 0, the routine internally loads 0.5 to F and therefore becomes non-normalized. If theflag is set to 1, the routine internally loads 0.8165 to F and is thereforenormalized.

The Hanning Window is computed using the winwts( ) function. The winwts( ) func-tion computes the weights array using the cosine function. This array is pointed to byvector w listed in the synopsis section shown above.

The input vector w contains the pre-computed cosine value cos ( Arg / n ) which maybe computed from the winwts( ) function.

hann( ) is a vector function. You may therefore use the stride arguments i, k and h todecimate both input and output for data congruence. For instance, suppose you use thewinwts( ) function to compute the FFT weights for a 16K point FFT. This would resultin an array whose length would be 16,384 points. If you were to later decide to com-pute an FFT of length 1,024 and run a Hanning Window on the results, you would notneed to rerun the winwts( ) function to generate new weights. Simply use the oldweights previously calculated and then stride by 16 (16,384/1024 = 16) on stride ele-ment h to obtain the correct window FFT weights. In this manner you may computewinwts( ) once, and later us them for varying length FFTs and windowing functions.

The Hanning window has a passband ripple of 0.0546 dB, a maximum stopband atten-uation of 44 dB, and a 31 dB main lobe relative to side lobe.

hann ( a, i, c, k, n, f, w, h )

2 3⁄( )



inl ( dataloc, fltmax, inlloc, min, max, n )

NAME Integral Non-linearity

DESCRIPTION The inl( ) function calculates the integral non-linearity based on the values supplied in input vector dataloc, input scalar fltmax, and the number of points supplied in input scalar n. The results are placed in output scalar min and max, while the results of the integral non-linearity calculation are output to vector inlloc.

ALGORITHM

pts = ( int ) ( n + 0.5 ) ;

devicelsb = ( dataloc[pts-1] - dataloc [0] ) / (pts - 1) ;

min = fltmax ;max = - fltmax ;

for ( i = 0; i < pts ; i++ ){ inlloc[i] = ( dataloc[i] - ( devicelsb * i + dataloc ) ) / devicelsb ; min = ( min < inlloc[i] ) ? min : inlloc[i] ; max = ( max > inlloc[i] ) ? max : inlloc[i] ;}

SYNOPSIS void inl ( dataloc, fltmax, inlloc, min, max, n )

float dm *dataloc ; /* Pointer to real input vector a */

float dm fltmax ; /* Maximum Boundaries a */

float dm *inlloc ; /* Pointer to output vector inlloc */

float dm *min ; /* Output scalar minimum */

float dm *max ; /* Output scalar maximum */


DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 48 + 5 * (INT) (N+0.5) cycles

NOTES The file tinl.c included in the distribution tape provides an example of this function’s use. The integral non-lineariry function is often used by designers of automated test equipment (ATE) to test the performance of A/D semiconductor devices.



iqunpk ( srcptr, dstptri, dstptrq, m, n, trans, isize, justify )

NAME Unpack an IQ Integer Packed Matrix

DESCRIPTION This function unpacks integer packed matrix srcptr[ ] [ ] into its I and Q component matricies. The routine unpacks an arbitrary number of data bits (up to 16 bits maxi-mum), and masks the remaining bits, thus allowing the use of upto a 16-bit A/D. The integer packed format is described below.

The I component is stored in output matrix dstptri[ ] [ ], while the Q component is stored in output matrix dstptrq[ ] [ ]. The number of rows in input matrix srcptr[ ] [ ] is defined by input scalar m, while the number of columns in input matrix srcptr[ ] [ ] is defined by input scalar n. Additionally, a transpose flag argument trans determines the strorage method, where trans = 0 is normal while trans = 1 is transposed.

ALGORITHM

SYNOPSIS Void iqunpk ( srcptr, dstptri, dstptrq, m, n, trans, isize,

justify )

int *srcptr ; /* Pointer to input matrix srcptr [ ] [ ] */

float *dstptri ; /* Pointer to output matrix dstptri [ ] */

float *dstptrq ; /* Pointer to output matrix dstptrq [ ] */

int m ; /* Number of rows in matrix srcptr [ ] [ ] */

int n ; /* Number of columns in matrix srcptr [ ] [ ] */

int trans ; /* Transpose Flag ( 0 = normal, 1 = transpose)*/

int isize ; /* Size of Integer to Unpack ( 16 bits max ) */

int justify ; /* Justify Flag ( 0 = left, 1 = right ) */

DOMAIN -3.4E+38 to 3.4E+38


Source Matrix Destination Matrix I, Destination Matrix Q⇒



EXECUTION TIME 49 + ( 3 + 6*N )*M cycles

NOTES The file tiqunpk.c included in the distribution tape provides an example of this func-tion’s use.

This function unpacks integer packed matrix srcptr[ ] [ ] into its I and Q component matricies. The integer packed format is described below:

X Bits I A/D 16-X Bits X Bits Q A/D 16-X Bits

31 0

Or

16-X Bits X Bits I A/D 16-X Bits X Bits Q A/D

31 0

where: X is the size if the integer to unpack (isize) and justify determines whether the data is left or right justified.

For example, for isize = 12 and justify = 0, the input data is expected to look like:

12 Bits I A/D 4 Bits 12 Bits Q A/D 4 Bits

31 20 19 16 15 4 3 0

Bits 0-3 and bits 16-19 are assumed to be zero.

The integer element is read and masked for both the I and Q data. Each data is then separated and converted to a floating-point format. The data is then stored in their respective destination matrices (dstptri and dstptrq).

Additionally, the transpose flag (trans) determines the storage method. If trans is set to zero, the element is stored in its respective position (ie, the destination matrix is the same as the source matrix, m by n). If trans is 1, the element is stored in its transposed position (the destination matrix is a transpose of the source matrix n by m). The justify falg determines the justification of the data. If justify is 0, the elements are left justi-fied. If justify is 1, the elements are right justified.

iqunpk ( srcptr, dstptri, dstptrq, m, n, trans, isize, justify )



kaiser ( a, i, c, alpha, n )

NAME Kaiser-Bessel Window Multiply

DESCRIPTION Computes a Kaiser-Bessel window on the values held in real input vector a . The results are stored in real output vector c.

ALGORITHM

SYNOPSIS void kaiser ( a, i, c, alpha, n)



float *c ; /* Pointer to resultant output vector c */

float alpha ; /* Kaiser window shape parameter */

int n ; /* Element count for output vector c */

DOMAIN -3.4E+38 to 3.4E+38


Ci Ai

I0 α i 2M i–( )( )

M-----------------------------------------

I0 α( )----------------------------------------------• =

i 0 1 2 … n, 1–, , ,{ }=

where

I0 x( )

x2---

k

k!----------

2

k 0=

∞

∑=

α scalar derived from alpha() function=



EXECUTION TIME 68 + 18*B + (41+18*B) * (N-1)

NOTES The file tkaiser.c included in the distribution tape provides an example of this func-tion’s use.

Note that B described in the execution time shown above is the number of iternationsneeded to complete the 0th order Bessel function and is indeterminate.

is a modified Bessel function of the 1st kind and oth order give by theform

and M, the midpoint is defined as

Alpha is the Kaiser window shape parameter as described by the library func-tion alpha( ).

The Kaiser-Bessel is often used in situations where great control is desired over the shape of the filter. In many situtations, use of the Hamming or Hanning windows result in implementations where the stopband and passband are considered unobtain-able due to the ripple factor .This shape value is adjustable and allows the window to obtain any desired ripple value or attenutation value. It has a stopband attenuation of

, and a passband of .

kaiser ( a, i, c, alpha, n )

I0

I0

x2---

2

k!----------

k 0=

∞

∑=

M N 1–( )2

-----------------=

α( )

δ

20 δ10log– 17.372δ



ldexp_wci ( x, n )

NAME Multiply by Power of 2

DESCRIPTION This function computes x multiplied by 2N. A range error may occur when x times is greater then 3.40E+38.

ALGORITHM

SYNOPSIS float ldexp_wci ( float x, int n )

DOMAIN -3.4E+38 to 3.40E+38



NOTES The file tldexp.c included in the distribution tape provides an example of this func-tion's use.

2n

return x 2n•=



linint ( xr, yr, xp, yp, slope, sflag, m, n )

NAME Linear Interpolation

DESCRIPTION This function calculates the linear interpolation using a reference grid formed by vec-tors xr [ ] and yr [ ]. The slope is computed from these vectors and placed in vector slope [ ]. A linear interpolation is computed for vector yp[ ] using the vectors slope [ ], xr [ ], yr [ ] and xp [ ]. This is illustrated below for a single xp point.

yp yr

yr2

yp1

yr1 xr

xr1 xr2

xp

xp1

ALGORITHM

See Pseudo Algorithm in Notes Section Below For A Better Description

SYNOPSIS void linint ( xr, yr, xp, yp, slope, sflag, m, n, )

float dm *xr ; /* Pointer to input vector xr [ ] */

float pm *yr ; /* Pointer to input vector yr [ ] */

float pm *xp ; /* Pointer to input vector xp [ ] */

float dm *yp ; /* Pointer to output vector yp [ ] */

float dm *slope ; /* Pointer to temp vector slope [ ] */

int sflag ; /* Slope Flag ( 1 = do slope, 0 = skip slope )*/

int n ; /* Number of elements in xr [ ] and yr [ ] */

int m ; /* Number of output elments in yp [ ] */

DOMAIN -3.4E+38 to 3.4E+38

slope xr2 xr1–yr2 yr1–-----------------------=

yp1 yr1 slope xp1 xr1–( )•+=




EXECUTION TIME for sflag = 1, 57 + 14 * ( n -1 ) + (14 + w) * m where w = 5 for xp [i] < xr [0] = 7 for xp [i] > = xr[n-1] = indeterminant, but < 4 *n

for sflag = 0, 57 + ( 14 + w) * m where w = 5 for xp [i] < xr [0] = 7 for xp [i] > = xr[n-1] = indeterminant, but < 4 *n

NOTES The file tlinint.c included in the distribution tape provides an example of this func-tion’s use.

The sflag determines if the slope needs to be computed. If sflag is 1 the slope is com-puted. If sflag is 0 the slope in not computed and the current values in slope [ ] are used. This provides for an initial call that computes slope [ ]. Subsequent calls skip slope [ ] computations saving overhead.

Values for vector xr [ ] are assumed to be in ascending order. The number of elements computed for slope [ ] is 1 less than the number of reference points in xr [ ] and yr [ ] (n - 1).

If xp1 should be bound beyond the bounds between xr[0] and xr[n-1] (the lowest and highest points respectively), then yp1 will be set to either yr[0] or yr[n-1].

if ( xp[i] < xr[0] ) yp[i] = yr [0]else if ( xp[i] >= xr[n-1] ) yp[i] = yr[n-1]else compute linear interpolation . .end if

linint ( xr, yr, xp, yp, slope, sflag, m, n )



linmag ( yr, i, yi, j, linmag, k, n )

NAME Vector Linear Magnitude

DESCRIPTION This function computes the linear magnitude from input complex vectors yr and yi. The results are stored to output vector linmag.

ALGORITHM

SYNOPSIS void linmag ( yr, i, yi, j, linmag, k, n )

float dm *yr ; /* Pointer to input vector yr (real) */

int i ; /* Element stride for input vector yr */

float pm *yi ; /* Pointer to input vector yi (imaginary) */

int j ; /* Element stride for input vector yi */

int dm *linmag ; /* Pointer to output vector linmag */

int k ; /* Element stride for output vector linmag */


DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 49 + 19*(N-1)

NOTES The file tlinmag.c included in the distribution tape provides an example of this func-tion’s use.

Note that vector yi is in pm memory. This function is tailored to use the output prod-ucts from the cfft () function call.

linmag m[ ]k yri m[ ]2 yij m[ ]2

+[ ]

12---

=

m 0 1 2 … n, 1–, , ,{ }=



log_wci ( x )

NAME Natural Logarithm

DESCRIPTION This function computes the natural logarithm of a floating-point number x. A domain error occurs if the argument is negative. A range error occurs if the argument is zero.

ALGORITHM

SYNOPSIS float log_wci ( float x )

DOMAIN 0.0 to 3.4 E+38



NOTES The file tlog.c included in the distribution tape provides an example of this function's use.

log10_wci ( x )

NAME Base 10 Logarithm

DESCRIPTION This function computes the base 10 logarithm of a floating-point number x. A domain error occurs if the argument is negative. A range error occurs if the argument is zero.

ALGORITHM

SYNOPSIS float log10_wci ( float x )




NOTES The file tlog10.c included in the distribution tape provides an example of this func-tion's use.

return loge x( )=

return log10 x( )=



logmag ( yr, i, yi, j, logmag, k, n )

NAME Vector Log Based Magnitude

DESCRIPTION This function computes the log based magnitude from input complex vectors yr and yi. The results are stored to output vector logmag.

ALGORITHM

SYNOPSIS void logmag ( yr, i, yi, j, logmag, k, n )

float dm *yr ; /* Pointer to input vector yr (real) */

int i ; /* Element stride for input vector yr */

float pm *yi ; /* Pointer to input vector yi (imaginary) */

int j ; /* Element stride for input vector yi */

int dm *logmag ; /* Pointer to output vector logmag */

int k ; /* Element stride for output vector logmag */


DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tlogmag.c included in the distribution tape provides an example of this func-tion’s use.

If the sum of the squares of input vectors yr[ ] and yi [ ] is 0.0, then the input value used is .

mlog ag m[ ]k 10 yri m[ ]2 yij m[ ]2

+[ ]log=

m 0 1 2 … n, 1–, , ,{ }=

3.4 38–×10



lsqreg ( a, b, &c, &d, n )

NAME Least-Squares Regression Fit

DESCRIPTION This routine calculates the linear least-squares regression fit of the elements within input vector a (the dependent x-coordinate values) with the corresponding elements within input vector b (the independent y-coordinate values), and places the resulting output slope in scalar c and the resulting output y-intecept in scalar d.

ALGORITHM

SYNOPSIS void lsqreg ( a, b, &c, &d, n)

float *a ; /* Pointer to x-coordinate input vector x */

float *b ; /* Pointer to y-coordinate input vector y */

float *c ; /* Pointer to resultant output slope c */

float *d ; /* Pointer to resultant output y-intercept */

int n ; /* Element Count */

DOMAIN -3.4E+38 to 3.4E+38


C slope

n AmBm Xm Ym

i 0=

n 1–

∑i 0=

n 1–

∑–

i 0=

n 1–

∑

n Am2

Am

i 0=

n 1–

∑

2

–

i 0=

n 1–

∑

-------------------------------------------------------------------------==

D y-intercept= y C x•( )–=

m 0 1 2 … n, 1–, , ,{ }=



EXECUTION TIME 99 + 5 * ( N - 1 )

NOTES The file tlsqreg.c included in the distribution tape provides an example of this func-tion’s use.

This routine calculates a linear regression line (a best-fit of the data) using the inputvariables contained within input vector a[ ] and input vector b[ ].

This routine does not support striding as statistically, the answers presented in outputscalars c and d will hold more significance as more data points are passed through thealgorithm. We wanted to discourage results which may not be statistically sound.

Recall from elementary algebra the traditional y-intecept, slope linear equation for-mula: where is the slope of the line and is the y-intercept. This formulacan be re-written as where is now the y-intercept and is now the slope.We can now derive the y-intercept (which we implement in this routine) using the cal-culated slope and the arithmetic mean of the y values ( input vector b[ ] values ) bysimply solving for as follows: where and are the arithmetic mean of they and x ( input vectors y[ ] and x[ ] ) input values respectively.

Note that this implementation is computationally faster than the traditional regressionformula usually found in statistical textbooks:

where the y-intercept is found to be a function of the

slope:

Note that in both methods the y-intercept calculation is the same, with being equal tothe arithmetic mean of all the x-values (input vector a[ ] ), while is equal to the arith-metic mean of all the y-values ( input vector b[ ] ).

In order for the least squares regression model to hold statistically true, a number ofassumptions regarding the input data in vectors [x] and [y] must also hold true:

1. For each value of input vector a, each corresponding value of vector y[ ] must be an independent value sampled at random.

2. The variances of the vector b values must be homoscedatic, i.e. have distribu-tions equal to one another.

To measure the accuracy of the predicted linear least sqaures regression use the see( )function to predict the standard error of estimate which will measure the variance ofthe y values ( input vector b[ ] ) around the true regression line.

lsqreg ( a, b, &c, &d, n )

y mx b+= mx by' a bx+= a b

a a y bx–= y x

slopexi x–( ) yi y–( )∑

xi x–( )2∑-------------------------------------------=

y-intercept y slope x•( )–=

xy



madd ( a, b, x, y, c )

NAME Matrix Add

DESCRIPTION This function computes the addition of input matrix a[ ] [ ] with input matrix b[ ] [ ] and stores the results to output matrix c[ ] [ ].

ALGORITHM

SYNOPSIS void madd ( a, b, x, y, c )


float dm *b ; /* Pointer to input matrix b [ ] [ ] */

int x ; /* Number of rows in matrix a[ ][ ] and b[ ][ ] */

int y ; /* Number of columns in matrix a[ ][ ] & b[ ][ ] */

float dm *c ; /* Pointer to output matrix c [ ] [ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmadd.c included in the distribution diskette provides an example of this function’s use.

The storage methodology for matrices is by rows. Matricies can be thought of as one long array (vector) where the beginning of each row is offset by the number of col-umns.

This algorithm computes the sum of two matrices of the same order: i.e., the number of rows and columns of input matrix a[ ] [ ] must equal the number of rows and col-umns of input matrix b[ ] [ ].

C11 C12 C13

C21 C22 C23

A11 A12 C13

A21 A22 A23

B11 B12 B13

B21 B22 B23

+=



mand ( a, b, x, y, c )

NAME AND Two Matrices

DESCRIPTION This function mand( ) logically ANDs elements of matrices a [ ] [ ] and b [ ] [ ], plac-ing the results in ouput matrix c [ ] [ ]. Matices a [ ] [ ] and b [ ] [ ] are of the same demension. Matrix c [ ] [ ] is a 0-1 matrix.

ALGORITHM

SYNOPSIS void mand ( a, b, x, y, c )



int x ; /* Number of rows in vector a[ ][ ] & b [ ] [ ] */

int y ; /* Number of columns in vector a[ ][ ] & b [ ] [ ]*/


DOMAIN -3.4 x 1038 to +3.4 x 1038


C11 C12

C21 C22

A11 A12

A21 A22

ANDB11 B12

B21 B22

=



EXECUTION TIME 33 + 4 * ( N -1 ) cycles

NOTES The file tmand.c included in the distribution diskette provides an example of this function’s use.


An example of a matrix AND is provided below.

a[ ][ ] = ;

b[ ][ ] =

x = 3, y = 3 ;

mand ( a, b, x, y, c ) ;


c[ ][ ] =

mand ( a, b, x, y, c )

1.0 1.7 5.55.0 1.6 1.01.0 5.0 6.0

0 1.7 5.55.0 1.6 1.01.0 0.0 6.0

0 1 11 1 11 0 1



maxm ( a, x, y )

NAME Maximum Value of a Matrix

DESCRIPTION This function finds the element in input matrix a[ ] [ ] with the maximum value and returns it from the function. The number of rows contained within input matrix a[ ] [ ] is defined by input integer x while the number of columns is defined by input scalar y.

ALGORITHM

SYNOPSIS float maxm ( a, x, y )

float *a ; /* Pointer to input/output matrix a[ ][ ] */

int x ; /* Number of rows in input/output vector a[ ][ ] */

int y ; /* Number of columns in input/output vector a[ ][ ]*/

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 10 + X * Y cycles

NOTES The file tmaxm.c included in the distribution diskette provides an example of this function’s use.

Matrices can be thought of as one long array (vector) where the beginning of each row is offset by the number of columns.

return maximum element in matrix A⇐



maxmc ( a, x, y, mx, my )

NAME Maximum Element of Matrix With Coordinates

DESCRIPTION This function finds the element in input matrix a[ ] [ ] with the maximum value along with its coordinates. The maximum value is returned from the function call, the maxi-mum x-coordinate is output in scalar mx, and the maximum y-coordinate in output scalar my. The number of rows contained within input matrix a[ ] [ ] is defined by input scalar x while the number of columns is defined by input scalar y.

ALGORITHM

SYNOPSIS float maxmc ( a, x, y, mx, my )


int x ; /* Number of rows in input vector a[ ][ ] */

int y ; /* Number of columns in input vector a[ ][ ] */

int mx ; /* X-coordinate of the maximum value */

int my ; /* Y-coordinate of the maximum value */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmaxmc.c included in the distribution diskette provides an example of this function’s use.


The coordinates reflects the largest elements. In other words, if the two largest ele-ments are equal, the coordinates are those of the second element.

The coordinate system begins with zero. If there are 7 elements in a column and 8 ele-ments in a row, then the column range is 0 to 6 and the row range is 0 to 7.

return maximum element of matrix A[ ][ ] with coordinates⇐



maxmgv ( a, i, c, n )

NAME Maximum Magnitude of Vector

DESCRIPTION This function finds the element in input vector a with the maximum magnitude and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void maxmgv ( a, i, c, n )



float *c ; /* Pointer to output maximum value scalar c */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmaxmgv.c included in the distribution diskette provides an example of this function’s use.

if C Ami then m 0 1 2 …n 1–, , ,{ }=<

C Ami=



maxmgvi ( a, i, c, d, f, n )

NAME Vector Maximum Magnitude with Index

DESCRIPTION This function finds the element in input vector a with the maximum magnitude and stores the result in output scalar c. The index of the maximum value of a is returned in output scalar d. A flag f controls the return of either the first or last maximum index is encountered.

ALGORITHM

D= Index of Maximum Magnitude Value of A

If f = 0 then the First Maximum Magnitude Returned

If f > 0 then the Last Maximum Magnitude Returned

SYNOPSIS void maxmgvi ( a, i, c, d, f, n )



float dm *c ; /* Pointer to output maximum value scalar c */

int dm d ; /* Pointer to index of maximum value d */

int f ; /* Flag to indicate first or last mag returned */


DOMAIN -3.4 x 1038 to +3.4 x 1038


if C Ami then m 0 1 2 …n 1–, , ,{ }=<

C Ami=




NOTES The file tmaxmgvi.c included in the distribution diskette provides an example of this function’s use.

The flag f indicates wether the index of the first or last magnitude is returned. If f is set to 0 then the index of the first maximum magnitude encountered in the input array is returned in d. If f is set to > 0 then the index of the last maximum magnitude encoun-tered in the input array is returned in d.

The search for the first or last magnitude is accomplished by searching backwards and forwards respectively.

The index is referenced to zero at the beginning of the array.

maxmgvi ( a, i, c, d, f, n )



maxv ( a, i, c, n )

NAME Vector Maximum Value

DESCRIPTION This function finds the element in input vector a with the maximum value and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void maxv ( a, i, c, n )



float *c ; /* Pointer to output scalar c */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmaxv.c included in the distribution diskette provides an example of this func-tion’s use.

if C Ami then m 0 1 2 …n 1–, , ,{ }=<

C Ami=



maxvi ( a, i, c, d, n )

NAME Vector Maximum Value With Index

DESCRIPTION This function finds the element in input vector a with the maximum value and stores the result in output scalar c. It also locates the index of the maximum values and stores it in output scalar d.

ALGORITHM

SYNOPSIS void maxvi ( a, i, c, d, n )




int *d ; /* Pointer to output maximum index scalar d */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmaxvi.c included in the distribution diskette provides an example of this function’s use.

This routine is slower than the maxv routine as the index of the largest element is returned along with the maximum value. If the input array a has two values that are computed to be the maximum, and both are equal, then the routine returns the index of the last largest element encountered during the search, as the maximum element.

if C Ami then m 0 1 2 …n 1–, , ,{ }=<

C Ami=

D index of max element=



mchol ( a, b, c, n )

NAME Choleski Decomposition

DESCRIPTION This function computes a Choleski decomposition of input matrix a[ ] [ ]. The lower triangular matrix is stored to output matrix b[ ] [ ] while the upper triangular matrix is stored to output matrix c[ ] [ ].

ALGORITHM

SYNOPSIS void mchol ( a, b, c, n )


float dm *b ; /* Pointer to lower output matrix b[ ][ ] */

float dm *c ; /* Pointer to upper output matrix c[ ][ ] */

int n ; /* Number of rows and column */

DOMAIN -3.4 x 1038 to +3.4 x 1038


A [ ] [ ] Blower triangular [ ] [ ],Cupper triangular [ ] [ ]⇒



EXECUTION TIME 116+(3(N-1)+(3+2N)N+(42+3(N-1)+(9+4(N-1))(N-2))(N-2) cycles

NOTES The file tmchol.c included in the distribution diskette provides an example of this function’s use.

The Choleski Decomposition is computed as follows:

where C [ ] [ ] is the transpose of B [ ] [ ].

An example of a Choleski Decomposition is provided as follows:


It is assumed that the input matrix is positive definite. No error checking is provided for this.

mchol ( a, b, c, n )

A [ ] [ ] = B[ ] [ ] C [ ] [ ] •

A

4 2– 4 22– 10 2– 7–

4 2– 8 42 7– 4 7

n=4,=

B

2 0 0 01– 3 0 0

2 0 2 01 2– 1 1

=

C

2 1– 2 10 3 0 2–0 0 2 10 0 0 1

=



mconst ( a, b, x, y )

NAME Constant Matrix

DESCRIPTION This function creates a constant output matrix a whose elements are all equal to the assigned constant supplied in input scalar constant b. The input integer scalar x con-tains the number of rows in input matrix a[ ] [ ]. The input integer scalar y contains the number of columns in input matrix a[ ] [ ].

ALGORITHM

SYNOPSIS void mconst ( a, b, x, y )

float dm *a ; /* Pointer to output matrix a [ ][ ] */

float dm b ; /* Input scalar b constant */



DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 9 + ( X * Y ) cycles

NOTES The file tmconst.c included in the distribution diskette provides an example of this function’s use.


Ax y, B=



mcpivot ( a, b, c, x )

NAME Complete Pivoting For a Matrix

DESCRIPTION This function performs a complete pivoting of input matrix a[ ] [ ]. The results are returned in output matrix a[ ] [ ]. Vector b[ ] is used to return the row interchanges. Vector c[ ] is used to return the column interchanges. The row and column dimensions of matrix a[ ] [ ] are x. The vector dimension of b[ ] and c[ ] are also are also defined by input scalar x.

ALGORITHM Complete pivoting attempts to arrange matrix a[ ] [ ] in an optimal order by swapping rows and columns. This optimal order arranges elements of a[ ] [ ] so that round off errors are reduced when using the mlu( ) and mfbsub( ) functions. When swapping rows and columns, the largest elements ( in magnitude) are placed along the diagonal.

SYNOPSIS void mcpivot ( a, b, c, x )

float dm *a ; /* Pointer to input/output matrix a[ ][ ] */

int dm *b ; /* Pointer to output vector b[ ] */

int dm *c ; /* Pointer to output vector c[ ] */

int x ; /* Number of rows and columns in matrix a[ ][ ]*/

DOMAIN -3.4 x 1038 to +3.4 x 1038




EXECUTION TIME 82 + 2X + (43+8X+(2+11X)X)X cycles

NOTES The file tmcpivot.c included in the distribution diskette provides an example of this function’s use.

Complete pivoting attempts to find the largest elements for the diagonal. First, the entire matrix is searched for the element with the largest magnitude. Its’ row and column is swapped with the 0th row and column. The 0th row and column are excluded and the minor sub-matrix is searched for the largest magnitude. Its row and column are swapped with the 1st row and column. This process is performed until the last row and column are left.

Given:

The element with the largest magnitude is -16.

The element with the largest magnitude in the minor sub-matrix (excluding row 0 and column 0) is 12.

The element with the largest magnitude in the minor sub-matrix (excluding rows 0 and 1 and columns 0 and 1) is 9.

The storage methodology for matrices is by rows. Matrices can be thought of as one long array (vector) where the beginning of each row is offset by the number of columns. Vector b [ ] and c [ ] need not be initialized. They are assigned within the function. For the example above, the vector for interchanges is interpreted as follows:

b[0] - Row 0 was swapped with row 3b[1] - Row 1 was swapped with row 2b[2] - Row 2 was swapped with row 3

The row vector interchange b[ ] is used by the function mfbsub() in solving for the unknowns in a set of simultaneous linear equations.

mcpivot ( a, b, c, x )

B 0 0 0 0 C,= 0 0 0 0 A

4 2 9 18 3 0 57 6 10 1216– 14 13 15

=,=

B 3 0 0 0 C,= 0 0 0 0 A

16– 14 13 158 3 0 57 6 10 124 2 9 1

=,=

B 3 2 0 0 C,= 0 3 0 0 A

16– 15 13 147 12 10 68 5 0 34 1 9 2

=,=

B 3 2 3 0 C,= 0 3 2 0 A

16– 15 13 147 12 10 64 1 9 28 5 0 3

=,=



mdeterm ( a, x )

NAME Determinant of a Matrix

DESCRIPTION This function computes the determinant of input matrix a [ ] [ ] and returns the result. The row and column dimension of matrix a [ ] [ ] is x.

ALGORITHM

SYNOPSIS void mdeterm ( a, x )

float dm *a ; /* Pointer to input matrix a [ ][ ] */

int x ; /* Number of rows and columns in matrix a[ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


Return Determinant mdeterm matrix A( )=



EXECUTION TIME 51 + (12 + (16 + 4 * X) * ( X-1 )) * ( X-1 ) + ( X-1) cycles

NOTES The file tmdeterm.c included in the distribution diskette provides an example of this function’s use.

This algorithm performs elementary row operations to reduce the matrix to upper tri-angular form. The determinant is the product of the diagonal elements.

An example of computing a matrix determinant is provided below.

determ = 2.0 * 4.0 * 6.0 = 48.0


Note that the diagonal elements must be non-zero. Use the library function mcpivot( ) to rearrange the matrix to get the diagonal elements to non-zero.

For small matrices (2x2 or 3x3) use mdetrm2( ) or mdetrm3( ).

The source matrix a [ ] [ ] is destroyed by the process. ( It is in upper-triangular form ).

The source matrix a [ ] [ ] has to be sqaure.

If at any time row reduction causes a diagonal elements to be zero, the determinant is zero.

mdeterm ( a, x )

If A2.0 1.0– 3.04.0 2.0 1.06.0– 1.0– 2.0

and x = 3=

A2 1– 30 4 5–6– 1– 2

from 2nd row subtract 2 * the first row (the reference row)=

A2 1– 30 4 5–6– 1– 2

from 3rd row subtract 3 * the first row (the reference row)=

A2 1– 30 4 5–6– 1– 2

from 3rd row subtract 1 * the second row (the reference row)=



mdetrm2 ( a )

NAME Determinant of a 2x2 Matrix

DESCRIPTION This function computes the determinant of a 2x2 input matrix a [ ] [ ] and returns the result.

ALGORITHM

SYNOPSIS void mdetrm2 ( a )

float dm *a ; /* Pointer to 2x2 input matrix a [ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmdetrm2.c included in the distribution diskette provides an example of this function’s use.

The determinant of a 2x2 matrix is computed as follows:


Return Determinant mdetrm2 A( )=

If A a bc d

then determ a d b c•–•==



mdetrm3 ( a )

NAME Determinant of a 3x3 Matrix

DESCRIPTION This function computes the determinant of a 3x3 input matrix a [ ] [ ] and returns the result.

ALGORITHM

SYNOPSIS void mdetrm3 ( a )

float dm *a ; /* Pointer to 3x3 input matrix a [ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmdetrm3.c included in the distribution diskette provides an example of this function’s use.

The determinant of a 3x3 matrix is computed as follows:


Return Determinant mdetrm3 A( )=

If A

x11 x12 x13x21 x22 x23x31 x32 x33

then determ =

A x11 x22 x33••( ) x12 x23 x31••( ) x13 x21 x32••( ) x11 x23 x32••( )– x12 x21 x33••( )– x13 x22 x31••( )–+ +=



meandev ( a, i, b, c, n )

NAME Mean Deviation

DESCRIPTION This routine calculates the mean deviation of the elements within input vector a from the mean input scalar b, and places the results in output scalar c.

ALGORITHM

SYNOPSIS void meandev ( a, i, b, c, n)



float *b ; /* Pointer to input mean scalar b */

float *c ; /* Pointer to resultant output scalar c */


DOMAIN -3.4E+38 to 3.4E+38



NOTES The file meandev.c included in the distribution tape provides an example of this func-tion’s use.

C 1 n------ Ami b–

i 0=

n 1–

∑• m 0 1 2 … n, 1–, , ,{ }==



memtest ( a, i, type, pattern, n )

NAME Memory Test

DESCRIPTION This function performs a sanity check on the memory location specified by the start address a for a block of memory defined by n. The type flag determines whether dm or pm memory is tested. ( 1 = dm, 0 = pm ).

memtest( ) writes a test pattern to memory and reads it back. If the written pattern is the same as the pattern read, a 1 is returned to indicate a passed test. If the written pat-tern does not match the read pattern, a 0 is returned to indicate a failed test.

ALGORITHM

SYNOPSIS void memtest ( a, i, type, pattern, n )

int a ; /* Start address of memory */

int i ; /* Memory stride */

int type ; /* Type of memory to test ( 1=dm or 0=pm ) */

int pattern ; /* Test pattern to read/write */

int n ; /* Length of block to be checked */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmemtest.c included in the distribution diskette provides an example of this function’s use.

Return 1 or 0⇐



mextr ( a, ax, ay, b, bx, by, ix, iy )

NAME Extract A Sub-Matrix From A Source Matrix

DESCRIPTION This function extracts an output sub-matrix a[ ] [ ] from an input source matrix b[ ] [ ] starting at a point described by input row integer ix and input column integer iy. The results are stored to output matrix a[ ] [ ].

ALGORITHM

SYNOPSIS void mextr ( a, ax, ay, b, bx, by, ix, iy )

float dm *a ; /* Pointer to output sub-matrix a[ ][ ] */

int ax ; /* Number of rows in sub-matrix a[ ][ ] */

int ay ; /* Number of columns in sub-matrix a[ ][ ] */

float dm *b ; /* Pointer to input source matrix b[ ][ ] */

int bx ; /* Number of rows in source matrix b[ ][ ] */

int by ; /* Number of columns in source matrix b[ ][ ] */

int ix ; /* Extract position for row start */

int iy ; /* Extract position for column start */

DOMAIN -3.4 x 1038 to +3.4 x 1038


A 0 00 0

ax 2 , ay=2, =,=

Bstart

0 0 00 1 40 7 9

bx 3 , by=3, =,=

ix 1 , iy=1, =

Afinish1 47 9

=




NOTES The file tmextr.c included in the distribution diskette provides an example of this function’s use.

This function extract a sub-matrix a[ ] [ ] from a source matrix b[ ] [ ]. Matrix a[ ] [ ] row and column must be less than the row and column of matrix b[ ] [ ]. Example as follows:

The coordinate system begins with zero. If there are 7 elements in a column and 8 ele-ments in a row, then the column is 0 to 6 and the row is 0 to 7.

The extract position must be a valid position (i.e., non-negative). However, it is allowed to have an extract position such that only a part of the sub-matrix is extracted (i.e. if ix > bx - ax).

Example as follows:


mextr ( a, ax, ay, b, bx, by, ix, iy )

A0 0 00 0 00 0 0

B,

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 1 2 3 0 0 00 0 4 5 6 0 0 00 0 7 8 9 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

Extract Position 2 2,( ) Afinal

1 2 34 5 67 8 9

=,=,= =

A0 0 00 0 00 0 0

B,

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 1 20 0 0 0 0 0 4 50 0 0 0 0 0 7 8

Extract Position 4 6,( ) Afinal

1 2 04 5 07 8 0

=,=,= =



mfbsub ( a, b, c, d, x )

NAME Matrix Forward-Back Substitution

DESCRIPTION This routine performs a forward and back substitution using matrices a[ ][ ], b[ ][ ] and vector c[ ]. The results are placed in input/output vector c[ ]. Vector d[ ] is used to order the input solution vector c[ ] to match the swapping that may have been done in mppivot ( ) or mcpivot ( ). The row and column dimensions of the matrices and the vector dimensions are x.

ALGORITHM This function uses matrices from the LU decomposition function mlu( ) to solve a set of simul-taneous linear equations for a set of unknowns. The L and U decomposed matrices are repre-sented by a[ ][ ] and b[ ][ ] respectively. A system of simultaneous linear equations can be written as: Ax = b, where A represents the coefficients of the equations in matrix form, x repre-sents a vector of unknowns and b represents a vector of the solutions.

Decomposing A = LU the above equation can be rewritten as: LUx = b. Setting y = Ux, the equation becomes Ly = b. Use forward substitution to solve for y. Once y is found, use back substitution to solve for x.

Forward Substitution - This solves for the temporary solution used in back substitution. Forward substitution uses the L component of the LU decompositon and solves for a temporary set of unknowns. These unknowns are then used by back substitution.

Back Substitution - Back substitution uses the U component of the LU Decomposition and solves for the system of unknowns.

SYNOPSIS void mfbsub ( a, b, c, d, x )

float dm *a ; /* Pointer to input matrix a */

float dm *b ; /* Pointer to input matrix b */

float dm *c ; /* Pointer to input matrix c */

float dm *d ; /* Pointer to input matrix d */

int x ; /* Number of rows,columns & vector length*/

DOMAIN -3.4E+38 to 3.4E+38




EXECUTION TIME 72 + ( 8 + 4 * X - 1) ( X - 1) + ( 28 + 4 * X -1) ( X - 1)

NOTES

The file tmfbsub.c included in the distribution tape provides an example of this function’s use.

Prior to Forward-Back substitution, the intial input solution vector c[] is ordered by the order vector d[].

For the example above, the input solution vector c[] was originally = { 1, 2, 3, 4 }.

0w + 1x - 3y - 4z = 11w - 1x + 0y + 4z = 22w + 1x + 2y + 3z = 36w + 2x + 4y + 8z = 4

The order vector d[ ] (from mppivot( )) = { 3, 2, 3, 3 }. The operation of vector d[ ] on c[ ] yields: C D

1 2 3 4 3 2 3 34 2 3 1 ^ Swap position 0 with 34 3 2 1 ^ Swap position 1 with 24 3 1 2 ^ Swap position 2 with 34 3 1 2 ^ Swap position 3 with 3

The full process for solving a system of simultaneous linear equations is to call mppivot( ) or mcpivot( ) to order the equation matrix, followed by mlu( ) to perform LU decomposition fol-lowed by mfbsub( ) to perform forward-back substitution

The reason this was broken up in this way is that pivoting and LU Decomposition need only be performed once. The Forward-Back substitution can be called for multiple input solution vec-tors c[ ]. This way a system of linear equations can be solved for many solutions without the expense of pivoting or decomposing every time.

If pivoting was not performed (via mcpivot( ) or mppivot( )), then set the order vector d[ ] = 0, 1, 2, 3, ... N. This ensures that no swapping will be performed.

mfbsub ( a, b, c, d, x )

6 2 4 8+ + +2 1 2 3+ + +0 1 3– 4–+1 1– 0 4+ +

wxyz

L⇒=

1 0 0 00.333 1 0 0

0 3 1 00.166 4– 0.4– 1

U

6 2 4 80 0.333 0.666 0.3330 0 5.00– 5.00–0 0 0 2

=,=

Back Substitution

6 2 4 80 0.333 0.666 0.3330 0 5– 5–0 0 0 2

=

wxyz

•

4.0001.6664.000–

6.400

w 4.2–=x 6.6=y 2.4–=z 3.2=

⇒=

Forward Substitution

1 0 0 00.333 1 0 0

0 3 1 00.166 4– 0.4– 1

y1y2y3y4

•

4312

y1 4=y2 1.666=y3 4–=y4 6.4=

⇒==



mident ( a, x )

NAME Martix Identity

DESCRIPTION This function creates an output identity matrix a [ ] [ ] whose diagonal elements are all 1s and all the other elements are 0s. The matrix created is square, with the size of the rows and columns being supplied by input scalar x.

ALGORITHM for i=0 to number of rows

for j=0 to number of columns

if (i=j)

a[i][j]=1

else

a[i][j]=0

SYNOPSIS void mident ( a, x )



DOMAIN 0 to 362


EXECUTION TIME 11 + ( 2 + X ) * X cycles

NOTES The file tmident.c included in the distribution diskette provides an example of this function’s use.

An example of an identity matrix is as follows:


If x=4 then a[ ][ ]

1 0 0 00 1 0 00 0 1 00 0 0 1

=



minm ( a, x, y )

NAME Minimum Value of a Matrix

DESCRIPTION This function finds the element in input matrix a[ ] [ ] with the minimum value and returns it from the function . The number of rows contained within input matrix a[ ] [ ] is defined by input integer x while the number of columns is defined by input scalar y.

ALGORITHM

SYNOPSIS float minm ( a, x, y )

float *a ; /* Pointer to input/output matrix a[ ][ ] */

int x ; /* Number of rows in input/output vector a[ ][ ] */

int y ; /* Number of columns in input/output vector a[ ][ ]*/

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tminm.c included in the distribution diskette provides an example of this function’s use.


return minimum element in matrix A⇐



minmc ( a, x, y, mx, my )

NAME Minimum Element of Matrix With Coordinates

DESCRIPTION This function finds the element in input matrix a[ ] [ ] with the minimum value along with its coordinates. The minimum value is returned from the function call, the mini-mum x-coordinate is output in scalar mx, and the minimum y-coordinate in output sca-lar my. The number of rows contained within input matrix a[ ] [ ] is defined by input scalar x while the number of columns is defined by input scalar y.

ALGORITHM

SYNOPSIS float minmc ( a, x, y, mx, my )




int mx ; /* X-coordinate of the minimum value */

int my ; /* Y-coordinate of the minimum value */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tminmc.c included in the distribution diskette provides an example of this function’s use.


The coordinates reflects the last smallest element. In other words, if the two smallest elements are equal, the coordinates are those of the second element.

The coordinate system begins with zero. If there are 7 elements in a column and 8 ele-ments in a row, then the column range is 0 to 6 and the row range is 0 to 7.

return minimum element of matrix A[ ][ ] with coordinates⇐



minmgv ( a, i, c, n )

NAME Minimum Magnitude of Vector

DESCRIPTION This function finds the element in input vector a with the minimum magnitude and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void minmgv ( a, i, c, n )





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tminmgv.c included in the distribution diskette provides an example of this function’s use.

if C Ami then m 0 1 2 …n 1–, , ,{ }=>

C Ami=



minmgvi ( a, i, c, d, f, n )

NAME Vector Minimum Magnitude with Index

DESCRIPTION This function finds the element in input vector a with the minimum magnitude and stores the result in output scalar c. The index of the minimum value of a is returned in output scalar d. A flag f controls the return of either the first or last minimum index encountered.

ALGORITHM

D= Index of Minimum Magnitude Value of A

If f = 0 then the First Minimum Magnitude Returned

If f > 0 then the Last Minimum Magnitude Returned

SYNOPSIS void minmgvi ( a, i, c, d, f, n )



float dm *c ; /* Pointer to output minimum value scalar c */

int dm d ; /* Pointer to index of minimum value d */

int f ; /* Flag to indicate first or last mag returned */


DOMAIN -3.4 x 1038 to +3.4 x 1038


if C Ami then m 0 1 2 …n 1–, , ,{ }=>

C Ami=




NOTES The file tminmgvi.c included in the distribution diskette provides an example of this function’s use.

The flag f indicates wether the first or last magnitude is returned if multiple minimum values are encountered. If f is set to 0 then the index of the first minimum magnitude encountered in the input array is returned in d. If f is set to > 0 then the index of the last minimum magnitude encountered in the input array is returned in d.

The search for the first or last magnitude is accomplished by searching backwards and forwards respectively.

The index is referenced to zero at the beginning of the array.

minmgvi ( a, i, c, d, f, n )



minsrt ( a, ax, ay, b, bx, by, ix, iy )

NAME Insert Sub-Matrix Into Source Matrix

DESCRIPTION This function inserts an input sub-matrix a[ ] [ ] into an input/output source matrix b[ ] [ ] at a point described by input row integer ix and input column integer iy. The results are stored to output matrix b[ ] [ ].

ALGORITHM

SYNOPSIS void minsrt ( a, ax, ay, b, bx, by, ix, iy )

float dm *a ; /* Pointer to input sub-matrix a[ ][ ] */

int ax ; /* Number of rows in sub-matrix a[ ][ ] */

int ay ; /* Number of columns in sub-matrix a[ ][ ] */

float dm *b ; /* Pointer to input source matrix b[ ][ ] */

int bx ; /* Number of rows in source matrix b[ ][ ] */

int by ; /* Number of columns in source matrix b[ ][ ] */

int ix ; /* Insert position for row start */

int iy ; /* Insert position for column start */

DOMAIN -3.4 x 1038 to +3.4 x 1038


A 1 47 9

ax 2 , ay=2, =,=

Bstart

0 0 00 0 00 0 0

bx 3 , by=3, =,=

ix 1 , iy=1, =

Bfinish

0 0 00 1 40 7 9

=




NOTES The file tminsrt.c included in the distribution diskette provides an example of this function’s use.

This function inserts submatrix a[ ] [ ] into source matrix b[ ] [ ]. Matrix a[ ] [ ] row and column must be less than the row and column of matrix b[ ] [ ]. Example as fol-lows:

The coordinate system begins with zero. If there are 7 elements in a column and 8 ele-ments in a row, then the column is 0 to 6 and the row is 0 to 7.

The insert position must be a valid position (i.e., non-negative). However, it is allowed to have an insert position such that only a part of the sub-matrix is inserted (i.e. if ix > bx - ax).

Example as follows:


minsrt ( a, ax, ay, b, bx, by, ix, iy )

A1 2 34 5 67 8 9

B,

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

Insert Position 3 3,( ) Bfinal

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 1 2 3 0 00 0 0 4 5 6 0 00 0 0 7 8 9 0 00 0 0 0 0 0 0 0

=,=,= =

A1 2 34 5 67 8 9

B,

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

Insert Position 3 6,( ) Bfinal

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 1 20 0 0 0 0 0 4 50 0 0 0 0 0 7 80 0 0 0 0 0 0 0

=,=,= =



minv ( a, i, c, n )

NAME Vector Minimum Value

DESCRIPTION This function finds the element in input vector a with the minimum value and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void minv ( a, i, c, n )





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tminv.c included in the distribution diskette provides an example of this func-tion’s use.

if C Ami then m 0 1 2 …n 1–, , ,{ }=>

C Ami=



minvi ( a, i, c, d, n )

NAME Vector Minimum Value With Index

DESCRIPTION This function finds the element in input vector a with the minimum value and stores the result in output scalar c. It also locates the index of the minimum value and stores it in output scalar d.

ALGORITHM

SYNOPSIS void minvi ( a, i, c, d, n )




int *d ; /* Pointer to output maximum index scalar d */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tminvi.c included in the distribution diskette provides an example of this function’s use.

This routine is slower than the minv routine as the index of the smallest element is returned along with the maximum value. If input array a has two values that are com-puted to be the minimum, and both are equal, then the routine returns the index of the last smallest element encountered during the search, as the minimum element.

if C Ami then m 0 1 2 …n 1–, , ,{ }=>

C Ami=

D index of minimum value=



minvrt ( a, b, x )

NAME Matrix Invert

DESCRIPTION This function inverts input matrix a[ ] [ ] and places the results in output matrix b[ ] [ ]. The dimensions of input matrix a[ ] [ ] and output matrix b[ ] [ ] are x.

This algorithm uses the Gaussian elimination with partial pivoting technique. Gauss-ian elimination is performed using LU decomposition.

ALGORITHM

SYNOPSIS void minvrt ( a, b, x )


float dm *b ; /* Pointer to input matrix b[ ][ ] */

int x ; /* Number of rows and columns in matricies */

a[ ][ ] and b[ ][ ]

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME See Execution Timing Listed at End of Discussion Section

NOTES The file tminvrt.c included in the distribution diskette provides an example of this function’s use.


This algorithm is broken into four section. The four sections are as follows:

1. Partial Pivoting - finding the largest elements along the diagonal by row.

2. LU Decompostion - partitioning the source matrix into its L and U compo-nents.

3. Forward Substitution - solving for temporary unknowns.

4. Back Substitution - solving for the inverse.

If A 5.0 8.0 1.00.0 2.0 1.04.0 3.0 1.0–

then A1–

5.0 11.0– 6.0–4.0– 9.0 5.0

8.0 17.0– 10.0–

==

where B[ ] A1–[ ]=



minvrt() computes the inverse of matrix a[][] and places the result in b[][]. The row and column dimensions of matri-ces a[][] and b[][] are x.

This algorithm uses the Gaussian Elimination with partial pivoting technique. Gaussian Elimination is performed using LU decomposition.

The inverse is performed as follows:-1b[n][n] = a[n][n] for n = 0 to x

An example of a matrix inversion is provided below.

float dm a[][] = 5.0 8.0 1.0 0.0 2.0 1.0

4.0 3.0 -1.0 int x = 3 ;int errror ;

error = minvrt ( a, b, x ) ;

The resulting values in matrix b[] is as follows:

b[] = 5.0 -11.0 -6.0 -4.0 9.0 5.0 8.0 -17.0 -10.0

If matrix a[][] is singular (ie. there is no inverse) then minvrt() returns a non-zero value. Otherwise minvrt() returns zero with the inverse matrix in b[][].

NOTES on Matrix Invert Routine for ADSP-210xx SHARC Family of Processors

The storage methodology for matrices is by rows. Matrices can be thought of as one long array (vector) where the begin-ning of each row is offset by the number of columns.

The algorithm is broken into four sections. These are:

1. Partial Pivoting - finding the largest elements along the diagonal by row.2. LU Decomposition - partitioning the source matrix into its L and U components3. Forward Substitution - solving for temporary unknowns4. Back Substitution - solving for the inverse

Partial Pivoting - This entails finding the largest element in the matrix or sub-matrix along the diagonal. The first pivot is found by searching the entire matrix for the largest element in the first column of each row (ignoring sign). Swapping the row with the largest element with the first row puts the element in the upper left corner. Now exclude the first row, the sub-matrix is searched for the next largest element in the second column. Perform swapping again positions the element in the next pivot (the next position along the diagonal). Repeat for the remaining elements along the diagonal.



Given:

a[][] = 5 8 1 0 2 1 4 3 -1

The element with the largest magnitude in col 0 is 5.

a[][] = 5 8 1 --> Swapped 1st row and 1st row 0 2 1 4 3 -1

The element with the largest magnitude in the minor sub-matrix (excluding row 0 and col 0) is 4.a[][] = 5 8 1 4 3 -1 --> Swapped 2nd row and 3rd row 0 2 1

LU Decomposition - This finds the L and U component of the source matrix after complete pivoting is done. The L component represents the lower triangular form and the U component the upper triangular form. The lower triangular form contains elements below the diagonal and the upper triangular form contains elements along the diagonal and above. The elements in L are the scalars (factors) used to reduce (by column) the non-pivot rows to zero. The elements in U are the results after row reduction has been performed using the scalars in L.

Since LU decomposition uses a pivot element (an element along the diagonal) as a divisor to compute a scalar, a check for a zero pivot is performed prior decomposition. If a zero pivot is found, then the function is aborted and a status flag of non-zero is returned.

An example of LU Decomposition is provided below: diagonal and above. The elements in L are the scalars (factors)used to reduce (by column) the non-pivot rows to zero. The elements in U are the results after row reduction has been per-formed using the scalars in L.

Given a[][] = 5 8 1 4 3 -1 --> Reduce these rows so that 0 2 1 --> col 0 is 0

Decompose a[][] = l[][] * u[][]

Scalar of -4/5 (-0.8) times the first row plus the second row.Scalar of -0/5 (0) times the first row plus the third row.

u[][] = 5.000 8.000 1.000 -0.800 -3.400 -1.800 --> Reduce these rows so that 0.000 2.000 1.000 --> col 1 is 0

1[][] = 1.000 0.000 0.000 -0.800 1.000 0.000 0.000 0.000 1.000

Scalar of 2/3.4 (0.58823529) times the second row plus the third row.



u[][] = 5.000000000 8.000000000 1.000000000 0.000000000 -3.400000000 -1.800000000 0.000000000 0.000000000 -0.058823529

1[][] = 1.000000000 0.000000000 0.000000000 -0.800000000 1.000000000 0.000000000 0.000000000 0.588235294 1.000000000

Normally LU decomposition would consist of two matrices (an L matrix and a U matrix). The L matrix has the form1 0 0 a 1 0 for a 3x3 matrix b c 1 where a, b, c are the scalars used in row operations to reduce the matrix to upper triangu-lar form.

The U matrix has the form

d e f0 g h for a 3x3 matrix0 0 iwhere d, e, f, g, h, i are the elements after row operations have reduce the matrix to upper triangular form.

See Richard Bronson Linear Algebra An Introduction., Academic Press, Inc. 1995 pp. 65-66.

For efficiency, the LU decomposition is placed together in a[][] in partioned form. That is, the original matrix a[][] is replaced with

d e f

__a| g h|__b c| i

So for the above example, the LU decomposition would look like the following in matrix a[][].

a[][] = 5.000000000 8.000000000 1.000000000 -0.800000000 -3.400000000 -1.800000000 0.000000000 0.588235294 -0.058823529

Forward Substitution - This solves for the solution used in back substitution. Forward substitution uses the L component found above and a corresponding column in the identity matrix to solve for a set of temporary unknowns. The process is completed for all remaining columns. These unknowns are then used by backsubstitution.

Back Substitution - This solves for the inverse solution in the identity matrix. Back substitution uses the U component found above and a corresponding column in the identity matrix containing the solution from forward substitution to solve for theinverse. The process is completed for all remaining columns.

Forward and back substitution uses the matrix from the LU decomposition to solve a set of simultaneous linear equations for a set of unknowns.



A system of simultaneous linear equations can be written as:

Ax = b

where: A - represents the coefficients of the equations in matrix form x - represents a vector of unknowns b - represents a vector of the solutions

Decomposing A = LU the above equation can be rewritten as: LUx = b Setting y = Ux, the equation becomes Ly = b

Use forward substitution to solve for y. Once y is found, use back substitution to solve for x.

Using the property that A*A-1 = I (that is matrix a[][] times its inverse equals the identity matrix), the LU decomposition with forward-back substitution can be used to solve for the inverse.

In this case the vector b represents a column in the identity matrix. Looping for each column solves for the inverse of matrix a[][].

Example: Given the set of simultaneous linear equations forthree unknowns (x, y, z).

5x + 8y + 1z = 14x + 3y - 1z = 00x + 2y + 1z = 0

This system has the matrix form:

| 5 8 1 | | x | | 1 || 4 3 -1 | * | y | = | 0 || 0 2 1 | | z | | 0 |

LU decomposition of the coefficient matrix yields

u[][] = 5.000000000 8.000000000 1.000000000 0.000000000 -3.400000000 -1.800000000 0.000000000 0.000000000 -0.058823529

l[][] = 1.000000000 0.000000000 0.000000000 -0.800000000 1.000000000 0.000000000

0.000000000 0.588235294 1.000000000

Using forward substitution, solve for a temporary set of unknowns (Ly = b)

| 1.000000000 0.000000000 0.000000000 | | y1 | | 1 || -0.800000000 1.000000000 0.000000000 | * | y2 | = | 0 || 0.000000000 0.588235294 1.000000000 | | y3 | | 0 |

y1 = 10.80000y1 + y2 = 00.588235294y2 + y3 = 0



y1 = 1y2 = -0.800000000y3 = 0.470588232

Using back substitution, solve for the set of unknowns (Ux = y)

| 5.000000000 8.000000000 1.000000000 | | x | | 1 || 0.000000000 -3.400000000 -1.800000000 | * | y | + | -0.80000000 || 0.000000000 0.000000000 -0.058823529 | | z | | 0.47058823 |

5.000000000x + 8.000y + 1.000000000z = 1- 3.400y - 1.800000000z = -0.80000000- 0.058823529z = 0.47058823

x = 5y = -4z = 8

Cycle Time = n =

3 - 481 27 - 94,6214 - 804 28 - 104,8565 - 1,264 29 - 115,8046 - 1,885 30 - 127,4897 - 2,691 31 - 139,9358 - 3,706 32 - 153,1669 - 4,954 33 - 167,20610 - 6,459 34 - 182,07911 - 8,245 35 - 197,80912 - 10,336 36 - 214,42013 - 12,756 37 - 231,93614 - 15,529 38 - 250,38115 - 18679 39 - 269,77916 - 22,230 40 - 290,15417 - 26,206 41 - 311,53018 - 30,631 42 - 333,93119 - 35,529 43 - 357,38120 - 40,924 44 - 381,90421 - 46,840 45 - 407,52422 - 53,301 46 - 434,26523 - 60,331 47 - 462,15124 - 67,954 48 - 491,20625 - 76,194 49 - 521,45426 - 85,075 50 - 552,919



mkron ( a, x, y, b, m, n, c )

NAME Computes the Kronecker Product of Two Matrices

DESCRIPTION This function computes the Kronecker Product of two input matrices a[ ] [ ] and b[ ] [ ], placing the results in output matrix c[ ] [ ]. The dimensions of input matrix a[ ] [ ] are defined by scalar inputs x and y, while the dimensions of input matrix b[ ] [ ] are defined by scalar inputs m and n. The resulting output matrix c[ ] [ ] is defined of dimensions x*m and y*n.

ALGORITHM

SYNOPSIS void mkron ( a, x, y, b, m, n, c )





int m ; /* Number of rows in matrix b[ ][ ] */

int n ; /* Number of columns in matrix b[ ][ ] */


DOMAIN -3.4 x 1038 to +3.4 x 1038


Kronecker A [ ] [ ] x B [ ] [ ]( ) C [ ] [ ]=



EXECUTION TIME 48 + ( 4 + ( 4 + ( 4 + 3 * N ) * Y ) * M ) * X cycles

NOTES The file tmkron.c included in the distribution diskette provides an example of this function’s use.

The Kronecker Product is computed as follows:

for i=0 to number of rows in a[ ][ ]

for j=0 to number of columns in a[ ][ ]

for k=0 to number of rows in b[ ][ ]

for l=0 to number of columns in b[ ][ ]

c[i+k] [j+1] = a[i][j] * b[k][i]

An example of a Kronecker Product is provided as follows:


mkron ( a, x, y, b, m, n, c )

A

1 1 1 1 21 1 1 1 21 1 1 1 21 1 9 10 20

B, 1 23 4

x, 4 y, 5 m, 2 n, 2= = = = = =

C

1 2 1 2 1 2 1 2 2 43 4 3 4 3 4 3 4 6 81 2 1 2 1 2 1 2 2 43 4 3 4 3 4 3 4 6 81 2 1 2 1 2 1 2 2 43 4 3 4 3 4 3 4 6 81 2 1 2 9 18 10 20 20 403 4 3 4 27 36 30 40 60 80

=



mlu ( a, b, c, x )

NAME Matrix LU Decomposition

DESCRIPTION This function computes the LU decomposition of input matrix a[ ] [ ]. The results are output to matrix b[ ] [ ] and matrix c[ ] [ ]. Output matrix b[ ] [ ] contains the lower tri-angular form (L) of the decomposition and output matrix c[ ] [ ] contains the upper tri-angular form (U) of the decomposition.

LU decomposition factorizes the source matrix a[ ] [ ] into L and U matrices such that a=LU. The lower triangular form (L) contains elements below a diagonal of ones and the upper triangular form (U) contains elements along the diagonal and above. The elements in L are the scalars (factors) used to reduce (by column) the non-pivot rows to zero. The elements in U are the results after row reduction has been performed using scalars in L.

LU decomposition is used primarily to slove for a system of simultaneous linear equa-tions. After decomposition, the L and U matrices are used by mfbsub( ) (forward and back substitution) to solve for the unknowns.

ALGORITHM

If A=

6 2 4 82 1 2 30 1 3– 4–1 1– 0 4

then

B=

6 2 4 80 0.333 0.6662 0.3330 0 5– 5–0 0 0 2

C=

1 0 0 00.333 1 0 0

0 3 1 00.166 4– 0.– 400 1

,



SYNOPSIS int mlu ( a, b, c, x )


float dm *b ; /* Pointer to output lower matrix b [ ] [ ] */

float dm *c ; /* Pointer to output upper matrix c [ ] [ ] */

int x ; /* Number of rows and columns in

matrices a [ ], b [ ] and c [ ] [ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME TBS cycles

NOTES The file tmlu.c included in the distribution diskette provides an example of this func-tion’s use.

LU decomposition does not perform pivoting. There are two pivoting functions avail-able (mcpivot( ) for complete pivoting and mppivot( ) for partial pivoting). It is rec-ommended that the library functions mcpivot( ) or mppivot ( ) be used prior to mlu( ) being called to reduce roundoff error.

Since mlu( ) uses a pivot element (an element along the diagonal) as a divisor to com-pute a scalar, a check for a zero is performed during decomposition. If a zero pivot is found, then the function is aborted and a status flag of 0 is returned. Otherwise, a value of 1 is returned to indicate a successful LU decomposition.

Matricies b[ ] [ ] and c[ ] [ ] are distinct and are initialzed prior to LU decomposition. Output matrix b[ ] [ ] is filled with input matrix a[ ] [ ] values and output matrix c[ ] [ ] is zeroed out.


mlu ( a, b, c, x )



mmax ( a, b, x, y, c )

NAME Maximum Value of Pair of Matrices

DESCRIPTION This function finds the element in input matrix a[ ] [ ] or input matrix b[ ] [ ] with the maximum value and returns it in output matrix c[ ] [ ] . The number of rows contained within input matrices a[ ] [ ] and b[ ] [ ] are defined by input scalar x while the number of columns is defined by input scalar y.

ALGORITHM

SYNOPSIS void mmax ( a, b, x, y, c )






DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 20 + 3 * ( X * Y - 1 ) cycles

NOTES The file tmmax.c included in the distribution diskette provides an example of this function’s use.


C11 C12

C21 C22

maxA11 A12

A21 A22

B11 B12

B21 B22

,

=



mmin ( a, b, x, y, c )

NAME Minimum Value of Pair of Matrices

DESCRIPTION This function finds the element in input matrix a[ ] [ ] or input matrix b[ ] [ ] with the minimum value and returns it in output matrix c[ ] [ ] . The number of rows contained within input matrices a[ ] [ ] and b[ ] [ ] are defined by input scalar x while the number of columns is defined by input scalar y.

ALGORITHM

SYNOPSIS void mmin ( a, b, x, y, c )






DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 20 + 3 * ( X * Y - 1 ) cycles

NOTES The file tmmin.c included in the distribution diskette provides an example of this function’s use.


C11 C12

C21 C22

minA11 A12

A21 A22

B11 B12

B21 B22

,

=



mmov ( a, x, y, b )

NAME Move a Source Matrix to a Destination Matrix

DESCRIPTION This function copies the contents of input matrix a[ ] [ ] to output matrix b[ ] [ ]. The number of rows in input matrices a[ ] [ ] and b[ ] [ ] is represented by scalar input x. The number of columns in input matrices a[ ] [ ] and b[ ] [ ] is represented by scalar input y.

ALGORITHM

SYNOPSIS void mmov ( a, x, y, b )


int x ; /* Number of rows in input matrices a[ ][ ] & b */

int y ; /* Number of columns in input matrix a[ ][ ]& b */

float dm *b ; /* Pointer to output matrix b[ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmmov.c included in the distribution diskette provides an example of this function’s use.

The matrices are assumbed to be identical.

Bx y, Ax y,=



mmov_dp ( a, x, y, b )

NAME Move a Source DM Matrix to a Destination PM Matrix

DESCRIPTION This function copies the contents of input matrix a[ ] [ ] to output matrix b[ ] [ ]. The number of rows in input matrices a[ ] [ ] and b[ ] [ ] is represented by scalar input x. The number of columns in input matrices a[ ] [ ] and b[ ] [ ] is represented by scalar input y. Input matrix a[ ] [ ] is in data memory while output matrix b[ ] [ ] is in pro-gram memory.

ALGORITHM

SYNOPSIS void mmov_dp ( a, x, y, b )




float pm *b ; /* Pointer to output matrix b[ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmmovpd.c included in the distribution diskette provides an example of this function’s use.

The matrices are assumed to be identical.

Bx y, Ax y,=



mmov_pd ( a, x, y, b )

NAME Move a Source PM Matrix to a Destination DM Matrix

DESCRIPTION This function copies the contents of input matrix a[ ] [ ] to output matrix b[ ] [ ]. The number of rows in input matrices a[ ] [ ] and b[ ] [ ] is represented by scalar input x. The number of columns in input matrices a[ ] [ ] and b[ ] [ ] is represented by scalar input y. Input matrix a[ ] [ ] is in program memory while output matrix b[ ] [ ] is in data memory.

ALGORITHM

SYNOPSIS void mmov_pd ( a, x, y, b )

float pm *a ; /* Pointer to input matrix a[ ][ ] */



float dm *b ; /* Pointer to output matrix b[ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmmovpd.c included in the distribution diskette provides an example of this function’s use.

The matrices are assumed to be identical.

Bx y, Ax y,=



mmul ( a, x, y, b, z, c )

NAME Matrix Multiply

DESCRIPTION This function computes the multiplication of input matrix a times input matrix b and stores the results to output matrix c. The dimensions of matrix a [ ] [ ] are x and y and the dimensions of input matrix b [ ] [ ] are y and z. The resulting ouput matrix c [ ] [ ] is of dimensions x and z.

ALGORITHM

SYNOPSIS void mmul ( a, x, y, b, z, c )







DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 48 + (5+(9+2*Y)*Z)*X cycles

NOTES The file tmmul.c included in the distribution diskette provides an example of this function’s use.


The first row of a [ ] [ ] times the first column of b [ ] [ ] is the first element pf c [ ] [ ] (row 1, column 1). The first row of a [ ] [ ] times the second row of b [ ] [ ] is the sec-ond element of c [ ] [ ] (row 1, column 2 ) ... etc.


C11 C12

C21 C22

A11 A12 A13

A21 A22 A23

B11 B12

B21 B22

B31 B32

•=



mmul_dpd ( a, x, y, b, z, c )



ALGORITHM

SYNOPSIS void mmul_dpd ( a, x, y, b, z, c )




float pm *b ; /* Pointer to input matrix b [ ] [ ] */



DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 34+(4+(6+Y)*Z)*X cycles

NOTES The file tmmuldpd.c included in the distribution diskette provides an example of this function’s use.




C11 C12

C21 C22

A11 A12 A13

A21 A22 A23

B11 B12

B21 B22

B31 B32

•=



mmul_dpp ( a, x, y, b, z, c )



ALGORITHM

SYNOPSIS void mmul_dpp ( a, x, y, b, z, c )




float pm *b ; /* Pointer to input matrix b [ ] [ ] */


float pm *c ; /* Pointer to output matrix c [ ] [ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmmuldpp.c included in the distribution diskette provides an example of this function’s use.




C11 C12

C21 C22

A11 A12 A13

A21 A22 A23

B11 B12

B21 B22

B31 B32

•=



mmul_pdd ( a, x, y, b, z, c )



ALGORITHM

SYNOPSIS void mmul_pdd ( a, x, y, b, z, c )

float pm *a ; /* Pointer to input matrix a [ ] [ ] */






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmmulpdd.c included in the distribution diskette provides an example of this function’s use.




C11 C12

C21 C22

A11 A12 A13

A21 A22 A23

B11 B12

B21 B22

B31 B32

•=



mmul_pdp ( a, x, y, b, z, c )



ALGORITHM

SYNOPSIS void mmul_pdp ( a, x, y, b, z, c )

float pm *a ; /* Pointer to input matrix a [ ] [ ] */





float pm *c ; /* Pointer to output matrix c [ ] [ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmmulpdp.c included in the distribution diskette provides an example of this function’s use.




C11 C12

C21 C22

A11 A12 A13

A21 A22 A23

B11 B12

B21 B22

B31 B32

•=



mnot ( a, x, y, c )

NAME Complement Matrix

DESCRIPTION This function mnot( ) logically complements the elements of matrix a [ ] [ ], placing the complemented elements in ouput matrix c [ ] [ ]. Matices a [ ] [ ] and c [ ] [ ] are of the same dimension. Matrix c [ ] [ ] is a 0-1 matrix.

ALGORITHM

SYNOPSIS void mnot ( a, x, y, c )


int x ; /* Number of rows in vector a[ ][ ] */

int y ; /* Number of columns in vector a[ ][ ] */


DOMAIN -3.4 x 1038 to +3.4 x 1038


ComplementC11 C12

C21 C22

A11 A12

A21 A22

=




NOTES The file tmnot.c included in the distribution diskette provides an example of this func-tion’s use.


An example of a matrix compliment is provided below.

a[ ][ ] = ;

x = 3, y = 3 ;

mnot ( a, x, y, c ) ;


c[ ][ ] =

mnot ( a, x, y, c )

1.0 1.7 5.55.0 1.6 1.00.0 5.0 6.0

1 1 11 1 10 1 1



mnull ( a, x, y )

NAME Create Null Matrix

DESCRIPTION This function creates an output null matrix a [ ] [ ] whose elements are all 0s. The number of rows is defined by input scalar x, while the number of columns is defined by input scalar y.

ALGORITHM

SYNOPSIS void mnull ( a, x, y )


int x ; /* Number of rows in output matrix a[ ][ ] */

int y ; /* Number of columns in output matrix a[ ][ ] */

DOMAIN 0 to 131,000



NOTES The file tmnull.c included in the distribution diskette provides an example of this function’s use.

An example of a null matrix is as follows:


Axy 0=

If x = 4 and y = 5 then a[][]

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

=

x 0 y 0>,>



modf_wci ( x, *n )

NAME Get Fraction and Integer

DESCRIPTION This function computes the integer and fractional parts of a floating-point number, x. The fractional value is returned from this function. The integer part is stored in the object pointed to by n.

ALGORITHM

SYNOPSIS float modf_wci ( float x )

DOMAIN -2.14E9 to 2.14E9



NOTES The file tmodf.c included in the distribution tape provides an example of this func-tion's use.

return x fix x–( ) sign x( )•=



mor ( a, b, x, y, c )

NAME OR Two Matrices

DESCRIPTION This function mor( ) logically ORs elements of matrices a [ ] [ ] and b [ ] [ ], placing the results in ouput matrix c [ ] [ ]. Matrices a [ ] [ ] and b [ ] [ ] are of the same demension. Matrix c [ ] [ ] is a 0-1 matrix.

ALGORITHM

SYNOPSIS void mor ( a, b, x, y, c )



int x ; /* Number of rows in vector a[ ][ ] & b [ ] [ ] */

int y ; /* Number of columns in vector a[ ][ ] & b [ ] [ ]*/


DOMAIN -3.4 x 1038 to +3.4 x 1038


C11 C12

C21 C22

A11 A12

A21 A22

ORB11 B12

B21 B22

=




NOTES The file tmor.c included in the distribution diskette provides an example of this func-tion’s use.


An example of a matrix OR is provided below.

a[ ][ ] = ;

b[ ][ ] =

x = 3, y = 3 ;

mor ( a, b, x, y, c ) ;


c[ ][ ] =

mor ( a, b, x, y, c )

0.0 1.7 5.55.0 1.6 1.01.0 0.0 6.0

0.0 1.7 5.55.0 1.6 0.01.0 0.0 6.0

0 1 11 1 11 0 1



morth ( a, x )

NAME Orthogonal Matrix Check

DESCRIPTION This function transposes input matrix a[ ] [ ] and determines if the matrix is orthogo-nal. If it is orthogonal, a value of 1 is returned. Otherwise a value of 0 is returned.

An orthgonal matrix is one whose transposed matrix times the original matrix equals the identity matrix. The matrix must be square.

ALGORITHM

SYNOPSIS void morth ( a, x )



DOMAIN -3.4 x 1038 to +3.4 x 1038


If A

0.5 0.5 0.5 0.50.5– 0.5– 0.5 0.50.5– 0.5 0.5– 0.50.5– 0.5 0.5 0.5–

then=

0.5 0.5 0.5 0.50.5– 0.5– 0.5 0.50.5– 0.5 0.5– 0.50.5– 0.5 0.5 0.5–

0.5 0.5– 0.5– 0.5–0.5 0.5– 0.5 0.50.5 0.5 0.5– 0.50.5 0.5 0.5 0.5–

•

1 0 0 00 1 0 00 0 1 00 0 0 1

=

Morth returns 0



EXECUTION TIME 30 + ( 8 + ( 12 + 2 * X ) * X ) * X cycles

NOTES The file tmorth.c included in the distribution diskette provides an example of this function’s use.

Transposing a matrix is computed as follows:


morth ( a, x )

B n[ ] m[ ] A m[ ] n[ ] for m=0 to x-1 and n=0 to x-1=



movavg ( a, i, w, c, k, n )

NAME Moving Average

DESCRIPTION This function computes a moving average on the input vector a, using input scalar w as the width of the sliding window. The sucessive moving average are placed in output vector c.

ALGORITHM

SYNOPSIS void movavg ( a, i, w, c, k, n )



int w ; /* Pointer to the length of sliding window w */




DOMAIN -3.4 x 1038 to +3.4 x 1038


Cmk1

w 1–------------- Ami

0

w

∑• m 0 1 2 …n, 1 }–, ,{==



EXECUTION TIME 52 + 3N-2W

NOTES The file tmovavg.c included in the distribution tape provides an example of this func-tion's use.

movavg() computes the mean of a moving window using vector [a] as an input. The result is stored in output vector [c]. Elements in a vector [a] are successively summed. The number of elements summed is defined by the window length. This sum is then divided by the window length.

Example:

[a] = { 2, 4, 0, 1, 3, 5, 3, 1, 4, 7, 1, };

Window size w = 9

c[0] = ( 2 + 4 + 0 + 1 + 3 + 5 + 3 + 1 + 4 ) / 9 = 2.55555

c[1] = ( 4 + 0 + 1 + 3 + 5 + 3 + 1 + 4 + 7 ) / 9 = 3.11111

c[2] = ( 0 + 1 + 3 + 5 + 3 + 1 + 4 + 7 + 1 ) / 9 = 2.77777

Given a population n and a window size w, there are answers stored to output vector [c].

The length of the window cannot exceed the population size divided by the striding. (i.e., w < = n / i).

The number of moving averages computed is determined by the total number of ele-ments minus the window size.

movavg ( a, i, w, c, k, n )

n w–( ) 1+



mppivot ( a, b, x )

NAME Partial Pivoting For a Matrix

DESCRIPTION This function performs partial pivoting of input matrix a[ ] [ ]. The results are returned in output matrix a[ ] [ ]. Vector b [ ] is used to return the row interchanges. The row and column dimensions of matrix a[ ] [ ] is x. The vector dimension of b[ ] is also x.

ALGORITHM Parital pivoting attempts to arrange matrix a[ ][ ] in an optimal order by swapping rows. This optimal order aranges elements of a[ ] [ ] so that round off erros are reduced when using the mlu( ) and mfbsub( ) functions. When swappping rows, the largest elements (magnitude) are placed along the diagonal.

SYNOPSIS void mppivot ( a, b, x )

float dm *a ; /* Pointer to input/output matrix a[ ][ ] */

int dm *b ; /* Pointer to output vector b[ ] */

int x ; /* Number of rows and columns in matrix a[ ][ ]*/

DOMAIN -3.4 x 1038 to +3.4 x 1038




EXECUTION TIME 62 + X + (24 + 11*X ) * X cycles

NOTES The file tmppivot.c included in the distribution diskette provides an example of this function’s use.

Partial pivoting attempts to find the largest elements for the diagonal among the rows. First, the elements in the first column are searched for the largest magnitude. Its’ row is swapped with the 0th row. The 0th row is excluded and the minor sub-matrix is searched for the largest magnitude in the 1st column. Its row is swapped with the 1st row. This process is performed until the last row is left.

Given:

The element with the largest magnitude is -16.

The element with the largest magnitude in the minor sub-matrix (excluding row 0 and column 0) is 6.

The element with the largest magnitude in the minor sub-matrix (excluding rows 0 and columns 0 and 1) is 9.

The storage methodology for matrices is by rows. Matrices can be thought of as one long array (vector) where the beginning of each row is offset by the number of col-umns. Vector b[ ] need not be initialized. It is assigned within the function. For the example above, the vector is interpreted as follows:

b[0] - Row 0 was swapped with row 3b[1] - Row 1 was swapped with row 2b[2] - Row 2 was swapped with row 3

The row vector interchange b[ ] is used by the function mfbsub() in solving for the unknowns in a set of simultaneous linear equations.

mppivot ( a, b, x )

B 0 0 0 0 A

4 2 9 18 3 0 57 6 10 1216– 14 13 15

=,=

B 3 0 0 0 A

16– 14 13 158 3 0 57 6 10 124 2 9 1

=,=

B 3 2 0 0 A

16– 14 13 157 6 10 128 3 0 54 2 9 1

=,=

B 3 2 3 0 A

16– 14 13 157 6 10 124 2 9 18 3 0 5

=,=



mpulse ( delta, i, sigma, j, factor, a, k, n )

NAME Monopulse Function

DESCRIPTION This function performs a monpulse computation on vectors delta and sigma using the scaling factor. The results are stored to output vector a.

ALGORITHM

SYNOPSIS void mpulse ( delta, i, sigma, j, factor, a, k, n )

complex pm *delta ; /* Pointer to input vector delta */

int i ; /* Vector delta element stride */

complex pm *sigma ; /* Pointer to input vector sigma */

int j ; /* Vector sigma element stride */

complex dm factor ; /* Complex input scalar factor */

float dm *a ; /* Pointer to Output Vector a[ ] */

int k ; /* Vector a element stride */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmpulse.c included in the distribution diskette provides an example of this function’s use.

Amk 0.5Re deltami s̃igmamj•[ ]

sigmamj2

factor•----------------------------------------------------------•=

where s̃igmamj Complex Conjugate of sigmamj=

m 0 1 2 …n, 1 }–, ,{=



mpulsep ( delta_r, delta_i, i, sigma_r, sigma_i, j, factor, a, k, n )

NAME Monopulse Function (Program Memory)

DESCRIPTION This function performs a monpulse computation on vectors delta and sigma using the scaling factor. The results are stored to output vector a.

ALGORITHM

SYNOPSIS void mpulsep ( delta_r, delta_i, i, sigma_r, sigma_i, j, factor,

a, k, n )

float pm *delta_r ; /* Pointer to input vector delta real */

float dm *delta_i ; /* Pointer to input vector delta imag */

int i ; /* Vector delta element stride */

float pm *sigma_r ; /* Pointer to input vector sigma real */

float dm *sigma_i ; /* Pointer to input vector sigma imag */

int j ; /* Vector sigma element stride */

complex dm factor ; /* Complex Input Scalar Factor */

float dm *a ; /* Pointer to Output Vector a[ ] */

int k ; /* Vector a element stride */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmpulsep.c included in the distribution diskette provides an example of this function’s use.

Amk 0.5Re deltami s̃igmamj•[ ]

sigmamj2

factor•----------------------------------------------------------•=

where s̃igmamj Complex Conjugate of sigmamj=

m 0 1 2 …n, 1 }–, ,{=



mskew ( a, x )

NAME Matrix Skew-Symmetric Check

DESCRIPTION This function performs a transpose of input matrix a and determines if the matrix is skew-symmetric. The input integer scalar x contains the number of rows and columns in sqaure input matrix a[ ] [ ].

If it is skew-symmetric a value of 1 is returned. Otherwise a value of zero is returned.

ALGORITHM

SYNOPSIS void mskew ( a, x )



/* Number of columns in matrix a[ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


return (1 or 0) Is

A11 A12 A13 …

A21 A22 A23 …

… … … …

Skew-Symmetric?=



EXECUTION TIME 10 + ( 3 + 5 * X ) * X cycles

NOTES The file tmskew.c included in the distribution diskette provides an example of this function’s use.

A skew-symmetric matrix is one whose transposed elements are equal to the negative of its corresponding elements (i.e. ). For the diagonal elements, the only value that satisfies this condition is zero. The matrix must be square.

Transposing is peformed as follows:

An example of a skew-symmetric matrix is as follows:


mskew ( a, x )

Amn A– nm=

Bnm Amn for m=0, x and n=0, x=

Amn

0 1 1– 0 0 0 01– 0 0 0 0 0 0

1 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 10 0 0 0 0 0 1–0 0 0 0 1– 1 0

=



msmul ( a, x, y, b, c )

NAME Matrix Scalar Multiply

DESCRIPTION This function computes the multiplication of input matrix a times input scalar b and stores the results to output matrix c [ ] [ ]. The dimensions of input matrix a[ ] [ ] and output matrix c[ ] [ ] are x and y.

ALGORITHM

SYNOPSIS void msmul ( a, x, y, b, c )




float dm *b ; /* Pointer to input scalar b [ ] [ ] */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmsmul.c included in the distribution diskette provides an example of this function’s use.


C11 C12 C13

C21 C22 C23

A11 A12 A13

A21 A22 A23

B•=



msqe ( a, i, b, j, c, n )

NAME Mean Squared Error

DESCRIPTION This routine calculates the mean squared error of the elements within input vector a from the mean input vector b, and places the results in output scalar c.

ALGORITHM

SYNOPSIS void msqe ( a, i, b, j, c, n)




int j ; /* Element stride for vector b */

float *c ; /* Pointer to resultant output scalar */


DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 53 +3*(N-1)

NOTES The file tmsqe.c included in the distribution tape provides an example of this func-tion’s use.

C 1 n

------- Ami Bmj–[ ]2

i 0=

n 1–

∑• m 0 1 2 … n, 1–, , ,{ }==



msub ( a, b, x, y, c )

NAME Matrix Subtract

DESCRIPTION This function computes the subtraction of input matrix b[ ] [ ] from input matrix a[ ] [ ] and stores the results to output matrix c[ ] [ ] .

ALGORITHM

SYNOPSIS void msub ( a, b, x, y, c )



int x ; /* Number of rows in matrix a[ ][ ] and b[ ][ ] */

int y ; /* Number of columns in matrix a[ ][ ] & b[ ][ ] */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmsub.c included in the distribution diskette provides an example of this func-tion’s use.


This algorithm computes the difference of two matrices of the same order: i.e, the number of rows and columns of input matrix a[ ] [ ] must equal the number of rows and columns of input matrix b[ ] [ ] .

C11 C12 C13

C21 C22 C23

A11 A12 A13

A21 A22 A23

B11 B12 B13

B21 B22 B23

–=



msym ( a, x )

NAME Matrix Symmetric Check

DESCRIPTION This function performs a transpose of input matrix a[ ] [ ] and determines if the matrix is symmetric.

If it is symmetric a value of 1 is returned. Otherwise a value of zero is returned.

The number of rows and columns in input matrix a[ ] [ ] is supplied in input scalar arguments x.

ALGORITHM

SYNOPSIS void msym ( a, x )



/* Number of columns in matrix a[ ][ ] */

DOMAIN -3.4 x 1038 to +3.4 x 1038


return (1 or 0) Is

A11 A12 A13 …

A21 A22 A23 …

… … … …

Symmetrical?=



EXECUTION TIME 18 + ( 3 + 4X)*X cycles

NOTES The file tmsym.c included in the distribution diskette provides an example of this function’s use.

A symmetric matrix is one whose transposed elements are equal to its corresponding elements (i.e. ). The matrix must be square.

Transposing is peformed as follows:

An example of a symmetric matrix is as follows:


msym ( a, x )

Amn Anm=

Bnm Amn for m=0, x and n=0, x=

Amn

1 1 1 0 0 0 01 1 0 0 0 0 01 0 1 0 0 0 00 0 0 0 0 0 00 0 0 0 1 0 10 0 0 0 0 1 10 0 0 0 1 1 1

=



mtrace ( a, x )

NAME Matrix Trace

DESCRIPTION This function computes the trace of input matrix a[ ] [ ] and returns the resulting sum. The input integer scalar x contains the number of rows and columns in input matrix a[ ] [ ].

ALGORITHM

SYNOPSIS float mtrace ( a, x )



DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 8 + X cycles

NOTES The file tmtrace.c included in the distribution diskette provides an example of this function’s use.

The trace is the sum of the diagonal elements of a square matrix


return trace a i[ ] i[ ]

i 0=

x

∑=

If a[][]

1 1.7 5.5 145 1.6 1 511 5 6 11 1.5 9 10

and x 4 then trace = 18.6==



mtrans ( a, b, x, y )

NAME Matrix Transpose

DESCRIPTION This function transposes the contents of input matrix a and places the results in output matrix b. The dimensions of matrix a[ ] [ ] are x and y and the dimensions of output matrix b[ ] [ ] are y and x.

ALGORITHM

SYNOPSIS void mtrans ( a, b, x, y )




Number of columns in matrix b[ ][ ]


Number of rows in matrix b[ ][ ]

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 32 + (3 + 2*Y) * X cycles

NOTES The file tmtrans.c included in the distribution diskette provides an example of this function’s use.


If A

A11 A12

A21 A22

A31 A32

then AT A11 A21 A31

A12 A22 A32

==



mulawc ( a, i, c, k, n )

NAME u-Law Compression

DESCRIPTION This routine performs an u-law compression on the elements in input vector a and out-puts the compressed results to output vector c.

ALGORITHM

SYNOPSIS void mulawc ( a, i, c, k, n )






DOMAIN 0 to 255



NOTES The file tmulawc.c included in the distribution tape provides an example of this func-tion’s use.

The mulawc() routine takes a linear 14-bit signed speech sample and compresses it according to CCITT (now ITU) recommendation G.711. The 8-bit compressed sample is output to vector c.

Cmk ulaw compression of Ami=

m 0 1 2 …n 1–, , ,{ }=



mulawe ( a, i, c, k, n )

NAME u-Law Expansion

DESCRIPTION This routine performs an u-law expansion on the elements in input vector a and out-puts the expanded results to output vector c.

ALGORITHM

SYNOPSIS void mulawe ( a, i, c, k, n )






DOMAIN 0 to 255



NOTES The file tmulawe.c included in the distribution tape provides an example of this func-tion’s use.

The mulawe() routine takes an 8-bit compressed speech sample and expands it according to CCITT (now ITU) recommendation G.711. The 14-bit signed sample is output to vector c.

Cmk ulaw expansion of Ami=

m 0 1 2 …n 1–, , ,{ }=



munity ( a, x, y )

NAME Unity Matrix

DESCRIPTION This function creates a constant output matrix a whose elements are all equal 1. The input integer scalar x contains the number of rows in input matrix a[ ] [ ]. The input integer scalar y contains the number of columns in input matrix a[ ] [ ].

ALGORITHM

SYNOPSIS void munity ( a, x, y )




DOMAIN 0 to 131,000



NOTES The file tmunity.c included in the distribution diskette provides an example of this function’s use.


Axy 1=

x 0 y 0>,>



mve ( a, i, c, n )

NAME Vector Mean

DESCRIPTION This function computes the mean of the elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void mve ( a, i, c, n )





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmve.c included in the distribution diskette provides an example of this func-tion’s use.

C 1n--- Ami

m 0=

n 1–

∑= m 0 1 2 …n, 1 }–, ,{=



mvemg ( a, i, c, n )

NAME Vector Mean of Magnitudes

DESCRIPTION This function computes the mean of the magnitudes of the elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void mvemg ( a, i, c, n )





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmvemg.c included in the distribution diskette provides an example of this function’s use.

C 1n--- Ami

m 0=

n 1–

∑= m 0 1 2 …n, 1 }–, ,{=



mvesq ( a, i, c, n )

NAME Vector Mean of Squares

DESCRIPTION This function computes the mean of the squares of all elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void mvesq ( a, i, c, n )





DOMAIN -1.8 x1019 to +1.8 x1019



NOTES The file tmvesq.c included in the distribution diskette provides an example of this function’s use.

The function can be used to compute Parseval’s theorem. Parseval’s theorem is often used as a way to compute the total signal power of a signal after being transformed by an FFT. This allows signal power to easily be calculated in the frequency domain.

C 1n--- A2

mim 0=

n 1–

∑= m 0 1 2 …n, 1 }–, ,{=



mvessq ( a, i, c, n )

NAME Vector Mean of Signed Squares

DESCRIPTION This function computes the mean of the signed squares of all elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void mvessq ( a, i, c, n )





DOMAIN -1.8 x 1019 to +1.8 x1019



NOTES The file tmvessq.c included in the distribution diskette provides an example of this function’s use.

C 1n--- Ami Ami•

m 0=

n 1–

∑= m 0 1 2 …n, 1 }–, ,{=



mvmul ( a, x, y, b, c )

NAME Matrix-Vector Multiply

DESCRIPTION This function multiplies input matrix a[ ] [ ] by input vector b, and stores the result in output vector c. The dimensions of input matrix a[ ] [ ] are supplied in input integer scalar arguments x and y while the length of input vector b is supplied in input integer scalar y. The resulting output vector c is of length x.

ALGORITHM

SYNOPSIS void mvmul ( a, x, y, b, c )



/* Number of columns in matrix a */

int y ; /* Number of elements in vector b */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tmvmul.c included in the distribution diskette provides an example of this function’s use.


The first row of a[ ] [ ] times vector b is the first element of vector c. The first row of a[ ] [ ] times vector b is the second elements of vector c, etc.

Cm Am k, Bk•

k 0=

y 1–

∑=

m 0 1 2 …n 1–, , ,{ }=



perm ( a, b )

NAME Permutation

DESCRIPTION This function computes the permutation of input scalar a and input scalar b. The result is returned in perm.

ALGORITHM

SYNOPSIS void perm ( a, b )

int dm a ; /* Input value a */

int dm b ; /* Input value b */

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 20 + B*2 cycles

NOTES The file tperm.c included in the distribution diskette provides an example of this func-tion’s use.

The of input scalar a must be positive. If not, a value of -99 is returned.

The value of input scalar b must be positive and less then the value of input scalar a. If not, a value of -99 is returned.

return a!a b–( )!

------------------=



polar ( a, i, c, k, n )

NAME Vector Cartesian To Polar Coordinates Conversion

DESCRIPTION This function converts Cartesian (rectangular) coordinates in real input vector a to polar coordinates and stores the results in real output vector c. All input coordinates are considered to be in radians.

ALGORITHM

SYNOPSIS void polar ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038


Re Cmk{ } Re2

Ami{ } Im2

Ami{ }+=

Im Cmk{ }Im Ami{ }

Re Ami{ }------------------------atan= m 0 1 2 …n, 1 }–, ,{=




NOTES The file tpolar.c included in the distribution diskette provides an example of this func-tion’s use.

Cartesian coordinates are sometimes refered to as rectangular or x,y coordinates. Polar coordinates are sometimes refered to as magnitude, phase or radius, angle coordinates.

Note that the synposis section does not list either the Cartesian coordinates in input vector a or the Polar coordinates in output vector c as type complex. However, for computational purposes you may consider these as complex types, each having a real and imaginary portion consisting of either an x,y or a magnitude, phase component. The algorithm expects the Cartesian and Polar coordinates to be supplied in pairs, analagous to the manner in which we treat complex pairs. They are dyads which are considered to be atomic.

Note that this algorithm is the opposite of rect, which converts Polar coordinates to rectangular coordinates.

polar ( a, i, c, k, n )



pow_wci ( x, y )

NAME Power

DESCRIPTION This function computes x raised to the power y. A domain error is returned if x is neg-ative and y is not an integral value.

ALGORITHM

SYNOPSIS float pow_wci ( float x, float y )

DOMAIN -3.4E+38 to 3.4E+38 ( range of y depends on x )



NOTES The file tpow.c included in the distribution tape provides an example of this function's use.

return xy=



rangev ( a, i, &c, n )

NAME Range of Vector

DESCRIPTION This routine computes the range between the maximum value and minimum value of the elements in input vector a and places the results in output scalar c.

ALGORITHM

SYNOPSIS void range ( a, i, &c, n )





DOMAIN -3.4E+38 to 3.4E+38



NOTES The file trangev.c included in the distribution tape provides an example of this func-tion’s use.

rangev() sucessively searches all elements within input vector [a] and computes the range between the maximum and minimum values.

C Max a[ ] Min a[ ]– from 0 to n-1=

m 0 1 2 … n, 1–, , ,{ }=



rect ( a, i, c, k, n )

NAME Vector Polar To Cartesian Coordinates Conversion

DESCRIPTION This function converts polar coordinates (magnitude, phase) in input vector a to Car-tesian (real, imaginary) coordinates and stores the results in output vector c. All input coordinates are considered to be in radians.

ALGORITHM

SYNOPSIS void rect ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038


Re Cmk{ } R= e Ami{ } Im Ami{ }cos•

Im Cmk{ } Re Ami{ } Im Ami{ }sin•=

m 0 1 2 …n 1–, , ,{ }=




NOTES The file trect.c included in the distribution diskette provides an example of this func-tion’s use.

Polar coordinates are sometimes refered to as magnitude, phase or radius, angle coor-dinates. Cartesian coordinates are sometimes refered to as rectangular or x,y coordi-nates.

Note that the synposis section does not list either the Polar coordinates in input vector a or the Cartesian coordinates in output vector c as type complex. However, for com-putational purposes you may consider these as complex types, each having a real and imaginary portion consisting of either an x,y or a magnitude, phase component. The algorithm expects the Polar and Polar coordinates to be supplied in pairs, analagous to the manner in which we treat complex pairs. They are dyads which are considered to be atomic.

Note that this algorithm is the opposite of polar, which converts Rectangular coordi-nates to Polar coordinates.

rect ( a, i, c, k, n )



rfft ( x, real, imag, sintab, costab, wstr, n )

NAME Fast Fourier Transform of Real Input/Output Data

DESCRIPTION The routines computes an out-of-place, Radix-2 decimation in frequency FFT of length 64 or greater on input data x. Input data is no destroyed during the course of the execution of this routines.

ALGORITHM

SYNOPSIS void rfft ( x, real, imag, sintab, costab, wstr, n )

float dm *x ; /* Pointer to real input data */

float dm *real ; /* Pointer to output real data */

float pm *imag ; /* Pointer to output imaginary data */

float dm *sintab ; /* Pointer to sine table */

float pm *costab ; /* Pointer to cosine table */


int n ; /* FFT Size (In Real Elements) */

DOMAIN -3.4E+38 to 3.4E+38



Cm e2πjmk– n⁄

m 0 1 2 … n, 1–, , ,{ }=

m 0=

h 1–

∑=



NOTES The file trfft.c included in the distribution tape provides an example of this function’suse.

This algorithm is the fastest performing routine amongst the rfft( ), rffts( ) andrfftip() routines. This algorithm operates using an out-of-place methodology. Notethat versions 2.5 and lower of the Wideband Optimized DSP Library containedan rfft( ) routine which has been renamed rfftip( ). The current routine rfft( ) dif-fers significantly from the previous rfft( ) routine found in earlier versions of thelibrary. See the final section of Chapter 4 for more details.

The real FFT is accomplished by doing an N/2 complex FFT of y(i), whereRE[y(i)]=x(2i) and Im[y(i)] = x(2i+1) for i=0,1,...,n-1. This is followed by some calcu-lation to derive X(k) from Y(k). This method has been refered to as “packing”. Inputdata is NOT destroyed during the course of this routine. Note that only N/2+1 FFTpoints are needed to deine the real FFT spectra because of the Hermitian symmetry(complex conjugate) of the real FFT.

x [n] is the real normal-ordered input in DMreal [n/2+1] are the real parts of fft stored in DMimag [n/2+1] are the imaginary parts of fft stored in PMsintab [n/2] are the imaginary parts of weight/twiddle data in DMcostab [n/2] are the real parts of weight/twiddle data in PM

The weight/twiddle tables are in sintab[ ] and costab[ ] respectively and are given thevalues:

for i = { 0, 1, ..., wstr*n/2-1 }

The last 3 timings on timing chart were extrapolated from the based cycle count of1,024 points using the following formula:

Cycle count for

This cycle count will probably be on the low side as there may not be enough on-chipmemory to accomodate the accurate measument of large FFTs.

Restrictions:

• The number of elements n must be an integral power of 2 and a minimum of 64.• The input vector x[ ] must be in dm memory and must be aligned to an address which is

an integral multiple of n.• The input vector x[ ] is strided by one.• The output vectors are real[ ] and imag[ ] are in dm and pm memory respectively.• The real and imaginary weights/twiddles (cosine and sine data) are defined in pm and

dm memory respectively.


sintab[i] π i wstrn2---•

⁄• and tab[i]cos π i wstrn2---•

⁄•cos=sin=

nn

2nlog•

10242

of1024log•---------------------------------------------- 1 024 cycle,•= n=

n2

nlog•10240

----------------------- 11 990,•=



SPECIAL NOTES Previous users have sometimes reported problems associated with implementing inter-rupt service routines (ISRs), when used in conjunction with the FFT routines (cfft ( ),cffti( ), rfft( ), rffti( ), etc. ). Observations related to the Wideband technical staff typ-ically include a description of the Wideband routine executing perfectly, but unable toreturn to an exact state after being interrupted by the ISR ( a “tumble into the weeds” ).


End users are usually cognizant that their ISR calling routine is responsible for savingand restoring the registers of the Wideband routines. However, end users sometimesforget to save and restore ( push and pop ) the mode 1 regiser, which is associated withbit reversing and the B ( base ) and L ( length ) registers associated with circularaddressing. In such circumstances where they are not saved and restored by the ISRthey are unable to return the proper length parameter ( L Register ) used for circularaddressing or the proper mode ( Mode 1 Register ) used in Bit Reversing. This resultsin the strange manefestations users sometimes report.






TABLE 9 Performance Timings For Real FFTs


64 805

128 1,430

256 2,774

512 5,686

1,024 11,990

2,048 25,590

4,096 54,806

8,192 117,302

16,384 268,576 (Extrapolated)


65,536 1,227,776 (Extrapolated)



rfft8 ( xr, yr, yi )

NAME 8-Point Real Fast Fourier Transform (Inline)

DESCRIPTION Computes the Fast Fourier Transform of the real elements stored in input vector xr. The results are stored to output vector yr and yi. This routine is coded fully inline for minimum execution time but is structured as an out-of-place routine.

ALGORITHM

SYNOPSIS void rfft8 ( xr, yr, yi )




DOMAIN -3.4E+38 to 3.4E+38



NOTES This is an 8-point radix-2 Fast Fourier Transform using parallel data memory/programmemory data accesses to maximize the throughput on the 21020/60/62 processor.

The real input data is in vector xr. This vector must be aligned on an address whichis an integer multiple of the FFT size, as required for 21K bit-reverse addressing.The input vector is in data memory.

The complex output is separated into real and imaginary parts, yr and yi. These vec-tors may have arbitrary address alignment; however yr is in the data memory and yi isin the program memory.

The file trfft8.c included in the distribution tape provides an example of this func-tion’s use.

The entire algorithm is coded inline for optimal execution speed.



m 0 1 2 … 7,, , ,{ }=

k 0=

7

∑=



SPECIAL NOTES Some end users have reported reported problems associated with implementing inter-rupt service routines (ISRs) used in conjunction with the rfft8( ) routine. Recordedobservations forwarded to the Wideband technical staff usually include a descriptionof the Wideband routine somehow “tumbling into the weeds” after being interupted byan ISR. This is because the exact state of the fft routine is not saved and restored wheninterupted by the ISR.

All Wideband Fast Fourier transform routines, both complex and real, forward andinverse, use the built-in bit reversing and circular addressing capabilites of theSHARC architecture. This is also true of other routines such as some of the FIR filters,which use the SHARC’s internal circular addressing capabilities.

End users are usually cognizant that their ISR calling routine is responsible for savingand restoring the registers of the Wideband routines. However, end users sometimesforget to save and restore ( push and pop ) the Mode1 register, which is associatedwith bit reversing and the B ( base ) and L ( length ) registers associated with circularaddressing. In circumstances where they are not saved and restored by the callingISR, the length parameter ( L Register ) used for circular addressing or the propermode ( Mode1 Register ) used in bit reversing are not properly restored. This resultsin the strange manefestations users sometimes report, as the fft routine, upon restora-tion continues to execute code but the Mode 1 Register or L Register are not restored,thus indicating a false count or mode.







ALGORITHM





DOMAIN -3.4E+38 to 3.4E+38










m 0 1 2 … 15,, , ,{ }=

k 0=

15

∑=












ALGORITHM





DOMAIN -3.4E+38 to 3.4E+38










m 0 1 2 … 31,, , ,{ }=

k 0=

31

∑=









rffti ( real, imag, x, sintab, costab, wstr, n )

NAME Inverse Fast Fourier Transform of Real Input/Output Data

DESCRIPTION Computes the inverse real FFT ( Fast Fourier Transform ) of length 64 or greater on input data X(n). X(n) is converted into U(n) + jV(n), where U(n) and V(n) are the real and imaginary parts of the FFT transform of x(2n) + jx(2n+1). An inverse FFT is then performed on U(n) + jV(n) to yield x(n).

ALGORITHM

SYNOPSIS void rffti ( real, imag, x, sintab, costab, wstr, n )

float dm *real ; /* Pointer to real input data */

float pm *imag ; /* Pointer to input imaginary data */

float dm *x ; /* Pointer to output data */

float dm *sintab ; /* Pointer to sine weight table */

float pm *costab ; /* Pointer to cosine weight table */

int wstr ; /* Weight stride */

int n ; /* Number of Elements */

DOMAIN -3.4E+38 to 3.4E+38



Xm Um Vj m 0 1 2 … n, 1–, , ,{ }=⋅

j 0=

h 1–

∑=



NOTES The file trffti.c included in the distribution tape provides an example of this function’suse.

The inverse FFT is accomplished by switching the real and imaginary parts, taking the forward FFT, then then switching the real and imaginary again. Note that for an N point IFFT, only N/2+1 FFT points are needed because of the Hermitian symmetry of the real FFT. Input data is destroyed during the course of the routine.

real [n/2+1] are real frequency domain data in DMimag [n/2+1] are imaginary frequency domain data in PMx [n] is the real normal-ordered output in DMsintab [n/2] are the imaginary part of weight/twiddle data in DMcostab [n/2] are the real part of weight/twiddle data in PM

The weight/twiddle tables are in sintab[ ] and costab[ ] respectively and are given thevalues:

for i = { 0, 1, ..., wstr*n/2-1 }

The last 3 timings on timing chart were extrapolated from the based cycle count of1,024 points using the following formula:

Cycle count for

This cycle count will probably be on the low side as there may be enough on-chipmemory to accomodate large FFTs.

Restrictions

• The number of elements n must be an integral power of 2 and a minimum of 64.

• The output vector x[ ] and input vector real[ ] must be in dm memory and must be aligned to an address which is an integral multiple of n.

• The output vector x[ ] is strided by one.

• The input vectors are real[ ] and imag[ ] are in dm and pm memory respectively.

• The real and imaginary weights/twiddles (cosine and sine data) are defined in pm and dm memory respectively


sintab[i] π i wstrn2---•

⁄•sin=

tab[i]cos π i wstrn2---•

⁄•cos=

nn

2nlog•

10242

of1024log•---------------------------------------------- 1 024 cycle,•= n=

n2

nlog•10240

----------------------- 13 794,•=



Previous users have sometimes reported problems associated with implementing inter-rupt service routines (ISRs) when used in conjunction with the FFT routines ( cfft( ),cffti( ), rfft( ), rffti( ) ). Observations related to the Wideband technical staff typicallyinclude a description of the Wideband routine executing perfectly, but unable to returnto an exact state after being interrupted by the ISR ( a veritable “tumble into theweeds.” )

The Wideband Fast Fourier transforms, both complex and real, forward and inverse,use the built-in bit reversing and circular addressing capabiliites of the SHARC archi-tecture. Also, other routines such as some of the FIR filters use the SHARC’s internalcircular addressing capabilities.







TABLE 10 Performance Timings For Real Inverse FFTs

Number of Input Points Processor Cycles

64 918

128 1,657

256 3,228

512 6,591

1,024 13,794

2,048 29,189

4,096 61,992

8,192 131,659



65,536 1,412,505 (Extrapolated)



rfftip ( data, wr, wi, wstr, tmp_dm, tmp_pm, n )

NAME Fast Fourier Transform of Real Input/Output Data In Place

DESCRIPTION This routine computes a Radix-2, in-place, decimation-in-time, Discrete Fourier Transform of real-valued input data using a Fast Fourier Transform (FFT) algorithm. The input vector data[ ] is a real-valued vector of n elements. On return from the rfftip ( ) call, data[ ] contains n/2 complex values representing the DFT of the input function. In previous versions of theWideband library ( 2.5 and below) this function was known as the rfft ( ) function. See the final section of Chapter 4 for more details.

To compute the inverse of the the rfftip ( ) function, reorder the output data from rfftip( ) using the vrfftip ( ) function, and then call the rffti ( ) function, supplying the results from the vrfftip ( ) function.

Restrictions

The number of elements n must be an integral power of 2 and a minimum of 64.

Vectors tmp_pm and tmp_dm are work buffers of minimum lenght n/2 and must be in program memory and data memory respectively.

Vectors wr and wi are in program memory and data memory respectively, and are given the values

wr[k] = cos [2 pi k / (wstr*n)], k = 0, 1, ..., wstr*n/2-1wr[k] = sin [2 pi k / (wstr*n)], k = 0, 1, ..., wstr*n/2-1

Vector data must be in data memory and must be aligned to an address which isan integral multiple of n.

ALGORITHM

SYNOPSIS void rfftip ( data, wr, wi, wstr, tmp_dm, tmp_pm, n )

float dm *data ; /* Pointer to real input/output data */



int dm wstr ; /* Weight stride */

float dm *tmp_dm ; /* Pointer to DM work buffer */

float dm *tmp_pm ; /* Pointer to PM work buffer */


Fk Fie2πij ik n⁄( )–

k 0 1 2 … n, 1–, , ,{ }=

i 0=

n 1–

∑=



DOMAIN -3.4E+38 to 3.4E+38



NOTES This function computes a Discrete Fourier Transform of real-valued input data using aFast Fourier Transform (FFT) algorithm. The input vector data is a real-valued func-tion on n elements. On return from the rfftip( ) call, data[] contains n/2 complex val-ues representing the DFT of the input function.

The general expression for the DFT is as follows:

F[ ] and f[ ] are complex-valued vectors. rffts( ) assumes that f[ ] is a real-valued vec-tor, i.e., Im(f) = 0. The DFT of a real-valued function has the following properties:

and

rfftip( ) makes use of these properties to “stuff” the complex-valued output functioninto the n-element real input vector. The DFT values are stored such that

To compute the inverse of the the rfftip( ) function, reorder the output data fromrfftip( ) using the vrfftip( ) function, and then call the rffti( ) function, supplying theresults from the vrfftip( ) function.

The rfft( ) algorithm executes faster than the rffts( ) algorithm, which in turn executesfaster than the rfftip( ) algorithm.

The file trfftip.c included in the distribution tape provides an example of this func-tion’s use.

rfftip ( data, wr, wi, wstr, tmp_dm, tmp_pm, n )

Fk fie2πj i k n⁄( )– k 0 1 2 … n, 1–, , ,{ }=

i 0=

n 1–

∑=

Fn k– Fk∗ k 0 1 … n 1–, , ,{ }== m Fn 2⁄( ) Im F0( ) 0==

Re F0( ) data 0[ ]=

Re Fk( ) data 2 k•[ ] k 1 2 … n 2 1–⁄, , ,{ }==

Im Fk( ) data 2 k 1+•[ ] k 1 2 … n 2 1–⁄, , ,{ }==

Re Fn 2⁄( ) data 1[ ]=



TABLE 11 Performance Timing for rfftip ( ) Routine


64 1,426

128 2,329

256 4,221

512 8,229

1,024 16,715

2,048 34,673

4,096 72,599

8,192 152,509



65,536 1,711,616 (Extrapolated)



SPECIAL NOTES Some end users have reported reported problems associated with implementing inter-rupt service routines (ISRs) used in conjunction with the fft routines. Recorded obser-vations forwarded to the Wideband technical staff usually include a description of theWideband routine somehow “tumbling into the weeds” after being interupted by anISR. This is because the exact state of the fft routine is not saved and restored wheninterupted by the ISR.






rffts ( x, tmppm, sintab, costab, wtstr, n )

NAME Fast Fourier Transform of Real Input/Output Data Using Sorenson Method

DESCRIPTION This routine computes a Radix-2 in-place, decimation-in-time, Discrete Fourier Transform of real-valued input data using a Fast Fourier Transform (FFT) algorithm. The input vector x [ ] is a real-valued vector of n elements. On return from the rffts ( ) call, x[ ] contains n/2 complex values representing the DFT of the input function. To compute the inverse of the the rffts ( ) function, reorder the output data from the rffts( ) function using the vrffts ( ) function, and then call the rffti ( ) function, supplying the results from the vrffts ( ) function.

Restrictions

The number of elements n must be an integral power of 2 and a minimum of 64.

Vector x must be in data memory and must be aligned to an address which is anintegral multiple of n.

ALGORITHM

SYNOPSIS void rffts ( x, tmppm, sintab, costab, wtstr, n )

float dm *x ; /* Pointer to real input/output data */

float pm *tmppm ; /* Pointer to scratch pm data */

float dm *sintab ; /* Pointer to sine weight table */

float pm *costab ; /* Pointer to Cosine weight table */

int dm *wtstr ; /* Weight stride */


DOMAIN -3.4E+38 to 3.4E+38


Fk Fie2πij ik n⁄( )–

k 0 1 2 … n, 1–, , ,{ }=

i 0=

n 1–

∑=




NOTES This function computes a Discrete Fourier Transform of real-valued input data using aFast Fourier Transform (FFT) algorithm. The input vector data is a real-valued func-tion on n elements. On return from the rfft( ) call, x contains n/2 complex values rep-resenting the DFT of the input function.

The general expression for the DFT is as follows:

F[] and f[] are complex-valued vectors. rffts( ) assumes that f[] is a real-valued vector,i.e., Im(f) = 0. The DFT of a real-valued function has the following properties:

and

rffts( ) makes use of these properties to “stuff” the complex-valued output functioninto the n-element real input vector. The DFT values are stored such that

Re ( F[0] ) = x[0]Re ( F[1] ) = x[1] . .Re ( F[n/2] ) = x[n/2]Im ( F[n/2-1] ) = x[n/2+1]Im ( F[n/2-2] ) = x[n/2+2] . .

Im ( F[1] ) = x[n-1]

This algorithm was adapted from the FORTRAN program developed by HenrikSorenson et al., in the IEEE Transaction on ASSP, June 1987.

To compute the inverse of the the rffts( ) function, reorder the output data from therffts( ) function using the vrffts( ) function, and then call the rffti( ) function, supply-ing the results from the vrffts( ) function.

The rfft( ) algorithm executes faster than the rffts( ) algorithm, which in turn executesfaster than the rfftip( ) algorithm.

The file trffts.c included in the distribution tape provides an example of this function’suse.

rffts ( x, tmppm, sintab, costab, wtstr, n )

Fk fie2πj i k n⁄( )– k 0 1 2 … n, 1–, , ,{ }=

i 0=

n 1–

∑=

Fn k– Fk∗ k 0 1 … n 1–, , ,{ }== Im Fn 2⁄( ) Im F0( ) 0==



TABLE 12 Performance Timing for rffts ( ) Routine


64 947

128 1,748

256 3,071

512 6,624

1,024 12,779

2,048 29,260

4,096 58,583

8,192 134,456



65,536 1,308,569 (Extrapolated)



SPECIAL NOTES Some end users have reported reported problems associated with implementing inter-rupt service routines (ISRs) used in conjunction with the fft routines. Recorded obser-vations forwarded to the Wideband technical staff usually include a description of theWideband routine somehow “tumbling into the weeds” after being interupted by anISR. This is because the exact state of the fft routine is not saved and restored wheninterupted by the ISR.






rmvesq ( a, i, c, n )

NAME Vector Root Mean Square

DESCRIPTION This function computes the root mean square of all elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void rmvesq ( a, i, c, n )





DOMAIN -1.8 x 1019 to +1.8 x1019



NOTES The file trmvesq.c included in the distribution diskette provides an example of this function’s use.

If all elements in input vector a are zero, a zero value is returned.

C 1n--- A2

mi

m 0=

n 1–

∑= m 0 1 2 …n, 1 }–, ,{=



rsqrt_wci ( x )

NAME Reciprocal Square Root

DESCRIPTION This function computes the reciprocal square root a floating-point number, x. The computed value is returned from this function. A domain error occurs if x < 0.0.

ALGORITHM

SYNOPSIS float rsqrt_wci ( float x )

DOMAIN 0.0 to 3.40E38



NOTES The file trsqrt.c included in the distribution tape provides an example of this func-tion's use.

return 1.0

x-------=



scvdiv ( a, i, b, c, k, n )

NAME Scalar Divide By Vector

DESCRIPTION This function divides input scalar b by input vector a and stores the result in output vector c.

ALGORITHM

SYNOPSIS void scvdiv ( a, i, b, c, k, n )



float b ; /* Value of input scalar b */



int n ; /* Count of floating-point elements */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tscvdiv.c included in the distribution diskette provides an example of this function’s use.

This routine does not contain logic to check for an attempted divide by zero. If an ele-ment of vector a [ ] is set to zero an attempt to divide by zero error will occur.

CmkB

Ami---------- m 0 1 2 …n, 1–, ,{ }==



scvpow ( a, i, b, c, k, n )

NAME Compute the Powers of a Scalar to a Vector

DESCRIPTION This function computes the results of scalar b raised to an element in input vector a [ ]. The results are stored in output vector c.

ALGORITHM

SYNOPSIS void scvpow ( a, i, b, c, k, n )



float b ; /* Value of the base b */




DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 50 + 61*N cycles for a < > 0 and b > 0

50 + 6*N cycles for a = 0 and b > 0

25 + 7*N cycles for b <= 0

NOTES The file tscvpow.c included in the distribution diskette provides an example of this function’s use.

1. For input scalar b equal to zero, elements in input vector a[ ] must be greater than zero.

2. For input scalar b less then zero, elements in input vector a[ ] must be equal to zero.

3. -99 is returned for conditions 1 or 2 not satisfied.

Cmk BAmi

m 0 1 2 …n 1–, , ,{ }==



scvsub ( a, i, b, c, k, n )

NAME Subtract Vector from Scalar

DESCRIPTION This function subtracts input vector a from input scalar b by and stores the result in output vector c.

ALGORITHM

SYNOPSIS void scvsub ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tscvsub.c included in the distribution diskette provides an example of this function’s use.

Cmk B Ami – m 0 1 2 …n, 1–, ,{ }==



see ( yi, sl, a, b, s, c, n )

NAME Standard Error of Estimate

DESCRIPTION This function computes the standard error of estimate of predicted y values of the lin-ear least-squares regression fit (see lsqreg( )) from the actual y values. Input vector c contains the actual measured input y values while input vector b contains the values predicted by the linear least-squares regression fit formula. The ideal y values are cal-culated from the scalar input y-intercept yi, the scalar input slope sl, and the x-coordi-nates values found within input vector a. The routine then calculates the standard error of estimate and places the results in output scalar d.

ALGORITHM

SYNOPSIS void see ( yi, sl, a, b, s, c, n )

float *yi ; /* Pointer to input scalar y-intercept yi */

int *sl ; /* Pointer to input scalar slope sl */

float *a ; /* Pointer to intput measured x values vector a */

float *b ; /* Pointer to intput measured x values vector b */

int s ; /* Element stride for input vectors [a] and [b] */

float *d ; /* Pointer to resultant output scalar error c */


DOMAIN -3.4E+38 to 3.4E+38


Bms yi sl Ams•( )+=

where yi= calculated y-intercept from the lsqreg() function

where sl= calculated slope from the lsqreg() function

D

n Cms Bms–( )2

i 0=

n 1–

∑n 2–

--------------------------------------------------=

m 0 1 2 …n, 1 }–, ,{=




NOTES The file tsee.c included in the distribution tape provides an example of this function's use.

This routine implements a mathematical technique to estimate the variance of the mea-sured data in input vector c from the ideal predicted data calculated from the lsqreg( ) regression equation routine. It can be thought of as a measure of the standard deviation of the measured population (input vector c) from the ideal population (input vector b) predicted by the linear regression line equation. As such, it can be used as an accurate predictor of the variation of the measured y data values (input vector c) about the pre-dicted ideal regression line, and hence shed light into the error contained within the sampled data.

Note that in order to execute this operation, you must first execute the lsqreq( ) routine to compute a y-intercept and x-coordinate vector. The resulting regression equation will yield two values, a, the y-intercept and b, the slope. When both values are used with the input value x, which represents the actual (observe value) x-coordinate, an ideal resulting y, which we represented as can be calculated. The formula is as fol-lows: .

We have renamed the variables a and b to yi and sl for clairity. This calculation is per-formed on each input x-coordinate value to an ideal array, which we call vector b, which contains the resulting idealized y values predicted by the regression equation. Each of these values, in turn, is compared to the actual measured y values to yield the final standard error fo the estimate.

Also note that the input scalar s contains the stride values for vectors a, and b, which should remain equal throughout the calculation.

An ideal value is calculated for each observed x value. The resulting values are placed in input array b.

see ( yi, sl, a, b, s, c, n )

y'y a bx+=

y'



sin_wci ( x )

NAME Sine

DESCRIPTION This function computes the sine of a floating-point number x where x is measured in radians. The computed value is returned from this function.

ALGORITHM

SYNOPSIS float sin_wci ( float x )

DOMAIN - 205,884 < x < 205,884 radians



NOTES The file tsin.c included in the distribution tape provides an example of this function's use.

return x( )sin=



sinc ( a, i, c, k, n )

NAME Sinc Function

DESCRIPTION This function computes the sinc function of the input elements contained within input vector a and stores the result to output vector c.

ALGORITHM

SYNOPSIS void sinc ( a, i, c, k, n )




int k ; /* Address stride in words for input vector a */


DOMAIN -1.8 x 1019 to +1.8 x1019



NOTES The file tsinc.c included in the distribution diskette provides an example of this func-tion’s use.

Note that A cannot be equal to zero or a divide by zero error will occur.

Cmk

Amisin

Ami----------------- m 0 1 2 …n, 1 }–, ,{==



sinh_wci

NAME Hyperbolic Sine

DESCRIPTION This function computes the hyperbolic sine of a floating-point number, x ( x is mea-sured in radians ). The computed value is returned from this function.

ALGORITHM

SYNOPSIS float sinh_wci ( float x )

DOMAIN -88 to +88



NOTES The file tsinh.c included in the distribution tape provides an example of this function's use.

return x( )sinh=



slidwin ( a, i, w, c, k, n )

NAME Sliding Window

DESCRIPTION This function computes a sliding window summation of the values contained within input vector a, defined using input scalar w as the width of the sliding window. The sliding window summations are placed in output vector c.

ALGORITHM

SYNOPSIS void slidwin ( a, i, w, c, k, n )



int w ; /* Pointer to the length of sliding window w */




DOMAIN -3.4 x 1038 to +3.4 x 1038


Cmk Ami

0

w

∑ m 0 1 2 …n, 1 }–, ,{==



EXECUTION TIME 38 + ( W - 1 ) + 3 * ( N - W )

NOTES The file tslidwin.c included in the distribution tape provides an example of this func-tion's use.

slidwin() computes the sum of a moving window using vector a as an input vector. The results are stored to output vector c. Elements in a vector a are successively summed. The number of elements summed is defined by the window length. This sum is then stored.

Example:

[a] = { 2, 4, 0, 1, 3, 5, 3, 1, 4, 7, 1, };

Window size w = 9

c[0] = ( 2 + 4 + 0 + 1 + 3 + 5 + 3 + 1 + 4 ) = 23

c[1] = ( 4 + 0 + 1 + 3 + 5 + 3 + 1 + 4 + 7 ) = 28

c[2] = ( 0 + 1 + 3 + 5 + 3 + 1 + 4 + 7 + 1 ) = 25

The length of the window cannot exceed the population size divided by the striding. (i.e., w < = n / i).

The number of elements computed is determined by the total number of elements minus the window size.

slidwin ( a, i, w, c, k, n )



snd ( a, i, b, c, n )

NAME Signal Noise Distortion

DESCRIPTION This function computes the signal to noise density of the input elements contained within input vector a with respect to the input signal scalar b and stores the result to output scalar c.

ALGORITHM

SYNOPSIS void snd ( a, i, b, c, n )



float b ; /* Value of input signal b */



DOMAIN -1.8 x 1019 to +1.8 x1019



NOTES The file tsnd.c included in the distribution diskette provides an example of this func-tion’s use.

C 2010

B( ) 2010

Ami∑ log•–log•=

m 0 1 2 …n, 1 }–, ,{=



sqrt_wci ( x )

NAME Square Root

DESCRIPTION This function computes the square root a floating-point number, x. The computed value is returned from this function. A domain error occurs if x < 0.

ALGORITHM

SYNOPSIS float sqrt_wci ( float x )




NOTES The file tsqrt.c included in the distribution tape provides an example of this function's use.

return x=



std ( a, i, c, n )

NAME Standard Deviation

DESCRIPTION This function computes the standard deviation of the input elements contained within input vector a and stores the result to output scalar c.

ALGORITHM

SYNOPSIS void std ( a, i, c, n )





DOMAIN -1.8 x 1019 to +1.8 x1019



NOTES The file tstd.c included in the distribution diskette provides an example of this func-tion’s use.

Cn ami

2ami∑

2

–∑n n 1–( )

--------------------------------------------------=

m 0 1 2 …n, 1 }–, ,{=



sve ( a, i, c, n )

NAME Vector Sum

DESCRIPTION This function computes the sum of the elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void sve ( a, i, c, n )





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tsve.c included in the distribution diskette provides an example of this func-tion’s use.

C Ami

m 0=

n 1–

∑= m 0 1 2 …n, 1 }–, ,{=



svemg ( a, i, c, n )

NAME Vector Sum Magnitudes

DESCRIPTION This function computes the sum of the magnitudes of the elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void svemg ( a, i, c, n )





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tsvemg.c included in the distribution diskette provides an example of this function’s use.

C Ami

m 0=

n 1–

∑=

m 0 1 2 …n, 1 }–, ,{=



svesq ( a, i, c, n )

NAME Vector Sum of the Squares

DESCRIPTION This function computes the sum of the squares of the elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void svesq ( a, i, c, n )





DOMAIN -1.8 x 1019 to +1.8 x 1019



NOTES The file tsvesq.c included in the distribution diskette provides an example of this func-tion’s use.

C A2mi

m 0=

n 1–

∑=

m 0 1 2 …n, 1 }–, ,{=



svessq ( a, i, c, n )

NAME Vector Sum of the Signed Squares

DESCRIPTION This function computes the sum of the signed squares of the elements in input vector a and stores the result in output scalar c.

ALGORITHM

SYNOPSIS void svessq ( a, i, c, n )





DOMAIN -1.8 x 1019 to +1.8 x 1019



NOTES The file tsvessq.c included in the distribution diskette provides an example of this function’s use.

C Ami Ami•

m 0=

n 1–

∑= m 0 1 2 …n, 1 }–, ,{=



tan_wci ( x )

NAME Tangent

DESCRIPTION This function computes the tangent of a floating-point number, x ( x is measured in radians ). The computed value is returned from this function.

ALGORITHM

SYNOPSIS float tan_wci ( float x )

DOMAIN - 205,884 < x < 205,884 radians



NOTES The file ttan.c included in the distribution tape provides an example of this function's use.

tanh_wci ( x )

NAME Hyperbolic Tangent

DESCRIPTION This function computes the hyperbolic tangent of a floating-point number, x ( x is measured in radians ). The computed value is returned from this function.

ALGORITHM

SYNOPSIS float tanh_wci ( float x )

DOMAIN -88 to +88



NOTES The file ttanh.c included in the distribution tape provides an example of this function's use.

return x( )tan=

return x( )tanh=



vaam ( a, i, b, j, c, k, d, l, e, h, n )

NAME Vector Add, Add, and Multiply

DESCRIPTION This function adds two input vectors a and b, adds two other input vectors c and d, then multiplies the two intermediate results, and stores the result in output vector e.

ALGORITHM

SYNOPSIS void vaam ( a, i, b, j, c, k, d, l, e, h, n )





float *c ; /* Pointer to input vector c */


float *d ; /* Pointer to input vector d */

int l ; /* Address stride in words for input vector d */

float *e ; /* Pointer to output vector e */

int h ; /* Address stride in words for output vector e */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvaam.c included in the distribution diskette provides an example of this func-tion’s use.

Emh Ami Bmj+( ) Cmk Dml+( )•=

m 0 1 2 …n 1–, , ,{ }=



vabs ( a, i, c, k, n )

NAME Vector Absolute Value

DESCRIPTION This function computes the absolute value of all elements of vector a and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vabs ( a, i, c, k, n )






DOMAIN -3.4E+38 to +3.4E+38



NOTES The file tvabs.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }==



vacos ( a, i, c, k, n )

NAME Vector Arccosine

DESCRIPTION This function computes the arc cosine ( inverse cosine ) for each element in input vec-tor a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vacos ( a, i, c, k, n )






DOMAIN -1.0 to +1.0


EXECUTION TIME 66 + (35 to 56)*N cycles

NOTES The file tvacos.c included in the distribution diskette provides an example of this func-tion’s use.

If the input argument supplied from in input vector a is less than 0.5 then the inner loop of this routine will execute in 36 cycles. If the input argument supplied in input vector a is greater than 0.5 then the inner-loop of this routine will execute in 56 cycles.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=acos=



vacosh ( a, i, c, k, n )

NAME Vector Inverse Hyperbolic Cosine

DESCRIPTION This function computes the inverse hyperbolic cosine for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vacosh ( a, i, c, k, n )






DOMAIN -1.0 to +1.0



NOTES The file tvacosh.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=acosh=



vacot ( a, i, c, k, n )

NAME Vector Arc Cotangent

DESCRIPTION This function computes the arc cotangent ( inverse cotangent ) for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vacot ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 48 + (60 to 77 to 63 to 53)*N cycles

NOTES The file tvacot.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amiacot m 0 1 2 …n 1–, , ,{ }==



vacoth ( a, i, c, k, n )

NAME Vector Arc Hyperbolic Cotangent

DESCRIPTION This function computes the arc hyperbolic cotangent ( inverse hyperbolic tangent ) for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vacoth ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvacoth.c included in the distribution diskette provides an example of this function’s use.

Cmk Amiacoth m 0 1 2 …n 1–, , ,{ }==



vacsc ( a, i, c, k, n )

NAME Vector Arc Cosecant

DESCRIPTION This function computes the arc cosecant ( inverse cosecant ) for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vacsc ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvacsc.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amiacsc m 0 1 2 …n 1–, , ,{ }==



vacsch ( a, i, c, k, n )

NAME Vector Arc Hyperbolic Cosecant

DESCRIPTION This function computes the arc hyperbolic cosecant ( inversehyperbolic cosecant ) for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vacsch ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvacsch.c included in the distribution diskette provides an example of this function’s use.

Cmk Amiacsch m 0 1 2 …n 1–, , ,{ }==



vadd ( a, i, b, j, c, k, n )

NAME Vector Add

DESCRIPTION This function adds two input vectors a and b, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vadd ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvadd.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami Bmj+ m 0 1 2 …n 1–, , ,{ }==



vam ( a, i, b, j, c, k, d, l, n )

NAME Vector Add and Multiply

DESCRIPTION This function adds two input vectors a and b, then multiplies the sum by input vector c and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vam ( a, i, b, j, c, k, d, l, n )







float *d ; /* Pointer to output vector d */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvam.c included in the distribution diskette provides an example of this func-tion’s use.

Dml Ami Bmj+( ) Cmk• m 0 1 2 …n 1–, , ,{ }==



vand ( a, i, b, j, c, k, n )

NAME Vector And

DESCRIPTION This function performs a 32-bit logical AND of each element in input vector a with each element in input vector b and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vand ( a, i, b, j, c, k, n )








DOMAIN No restrictions

ACCURACY 32-bits


NOTES The file tvand.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami AND Bmj( )= m 0 1 2…n 1 }–, ,{=



var_wci ( a, i, c, n )

NAME Variance

DESCRIPTION This function computes the variance of the input elements contained within input vec-tor a and stores the result to output scalar c.

ALGORITHM

SYNOPSIS void var_wci ( a, i, &c, n )





DOMAIN -1.8 x 1019 to +1.8 x1019



NOTES The file tvarwci.c included in the distribution diskette provides an example of this function’s use.

Cn ami

2ami∑

2

–∑n n 1–( )

--------------------------------------------------=

m 0 1 2 …n, 1 }–, ,{=



vascm ( a, i, b, j, c, d, l, n )

NAME Vector Add and Scalar Multiply

DESCRIPTION This function adds two input vectors, a and b, then multiplies the sum by an input sca-lar c and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vascm ( a, i, b, j, c, d, l, n )





float c ; /* Value of input scalar c */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvascm.c included in the distribution diskette provides an example of this function’s use.

Dml Ami Bmj+( ) C• m 0 1 2 …n 1–, , ,{ }==



vascs ( a, i, b, j, c, d, l, n )

NAME Vector Add and Scalar Subtract

DESCRIPTION This function adds two input vectors a and b, then subtracts input scalar c from the sum, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vascs ( a, i, b, j, c, d, l, n )









DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvascs.c included in the distribution diskette provides an example of this func-tion’s use.

Dml Ami Bmj+( ) C– m 0 1 2 …n 1–, , ,{ }==



vasec ( a, i, c, k, n )

NAME Vector Arcsecant

DESCRIPTION This function computes the arcsecant ( inverse secant ) for each element in input vec-tor a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vasec ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvasec.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amiasec m 0 1 2 …n 1–, , ,{ }==



vasech ( a, i, c, k, n )

NAME Vector Arc Hyperbolic Secant

DESCRIPTION This function computes the arc hyperbolic secant ( inverse hyperbolic secant ) for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vasech ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 37 + ( 7 to 9 to 60)*N cycles

NOTES The file tvasech.c included in the distribution diskette provides an example of this function’s use.

Cmk Amiasech m 0 1 2 …n 1–, , ,{ }==



vashift ( a, i, c, k, n, h )

NAME Vector Arithmetic Shift

DESCRIPTION This function performs an arithmetic shift of h bits on each 32-bit element in input vector a and stores the results in output vector c. If h is negative the bits are right-shifted with a sign fill into the most significant bit (MSB). If h is positive the bits are left-shifted with a zero fill into the least significant bit (LSB).

ALGORITHM if then

Sign Fill MSB

else

Zero Fill LSB

SYNOPSIS void vashift ( a, i, c, k, n, h )






int h ; /* Length of arithmetic shift */

DOMAIN H must be between -32 to +32.



NOTES The file tvashift.c included in the distribution diskette provides an example of this function’s use. Note that each right-shift or left-shift is shifted through the carry bit.

H 0< m 0 1 2 …n, 1 }–, ,{=

Cmk Ami H»=

Cmk Ami H«=



vasin ( a, i, c, k, n )

NAME Vector Arcsine

DESCRIPTION This function computes the arcsine ( inverse sine ) for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vasin ( a, i, c, k, n )






DOMAIN -1.0 to +1.0



NOTES The file tvasin.c included in the distribution diskette provides an example of this func-tion’s use.

If the input argument supplied in input vector a is less than 0.5 then the inner-loop of this routine will execute in 36 cycles. If the input argument supplied in input vector a is greater than 0.5 then the inner-loop of this routine will execute in 56 cycles.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=asin=



vasinh ( a, i, c, k, n )

NAME Vector Inverse Hyperbolic Sine

DESCRIPTION This function computes the inverse hyperbolic sine for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vasinh ( a, i, c, k, n )






DOMAIN -1.0 to +1.0



NOTES The file tvasinh.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=asinh=



vasm ( a, i, b, j, c, k, d, l, e, h, n )

NAME Vector Add, Subtract and Multiply

DESCRIPTION This function adds two input vectors, a and b, adds two other input vectors, c and d, then multiplies the two intermediate results, and stores the result in output vector e.

ALGORITHM

SYNOPSIS void vasm ( a, i, b, j, c, k, d, l, e, h, n )












DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvasm.c included in the distribution diskette provides an example of this func-tion’s use.

Emh Ami Bmj+( ) Cmk Dml–( )• =

m 0 1 2 …n 1–, , ,{ }=



vaspec ( a, i, b, j, c, k, n )

NAME Vector Accumulating Auto-spectrum

DESCRIPTION Computes an accumulating autospectrum of vector elements found within input vec-tor a[ ] and vector b[ ]. The accumulated result is stored in output vector c.

ALGORITHM

SYNOPSIS void vaspec ( a, i, b, j, c, k, n )


int i ; /* Stride of a [ ] */


int j ; /* Stride of b [ ] */


int k ; /* Stride of c [ ] */


DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvaspec.c included in the distribution tape provides an example of this func-tion’s use.

The stride of vectors a and c must always be 1.

If you wish to clear the auto-spectrum results before they are added to output vector cuse the vclr( ) function. If the results are not cleared using vclr( ), autospectrum resultsare added to output vector c, thus computing an accumulating autospectrum.

The vaspec( ) function is intended to compliment the outputs from the cfft( ) and rfft( )functions.

Cmk Cmk Ami– Bmj+⇐

m 0 1 2 …n 1–, , ,{ }=



vatan ( a, i, c, k, n )

NAME Vector Arctangent

DESCRIPTION This function computes the arctangent ( inverse tangent ) for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vatan ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvatan.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=atan=



vatan2 ( y, i, x, j, c, k, n )

NAME Vector Arctangent, 2 Arguments

DESCRIPTION This function computes the arctangent ( inverse tangent ) for each element in input vectors y and x, and stores the result in output vector c. All angles in vector c are returned in radians. Vector y is divided by vector x to obtain the tangent for the argu-ment to the inverse tangent computation.

ALGORITHM

SYNOPSIS void vatan2 ( y, i, x, j, c, k, n )

float dm *y ; /* Pointer to input vector y */


float dm *x ; /* Pointer to input vector x */





DOMAIN -1.0 x 109 to +1.0 x 109 , Bmj should not equal zero.


EXECUTION TIME 50 + 70 (max) *N cycles

NOTES The file tvatan2.c included in the distribution diskette provides an example of this function’s use.

Cmk 2YmiXmj---------

m 0 1 2 …n 1–, , ,{ }=atan=



vatanh ( a, i, c, k, n )

NAME Vector Inverse Hyperbolic Tangent

DESCRIPTION This function computes the inverse hyperbolic tangent for each element in vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vatanh ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvatanh.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=atanh=



vavexp ( a, i, b, c, k, n )

NAME Vector Exponential Average

DESCRIPTION This function computes the exponential average of input vector a weighted by input scalar b. The results are stored to output vector c.

ALGORITHM

SYNOPSIS void vavexp ( a, i, b, c, k, n )


int i ; /* Element stride for inoput vector a */

float *b ; /* Scalar Weighting Factor */

float *c ; /* Pointer to input/output vector c */

int k ; /* Element stride for input/output vector c */


DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvavexp.c included in the distribution tape provides an example of this func-tion’s use.

Note that vector c is both an input and output vector. If you wish to clear this vector before use, use the vclr( ) function which will initialize the contents of input vector c with zeroes.

Input scalar b, the scalar weighting factor, should not equal zero to avoid a divide by zero error.

Cmk

Cmk B 1–( ) Ai+•

B------------------------------------------------ m 0 1 2 … n, 1–, , ,{ }==



vavlin ( a, i, b, c, k, n )

NAME Vector Linear Average

DESCRIPTION This function computes the linear average of input vector a weighted by input scalar b. The results are stored to output vector c.

ALGORITHM

SYNOPSIS void vavlin ( a, i, b, c, k, n )



float *b ; /* Scalar Weighting Factor */

float *c ; /* Pointer to input/output vector c */

int k ; /* Element stride for input/output vector c */


DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvavlin.c included in the distribution tape provides an example of this func-tion’s use.

Note that vector c is both an input and output vector. If you wish to clear this vector before use, use the vclr( ) function which will initialize the contents of input vector c with zeroes.

Input scalar b, the scalar weighting factor, should not equal zero to avoid a divide by zero error.

Cmk

Cmk B Ai+•

B 1.0+---------------------------------- m 0 1 2 … n, 1–, , ,{ }==



vbsort ( a, b, n )

NAME Vector Perform Bubble Sort

DESCRIPTION This function performs a bubble sort on input/ouput vector a. The sort order is deter-mined by the sort flag b. For b=1, input/output vector a is sorted in ascending order, smallest to largest. For b=0, input/ouput vector a is sorted in descending order, largest to smallest. The output is stored in vector a.

ALGORITHM

SYNOPSIS void vbsort ( a, b, n )

float dm *a ; /* Pointer to input\output vector a */

float b ; /* Sort flag */


DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 45 + N*(6 + 7*N) cycles

NOTES The file tvbsort.c included in the distribution diskette provides an example of this function’s use.

If B 1 then A is sorted smallest to largest=

If B 0 then A is sorted largest to smallest=



vcbrt ( a, i, c, k, n )

NAME Vector Cube Root

DESCRIPTION This function computes the cube root of each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vcbrt ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvcbrt.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami3 m 0 1 2 …n 1–, , ,{ }==



vceil ( a, i, c, k, n )

NAME Vector Ceiling (Round Up To Nearest Integral Value)

DESCRIPTION This function computes the smallest integral value greater than or equal to the float-ing-point number in input vector a and stores a floating-point representation of this integral result in output vector c.

ALGORITHM

SYNOPSIS void vceil ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvceil.c included in the distribution diskette provides an example of this func-tion’s use. This function is the opposite of vfloor( ).

Cmk smallest integer Ami≥= m 0 1 2 …n 1–, , ,{ }=



vclip ( a, i, b, c, d, l, n )

NAME Vector Clip

DESCRIPTION This function uses a lower-threshold scalar b and an upper-threshold scalar c to clip each element in input vector a and stores the results in output vector d.

ALGORITHM

SYNOPSIS void vclip ( a, i, b, c, d, l, n )



float b ; /* Value of input lower-threshold scalar b */

float c ; /* Value of input upper-threshold scalar c */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvclip.c included in the distribution diskette provides an example of this func-tion’s use.

if Ami B then Dml B m 0 1 2 … n, 1–, , ,{ }==<

if Ami C then Dml C=>

else Dml Ami=



vclr ( c, k, n )

NAME Vector Clear

DESCRIPTION This function sets all elements of input-output vector c to 0.0.

ALGORITHM

SYNOPSIS void vclr ( c, k, n )

float *c ; /* Pointer to input-output vector c */



DOMAIN 0 to 131,000



NOTES The file tvclr.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk 0.0= m 0 1 2 …n 1–, , ,{ }=

0 k 0 n<,<



vcmerg ( a, i, b, j, c, k, d, l, n )

NAME Vector Compressed Merge

DESCRIPTION This function merges two input vectors a and b based on the values contained within gating vector c. The results are stored in output vector d. Either vector a or vector b’s element is stored into output vector d based on the value of the corresponding element in vector c.

ALGORITHM if then

else

SYNOPSIS void vcmerg ( a, i, b, j, c, k, d, l, n )










DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvcmerg.c included in the distribution diskette provides an example of this function’s use.

Cmk 0.0≠

Dml Bmj=

Dml Amj=

m 0 1 2 …n, 1 }–, ,{=



vcmprs ( a, i, b, j, c, k, n )

NAME Vector Compress

DESCRIPTION This function compresses the elements of input vector a based upon the values con-tained within input vector b. The results are stored in vector c.

ALGORITHM if then

SYNOPSIS void vcmprs ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvcmprs.c included in the distribution diskette provides an example of this function’s use.

Bmj 0.0≠

Cmk Ami=

m 0 1 2 …n, 1 }–, ,{=



vcos ( a, i, c, k, n )

NAME Vector Cosine

DESCRIPTION This function computes the cosine for each element in input vector a, and stores the result in output vector c. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vcos ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvcos.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=cos=



vcos_dp ( a, i, c, k, n )

NAME Vector Cosine_dp

DESCRIPTION This function computes the cosine for each element in input vector a, and stores the result in output vector c. Vector a is in program memory while vector c is in data memory. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vcos_dp ( a, i, c, k, n )

float pm *a ; /* Pointer to input vector a */





DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvcos.c included in the distribution diskette provides an example of this func-tion’s use.

A separate example routine was not constructed to show the use of the program mem-ory to data memory data flow.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=cos=



vcosh ( a, i, c, k, n )

NAME Vector Hyperbolic Cosine

DESCRIPTION This function computes the hyperbolic cosine for each element in input vector a, and stores the result in output vector c.All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vcosh ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvcosh.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=cosh=



vcot ( a, i, c, k, n )

NAME Vector Cotangent

DESCRIPTION This function computes the cotangent for each element in input vector a, and stores the result in output vector c. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vcot ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvcot.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=cot=



vcoth ( a, i, c, k, n )

NAME Vector Hyperbolic Cotangent

DESCRIPTION This function computes the hyperbolic cotangent for each element in input vector a, and stores the result in output vector c. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vcoth ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvcoht.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amicoth m 0 1 2 …n 1–, , ,{ }==



vcsc ( a, i, c, k, n )

NAME Vector Cosecant

DESCRIPTION This function computes the cosecant for each element in input vector a, and stores the result in output vector c. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vcsc ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvcsc.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=csc=



vcsch ( a, i, c, k, n )

NAME Vector Hyperbolic Cosecant

DESCRIPTION This function computes the hyperbolic cosecant for each element in input vector a, and stores the result in output vector c. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vcsch ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvcsch.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amicsch m 0 1 2 …n 1–, , ,{ }==



vcube ( a, i, c, k, n )

NAME Vector Cube

DESCRIPTION This function computes the cube of input vector a and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vcube ( a, i, c, k, n )






DOMAIN -3.4 x 1012 to +3.4 x 1012



NOTES The file tvcube.c included in the distribution diskette provides an example of this function’s use.

Cmk A3mi m 0 1 2 …n 1–, , ,{ }==



vdblina ( a, i, c, k, n )

NAME Vector Convert Decibels To Linear Volt Units

DESCRIPTION This function converts each input element in vector a, assumed to be in Decibels units to linear volt units and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vdblina ( a, i, c, k, n )






DOMAIN Arguments greater than 770 will result in an overflow condition as 1038.5 is approxi-mately 3.4 x 1038, the positive upper-limit of the processor.



NOTES The file tvdblina.c included in the distribution diskette provides an example of this function’s use.

Input units to this routine are considered to be in Decibels while output units are con-sidered to be in linear volts.

Cmk 10Ami 0.05•[ ]

m 0 1 2 …n 1–, , ,{ }==



vdblinp ( a, i, c, k, n )

NAME Vector Convert Decibels To Linear Power Units

DESCRIPTION This function converts each input element in vector a, assumed to be in Decibels units to linear power units and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vdblinp ( a, i, c, k, n )






DOMAIN Arguments greater than 385 will result in an overflow condition as 1038.5 is approxi-mately 3.4 x 1038, the positive upper-limit of the processor.



NOTES The file tvdblinp.c included in the distribution diskette provides an example of this function’s use. Input units to this routine are considered to be in decibels while output units are considered to be in linear power units.

Cmk 10Ami 0.10•[ ]

m 0 1 2 …n 1–, , ,{ }==



vdeg ( a, i, c, k, n )

NAME Vector Degrees to Radians

DESCRIPTION This function converts each element in input vector a, assumed to be in degrees, to radians, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vdeg ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvdeg.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk 0.01745329 Ami• m 0 1 2 …n 1–, , ,{ }==



vdist ( a, i, b, j, c, k, n )

NAME Vector Distance

DESCRIPTION This function computes the square root of the sum of squares of input vectors a and b and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vdist ( a, i, b, j, c, k, n )








DOMAIN (a2 + b2) should not exceed the domain of [0.0 to +3.4 x 1038]



NOTES The file tvdist.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami2 Bmj

2+ m 0 1 2 …n 1–, , ,{ }==



vdiv ( a, i, b, j, c, k, n )

NAME Vector Divide

DESCRIPTION This function divides input vector a by input vector b, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vdiv ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038, Bmj cannot = 0.0



NOTES The file tvdiv.c included in the distribution diskette provides an example of this func-tion’s use.

This routine does not contain any code to detect an attempt to divide by zero, as opti-mum performance is the objective.

Use the routine vdivz( ) should you wish to check for the presence of zero in the denominator.

Cmk

Ami

Bmj---------- m 0 1 2 …n 1–, , ,{ }==



vdivz ( a, i, b, j, c, d, l, n )

NAME Vector Divide with Zero Check

DESCRIPTION This function divides input vector a by input vector b, and stores the result in output vector d. If input b is zero then scalar c is stored into output vector d.

ALGORITHM

SYNOPSIS void vdivz ( a, i, b, j, c, d, l, n )









DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvdivz.c included in the distribution diskette provides an example of this func-tion’s use.This routine is slightly slower than vdiv( ) as extra logic is included to check for an attempt to divide by zero.

if Bmj 0.0 then m 0 1 2 …n 1–, , ,{ }=≠

Dml

Ami

Bmj---------- else=

Dml C=



venvlp ( a, i, b, j, c, k, d, l, n )

NAME Vector Envelope

DESCRIPTION This function computes an envelope around input vector c. If an input vector c element either exceeds the upper-limit specified in vector a or falls below the lower-limit spec-ified in vector b, it is copied to output vector d. Otherwise a zero is stored into output vector d.

ALGORITHM if then

else

SYNOPSIS void venvlp ( a, i, b, j, c, k, d, l, n )

float *a ; /* Pointer to upper-limit threshold input vector a */


float *b ; /* Pointer to lower-limit threshold input vector b */







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvenvlp.c included in the distribution diskette provides an example of this function’s use.

The lower-limit threshold b should not exceed the upper-limit threshold a.

Bmj Cmk Ami> >

Dml Cmk=

Dml 0.0=

m 0 1 2 …n, 1 }–, ,{=



veq ( a, i, b, j, c, k, n )

NAME Vector Equal

DESCRIPTION This function compares each element in input vectors a and b for equality. Either 1.0 or 0.0 is stored in output vector c depending on the results of the comparison.

ALGORITHM if then

else

SYNOPSIS void veq ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tveq.c included in the distribution diskette provides an example of this func-tion’s use.

Ami Bmj=

Cmk 0.0=

Cmk 1.0=

m 0 1 2 …n, 1 }–, ,{=



vexp ( a, i, c, k, n )

NAME Vector Natural Exponential

DESCRIPTION This function computes the natural exponential for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vexp ( a, i, c, k, n )






DOMAIN -88.0 to +88.0



NOTES The file tvexp.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk eAmi

m 0 1 2 …n 1–, , ,{ }==



vexp10 ( a, i, c, k, n )

NAME Vector Base 10 Exponential

DESCRIPTION This function computes the base 10 exponential for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vexp10 ( a, i, c, k, n )






DOMAIN -38.0 to +38.0



NOTES The file tvexp10.c included in the distribution diskette provides an example of this function’s use.

Cmk 10Ami

m 0 1 2 …n 1–, , ,{ }==



vexp2 ( a, i, c, k, n )

NAME Vector Base 2 Exponential

DESCRIPTION This function computes the base 2 exponential for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vexp2 ( a, i, c, k, n )






DOMAIN -127.0 to +127.0



NOTES The file tvexp2.c included in the distribution diskette provides an example of this function’s use.

Cmk 2Ami

m 0 1 2 …n 1–, , ,{ }==



vfill ( a, c, k, n )

NAME Vector Fill

DESCRIPTION This function sets all elements of output vector c to a value specified by input scalar a.

ALGORITHM

SYNOPSIS void vfill ( a, c, k, n )

float a ; /* Value of specified scalar fill a */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvfill.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk A= m 0 1 2 …n 1–, , ,{ }=



vfix ( a, i, c, k, n )

NAME Vector Floating-Point To Integer Conversion

DESCRIPTION This function truncates each real element in input vector a to a 32-bit integer and stores the results in integer output vector c. Positive values of a are truncated to the next smallest integer value. Negative values of a are truncated to the next larger inte-ger.

ALGORITHM

SYNOPSIS void vfix ( a, i, c, k, n )






DOMAIN Floating-Point value of Ami must be within -2,147,483,648 to +2,147,483,647



NOTES The file tvfix.c included in the distribution diskette provides an example of this func-tion’s use. The routine is the opposite of vfloat( ). All conversions using vfix( ) trun-cate the fractional portion of the number.

There is no error checking in this routine. For negative values of a outside the domain, a +1 is returned. For positive values of a outside the domain, a -1 is returned.

For faster functions which are related but may have special application rules, see vnfix( ), vpfix( ) and virnd( ).

Cmk fix Ami( )= m 0 1 2 …n 1–, , ,{ }=



vfloat ( a, i, c, k, n )

NAME Vector Integer To Floating-Point Conversion

DESCRIPTION This function converts each integer element in integer input vector a to a 32-bit real and stores the results in floating-point output vector c.

ALGORITHM

SYNOPSIS void vfloat ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvfloat.c included in the distribution diskette provides an example of this function’s use.

This routine is the opposite of vfix( ).

Cmk float Ami( )= m 0 1 2 …n 1–, , ,{ }=



vfloor ( a, i, c, k, n )

NAME Vector Floor (Round Down To Nearest Integral Value)

DESCRIPTION This function computes the largest integral value less than or equal to the floating-point number in input vector a and stores a floating-point representation of this integer result in output vector c.

ALGORITHM

SYNOPSIS void vfloor ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvfloor.c included in the distribution diskette provides an example of this function’s use.

This function is the opposite of vceil( ).

There is no error checking in this routine.

Cmk l earg st integer Ami≤= m 0 1 2 …n 1–, , ,{ }=



vfrac ( a, i, c, k, n )

NAME Vector Truncate To Fraction

DESCRIPTION This function computes the fractional portion of each element in input vector a and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vfrac ( a, i, c, k, n )






DOMAIN Floating-Point value of Ami must be within -2,147,483,648 to +2,147,483,647



NOTES The file tvfrac.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami – float fix Ami( )( )= m 0 1 2 …n 1–, , ,{ }=



vgathr ( a, b, j, c, k, n )

NAME Vector Gather

DESCRIPTION This function copies selected elements of input vector a to output vector c using the integer elements in input vector b to determine which elements are to be copied to out-put vector c.

ALGORITHM

SYNOPSIS void vgathr ( a, b, j, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038, Bmj must be within the valid range of integers.



NOTES The file tvgathr.c included in the distribution diskette provides an example of this function’s use.

This function is the opposite of the scatter function, vscatr( ).

This function performs the same operation as vindex( ) execpt that vgathr( ) expects input vector b to be type integer. Note that only vectors band c may stride.

Cmk ABmj= m 0 1 2 …n, 1 }–, ,{=



vge ( a, i, b, j, c, k, n )

NAME Vector Greater Than Or Equal

DESCRIPTION This function compares each element in input vectors a and b for a greater than or equal condition. Either 1.0 or 0.0 is stored in output vector c depending on the results of the comparison.

ALGORITHM if then

else

SYNOPSIS void vge ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvge.c included in the distribution diskette provides an example of this func-tion’s use.

Ami Bmj≥

Cmk 0.0=

Cmk 1.0=

m 0 1 2 …n, 1 }–, ,{=



vgenp ( a, i, b, j, c, k, n, m )

NAME Vector Generation by Linear Interpolation and Extrapolation

DESCRIPTION This function generates a output vector c for the control points specified in input vec-tor b and the data values specified in input vector a.

ALGORITHM for i = 0 to n - 1 if i <= ( int ) ( b[0] ) c[i] = a[0]

else if i > ( int ) ( b[i-1} )

c[i] = a[i-1]

else

for k = 0 to m - 2

if [ ( int ) ( b[k] ) < i <= ( int ) ( b[k+1] ) ]

c[i] = a[k] +

( i - b[i] ) *

( a[i+1] - a[i] ) /

( b[i+1] - b[i] )

where: n = Number of points to interpolate

m = Number of elements in vectors a[ ] and b [ ]

SYNOPSIS void vgenp ( a, i, b, j, c, k, n, m )



float *b ; /* Pointer to input vector */

int j ; /* Element stride for input vector b */



int n ; /* Number of points to interpolate */

int m; /* Element count in vectors a and b */

DOMAIN -3.4E+38 to 3.4E+38





NOTES The file tvgenp.c included in the distribution tape provides an example of this func-tion’s use.

vgenp ( a, i, b, j, c, k, n, m )



vgt ( a, i, b, j, c, k, n )

NAME Vector Greater Than

DESCRIPTION This function compares each element in input vectors a and b for a greater than condi-tion. Either 1.0 or 0.0 is stored in output vector c depending on the results of the com-parison..

ALGORITHM if then

else

SYNOPSIS void vgt ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvgt.c included in the distribution diskette provides an example of this func-tion’s use.

Ami Bmj>

Cmk 0.0=

Cmk 1.0=

m 0 1 2 …n, 1 }–, ,{=



vhist ( a, i, c, n, vmin, vmax, nbin, oor )

NAME Vector Histogram

DESCRIPTION This function computes an accumulating histogram of vector elements found in inputreal vector a. The output of the histogram is placed in output integer vector c. The out-of-range count is output into integer oor.

ALGORITHM

SYNOPSIS void vhist ( a, i, c, n, vmin, vmax, nbin, oor)





float vmin ; /* Minimum range of histogram */

float vmax ; /* Maximum range of histogram */

int nbin ; /* Number of desired bins in histogram */

int *oor ; /* Pointer to out-of-range count */

DOMAIN -3.4E+38 to 3.4E+38


oor 0=

delta vmax vmin–( ) nbin⁄ 1–=

bin int a k[ ] vmin–( ) delta⁄( )=

if bin 0≥( )and bin nbin-1<( )then

++c k[ ]

else

++ *oor( )

m 0 1 2 … n, 1–, , ,{ }=



EXECUTION TIME 47 + 15*N (Maximum)

47 + 8*N (Minimum)

NOTES The file tvhist.c included in the distribution tape provides an example of this func-tion’s use.

"Bins" numbered 0 through nbin-1 are defined by dividing the interval from vmin tovmax into nbin equal segments. The bins are arranged so that the minimum value of 0corresponds to vmin and the maximum value of bin nbin-1 corresponds to vmax.

For each element in input vector a, the bin whose boundaries contain the element'svalue is determined, and that bin's histogram counter is incremented. An out-of-rangecounter (oor) is maintained to indicate the number of elements which were not con-tained within any bin boundaries.

vhist( ) does not clear the histogram results in output vector c. Instead the histogramresults are added to output vector c, enabling repeated calls to vhist( ) to perform anaccumulating histogram. Function vclr( ) may be used to initially clear the histogramresults vector.

Also note the following special considerations:

Arguments nbin, n, and vmax should not vary between accumulating histogram calls.

The size of the bin specified by the operator cannot be less than 1. There is no errorchecking for this type of error within the code.

The stride of output vector c is always one.

The sum of the elements within the bin plus the number of out-of-range (oor) elementsshould always equal the number of elements supplied in nbin by the operator.

Typical main loop execution time is between 8 and 15 cycles.

Since output vector c is an integer, then the number of floating-point elements pro-cessed must be less then 232.

vhist ( a, i, c, n, vmin, vmax, nbin, oor )



vhisti ( a, i, c, n, vmin, vmax, nbin, oor )

NAME Vector Integer Histogram

DESCRIPTION This function computes an accumulating integer histogram on the vector elementsfound in input integer vector a. The output of the histogram is placed in output integervector c. The out-of-range count is output into integer oor.

ALGORITHM

SYNOPSIS void vhisti ( a, i, c, n, vmin, vmax, nbin, oor )




int n ; /* Count of integer elements */

int vmin ; /* Minimum range of histogram */

int vmax ; /* Maximum range of histogram */

int nbin ; /* Number of desired bins in histogram */

int *oor ; /* Pointer to out-of-range count */

DOMAIN -2,147,483,648 to 2,147,483,647


oor 0=

delta vmax vmin–( ) nbin⁄ 1–=

bin int a k[ ] vmin–( ) delta⁄( )=

if bin 0≥( )and bin nbin-1<( )then

++c k[ ]

else

++ *oor( )

m 0 1 2 … n, 1–, , ,{ }=




NOTES The file tvhisti.c included in the distribution tape provides an example of this func-tion’s use.

"Bins" numbered 0 through nbin-1 are defined by dividing the interval from vmin tovmax into nbin equal segments. The bins are arranged so that the minimum value of 0corresponds to vmin and the maximum value of bin nbin-1 corresponds to vmax.

For each element in input vector a, the bin whose boundaries contain the element'svalue is determined, and that bin's histogram counter is incremented. An out-of-rangecounter (oor) is maintained to indicate the number of elements which were not con-tained within any bin boundaries.

vhisti( ) does not clear the histogram results in output vector c. Instead the histogramresults are added to output vector c, enabling repeated calls to vhist( ) to perform anaccumulating histogram. Function vclr( ) may be used to initially clear the histogramresults vector.

Also note the following special considerations:

Arguments nbin, n, vmin and vmax should not vary between accumulating histogramcalls.

The size of the bin specified by the operator cannot be less than 1. There is no errorchecking for this type of error within the code.

The stride of output vector c is always one.

The sum of the elements within the bin plus the number of out-of-range (oor) elementsshould always equal the number of elements supplied in nbin by the operator.

Typical main loop execution time is between 9 and 16 cycles.

vhisti ( a, i, c, n, vmin, vmax, nbin, oor )



viclip ( a, i, b, c, d, l, n )

NAME Vector Inverted Clip

DESCRIPTION This function computes an inverted clip using a lower-threshold scalar b and an upper-threshold scalar c to clip each element in input vector a and stores the results in output vector d.

ALGORITHM

SYNOPSIS void viclip ( a, i, b, c, d, l, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tviclip.c included in the distribution diskette provides an example of this func-tion’s use.

if B Ami C then m 0 1 2 …n 1–, , ,{ }=< <

if Ami 0 then >

Dml C=

else

Dml B=

else

Dml Ami=



vimag ( a, i, c, k, n )

NAME Vector Copy Imaginary Parts

DESCRIPTION This function creates a real output vector c based on the imaginary portion of complex input vector a.

ALGORITHM

SYNOPSIS void vimag ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvimag.c included in the distribution diskette provides an example of this function’s use.

Cmk Im Ami{ } m 0 1 2 …n 1–, , ,{ }==



vindex ( a, b, j, c, k, n )

NAME Vector Index

DESCRIPTION This function copies selected elements of input vector a to output vector c using the integer elements in input vector b to determine which elements are to be copied to out-put vector c. The values within vector b are assumed to be type real and are converted to type integer.

ALGORITHM

SYNOPSIS void vindex ( a, b, j, c, k, n )







DOMAIN 0 to +3.4 x 1038, Bmj must be within the valid range of integers before conversion.



NOTES The file tvindex.c included in the distribution diskette provides an example of this function’s use.

This function is the opposite of the scatter function, vscatr( ). This function performs the same operation as vgathr( ) execpt that vgathr( ) expects input vector b to be type integer. Note that only vectors b and c may stride.

Cmk Aint Bmj ][= m 0 1 2 … n 1–, , , ,{ }=



vintb ( a, i, b, j, c, d, l, n )

NAME Vector Linear Interpolate

DESCRIPTION Linearly interpolates between elements of two input vectors a and b according to a floating-point input scalar c and stores the result to output vector d.

ALGORITHM

SYNOPSIS void vintb ( a, i, b, j, c, d, l, n )





float *c ; /* Pointer to input scalar c */

float *d; /* Pointer to input vector d */

int l ; /* Element stride for vector d */

int n ; /* Element count

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvintb.c included in the distribution tape provides an example of this func-tion’s use.

Dml Ami C Bmj Ami–( ) • m 0 1 2 … n, 1–, , ,{ }=+=



vipimul ( a, i, c, k, n )

NAME Vector Multiply By

DESCRIPTION This function multiplies the contents of input vector a by the constant . The results are placed in output vector c.

ALGORITHM

SYNOPSIS void vipmul ( a, i, c, k, n )






DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvipimul.c included in the distribution tape provides an example of this func-tion’s use.

This routine computes as the sum of two number toachieve 40-bit precision. To use this routine as a scalar multiplier where the number of points to be computed is one (n = 1), set the stride of the input vector a to 0.

1 π⁄

1 π⁄

Cmk Ami1π---•=

1 π⁄



virnd ( a, i, c, k, n )

NAME Vector Round Floating Point to Integer

DESCRIPTION This function rounds the floating point value of each element in input vector a and stores the results in output vector c. The floating point value is converted to integer form and stored in output vector c.

ALGORITHM

SYNOPSIS void virnd ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvirnd.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk round Ami( )= m 0 1 2 …n 1–, , ,{ }=



vle ( a, i, b, j, c, k, n )

NAME Vector Less Than Or Equal

DESCRIPTION This function compares each element in input vectors a and b for a less than or equal-condition. Either 1.0 or 0.0 is stored in output vector c depending on the results of the comparison.

ALGORITHM if then

else

SYNOPSIS void vle ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvle.c included in the distribution diskette provides an example of this func-tion’s use.

Ami Bmj≤

Cmk 0.0=

Cmk 1.0=

m 0 1 2 …n, 1 }–, ,{=



vlim ( a, i, b, c, d, l, n )

NAME Vector Limit

DESCRIPTION This function compares each element of input vector a to input scalar b. For elements of vector a less than or equal to scalar b the negated value of input scalar c is stored in the corresponding element of output vector d. Otherwise, the value of input scalar c is stored into the corresponding element of d.

ALGORITHM

SYNOPSIS void vlim ( a, i, b, c, d, l, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvlim.c included in the distribution diskette provides an example of this func-tion’s use.

if Ami B then Dml C– m 0 1 2 …n 1–, , ,{ }==<

else Dml C=



vlindba ( a, i, c, k, n )

NAME Vector Convert Linear Volt Units To Decibels

DESCRIPTION This function converts each input element invector a, assumed to be in linear volt units to Decibels and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vlindba ( a, i, c, k, n )






DOMAIN a > 0 to +3.4 x 1038



NOTES The file tvlindba.c included in the distribution diskette provides an example of this function’s use.

Input units to this routine are considered to be in linear volt units while output units are considered to be in decibels.

Cmk 20 10 Ami( )log• m 0 1 2 …n 1–, , ,{ }==



vlindbp ( a, i, c, k, n )

NAME Vector Convert Linear Power Units To Decibels

DESCRIPTION This function converts each input element in vector a, assumed to be in linear power units to Decibels and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vlindbp ( a, i, c, k, n )






DOMAIN a > 0 to + 3.4 x 1038



NOTES The file tvlindbp.c included in the distribution diskette provides an example of this function’s use.

Input units to this routine are considered to be in linear power units while output units are considered to be in decibels.

Cmk 10 10 Ami( )log• m 0 1 2 …n 1–, , ,{ }==



vlmerg ( a, i, b, j, c, k, d, l, n )

NAME Vector Logical Merge

DESCRIPTION This function logically merges two input vectors a and b based on the values con-tained within gating vector c. The results are stored in output vector d. Either vector a or vector b’s element is stored into output vector d based on the value of the corre-sponding element in vector c.

ALGORITHM if then

else

SYNOPSIS void vlmerg ( a, i, b, j, c, k, d, l, n )










DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvlmerg.c included in the distribution diskette provides an example of this function’s use.

C 0.0≠

Dml Ami=

Dml Bmj=

m 0 1 2 …n, 1 }–, ,{=



vlog ( a, i, c, k, n )

NAME Vector Natural Logarithm

DESCRIPTION This function computes the natural logarithm for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vlog ( a, i, c, k, n )






DOMAIN A > 0.0 to +3.4 x1038



NOTES The file tvlog.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amilog m 0 1 2 …n 1–, , ,{ }==



vlog10 ( a, i, c, k, n )

NAME Vector Base 10 Logarithm

DESCRIPTION This function computes the base 10 logarithm for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vlog10 ( a, i, c, k, n )






DOMAIN A > 0.0 to +3.4 x1038



NOTES The file tvlog10.c included in the distribution diskette provides an example of this function’s use.

Cmk 10Amilog m 0 1 2 …n 1–, , ,{ }==



vlog2 ( a, i, c, k, n )

NAME Vector Base 2 Logarithm

DESCRIPTION This function computes the base 2 logarithm for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vlog2 ( a, i, c, k, n )






DOMAIN A > 0.0 to +3.4 x1038



NOTES The file tvlog2.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk 2Amilog m 0 1 2 …n 1–, , ,{ }==



vlshift ( a, i, c, k, n, h )

NAME Vector Logical Shift

DESCRIPTION This function performs a logical shift of h bits on each 32-bit element in input vector a and stores the results in output vector c. If h is negative the bits are right-shifted with a zero fill into the most significant bit (MSB). If h is positive the bits are left-shifted with a sign fill into the least significant bit (LSB).

ALGORITHM if then

Zero Fill MSB

else

Sign Fill LSB

SYNOPSIS void vlshift ( a, i, c, k, n, h )






int h ; /* Length of logical shift */

DOMAIN H must be between -32 to +32.

ACCURACY 32-bits


NOTES The file tvlshift.c included in the distribution diskette provides an example of this function’s use.

H 0<

Cmk Ami H»=

Cmk Ami H«=

m 0 1 2…n 1 }–, ,{=



vlt ( a, i, b, j, c, k, n )

NAME Vector Less Than

DESCRIPTION This function compares each element in input vectors a and b for a less than condition. Either 1.0 or 0.0 is stored in output vector c depending on the results of the compari-son.

ALGORITHM if then

else

SYNOPSIS void vlt ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvlt.c included in the distribution diskette provides an example of this func-tion’s use.

Ami Bmj<

Cmk 0.0=

Cmk 1.0=

m 0 1 2…n 1 }–, ,{=



vlthr ( a, i, b, c, k, n )

NAME Vector Lower Threshold

DESCRIPTION This function computes an output vector c based on a comparison of input vector a and threshold scalar b.

ALGORITHM if then

else

SYNOPSIS void vlthr ( a, i, b, c, k, n )



float b ; /* Value of input threshold scalar b */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvlthr.c included in the distribution diskette provides an example of this func-tion’s use.

Ami Bmj>

Cmk Ami=

Cmk B=

m 0 1 2…n 1 }–, ,{=



vma ( a, i, b, j, c, k, d, l, n )

NAME Vector Multiply and Add

DESCRIPTION This function multiplies two input vectors a and b, then adds the product to input vec-tor c and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vma ( a, i, b, j, c, k, d, l, n )










DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvma.c included in the distribution diskette provides an example of this func-tion’s use.

Dml Ami Bmj•( ) Cmk m 0 1 2 …n 1–, , ,{ }=+=



vmags ( a, i, b, j, c, k, n )

NAME Vector Magnitude Squared

DESCRIPTION This function computes the magnitude squared of real input vectors a and b and stores the result in real output vector c.

ALGORITHM

SYNOPSIS void vmags ( a, i, b, j, c, k, n )





float dm *c ; /* Pointer to real output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmags.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami( )2Bmj( )2

m 0 1 2 …n 1–, , ,{ }=+=



vmax ( a, i, b, j, c, k, n )

NAME Vector Maximum

DESCRIPTION This function computes the maximum of each element of input vectors a and b and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vmax ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmax.c included in the distribution diskette provides an example of this func-tion’s use.

if Ami Bmj then m 0 1 2 …n 1–, , ,{ }=≥

Cmk Ami else=

Cmk Bmj=



vmax3 ( a, i, b, j, c, k, d, l, n )

NAME Vector Maximum of 3 Vectors

DESCRIPTION This function performs a vector maximum operation on input vectors a, b, and c. The results are stored in output vector d. An element from vector a is compared to corre-sponding element in vectors b and c. The maximum element is stored to output vector d.

ALGORITHM

SYNOPSIS void vmax3 ( a, i, b, j, c, k, d, l, n )








int l ; /* Address stride in words for output vector c */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmax3.c included in the distribution diskette provides an example of this function’s use.

Dml Ami=

if Dml Bmj then <

Dml Bmj =

if Dml Cmk then<

Dml Cmk =

m 0 1 2 …n 1–, , ,{ }=



vmaxmg ( a, i, b, j, c, k, n )

NAME Vector Maximum Magnitude

DESCRIPTION This function computes the magnitude of each element of input vectors a and b and stores the maximum magnitude in output vector c.

ALGORITHM

SYNOPSIS void vmaxmg ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmaxmg.c included in the distribution diskette provides an example of this function’s use.

if Ami Bmj then m 0 1 2 …n 1–, , ,{ }=≥

Cmk Ami else=

Cmk Bmj=



vmin ( a, i, b, j, c, k, n )

NAME Vector Minimum

DESCRIPTION This function computes the minima of each element of input vectors a and b and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vmin ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmin.c included in the distribution diskette provides an example of this func-tion’s use.

if Ami Bmj then m 0 1 2 …n 1–, , ,{ }=<

Cmk Ami else=

Cmk Bmj=



vmin3 ( a, i, b, j, c, k, d, l, n )

NAME Vector Minimum of 3 Vectors

DESCRIPTION This function performs a vector minimum operation on input vectors a, b, and c. The results are stored in output vector d. An element from vector a is compared to corre-sponding element in vectors b and c. The minimum element is stored to output vector d.

ALGORITHM

SYNOPSIS void vmin3 ( a, i, b, j, c, k, d, l, n )








int l ; /* Address stride in words for output vector c */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmin3.c included in the distribution diskette provides an example of this function’s use.

Dml Ami=

if Dml Bmj then >

Dml Bmj =

if Dml Cmk then>

Dml Cmk =

m 0 1 2 …n 1–, , ,{ }=



vminmg ( a, i, b, j, c, k, n )

NAME Vector Minimum Magnitude

DESCRIPTION This function computes the magnitude of each element of input vectors a and b and stores the minimum magnitude in output vector c.

ALGORITHM

SYNOPSIS void vminmg ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvminmg.c included in the distribution diskette provides an example of this function’s use.

if Ami Bmj then m 0 1 2 …n 1–, , ,{ }=≤

Cmk Ami else=

Cmk Bmj=



vmma ( a, i, b, j, c, k, d, l, e, h, n )

NAME Vector Multiply, Multiply, and Add

DESCRIPTION This function multiplies two input vectors a and b, then multiplies another two input vectors c and d, then adds the products and stores the result in output vector e.

ALGORITHM

SYNOPSIS void vmma ( a, i, b, j, c, k, d, l, e, h, n )












DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmma.c included in the distribution diskette provides an example of this function’s use.

Emh Ami Bmj•( ) Cmk Dml•( )+=

m 0 1 2 …n 1–, , ,{ }=



vmms ( a, i, b, j, c, k, d, l, e, h, n )

NAME Vector Multiply, Multiply, and Subtract

DESCRIPTION This function multiplies two input vectors a and b, then multiplies another two input vectors c and d, then subtracts the products and stores the result in output vector e.

ALGORITHM

SYNOPSIS void vmms ( a, i, b, j, c, k, d, l, e, h, n )












DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmms.c included in the distribution diskette provides an example of this function’s use.

Emh Ami Bmj•( ) Cmk Dml•( )–=

m 0 1 2 …n 1–, , ,{ }=



vmmul ( a, b, x, y, c )

NAME Vector Matrix Multiply

DESCRIPTION This function multiplies input vector a by input matrix b, and stores the result in out-put vector c. The length of vector a is supplied in input integer scalar x and the dimen-sions of matrix b is supplied in input integer scalars x and y. The resulting output vector c is of length y.

ALGORITHM

SYNOPSIS void vmmul ( a, b, x, y, c )


float *b ; /* Pointer to input matrix b */

int x ; /* Number of rows in matrix b */

/* Number of elements in vector a */

int y ; /* Number of columns in matrix b */


DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 29 + ( 6 + 2 ( X-1 ) ) * Y cycles

NOTES The file tvmmul.c included in the distribution diskette provides an example of this function’s use.

Vector a times the first column of matrix b[ ] [ ] is the first element of vector c. Vector a times the second column of matrix b[ ] [ ] is the second elements of vector c, etc.

Cm Ak Bk m,•

k 0=

x 1–

∑=

m 0 1 2 …y 1–, , ,{ }=



vmov ( a, i, c, k, n )

NAME Vector Move

DESCRIPTION This function copies all elements of input vector a to output vector c.

ALGORITHM

SYNOPSIS void vmov ( a, i, c, k, n )

float a ; /* Pointer to input vector a */





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmov.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami= m 0 1 2 …n 1–, , ,{ }=



vmov_dp ( a, i, c, k, n )

NAME Vector Move Data Memory to Program Memory

DESCRIPTION This function copies all elements of input vector a to output vector c. Vector a is in data memory while vector c is in program memory.

ALGORITHM

SYNOPSIS void vmov_dp ( a, i, c, k, n )



float pm *c ; /* Pointer to output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmovdp.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami= m 0 1 2 …n 1–, , ,{ }=



vmov_pd ( a, i, c, k, n )

NAME Vector Move Program Memory to Data Memory

DESCRIPTION This function copies all elements of input vector a to output vector c. Vector a is in program memory while vector c is in data memory.

ALGORITHM

SYNOPSIS void vmov_pd ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038




Cmk Ami= m 0 1 2 …n 1–, , ,{ }=



vmov_pp ( a, i, c, k, n )

NAME Vector Move Program Memory to Program Memory

DESCRIPTION This function copies all elements of input vector a to output vector c. Vector a is in program memory while vector c is in program memory.

ALGORITHM

SYNOPSIS void vmov_pd ( a, i, c, k, n )



float pm *c ; /* Pointer to output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038




A separate example routine was not constructed to show the use of program memory to program memory data flow.

Cmk Ami= m 0 1 2 …n 1–, , ,{ }=



vms ( a, i, b, j, c, k, d, l, n )

NAME Vector Multiply and Subtract

DESCRIPTION This function multiplies two input vectors a and b, then subtracts input vector c from the product and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vms ( a, i, b, j, c, k, d, l, n )










DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvms.c included in the distribution diskette provides an example of this func-tion’s use.

Dml Ami Bmj•( ) Cmk m 0 1 2 …n 1–, , ,{ }=–=



vmsca ( a, i, b, j, c, d, l, n )

NAME Vector Multiply and Scalar Add

DESCRIPTION This function multiplies two input vectors a and b, then adds input scalar c to the prod-uct and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vmsca ( a, i, b, j, c, d, l, n )









DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmsca.c included in the distribution diskette provides an example of this function’s use.

Dml Ami Bmj•( ) C m 0 1 2 …n, 1–, ,{ }=+=



vmscs ( a, i, b, j, c, d, l, n )

NAME Vector Multiply and Scalar Subtract

DESCRIPTION This function multiplies two input vectors a and b, then subtracts input scalar c from the product and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vmscs ( a, i, b, j, c, d, l, n )









DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmscs.c included in the distribution diskette provides an example of this function’s use.

Dml Ami Bmj•( ) C– m 0 1 2 …n, 1–, ,{ }==



vmul ( a, i, b, j, c, k, n )

NAME Vector Multiply

DESCRIPTION This function multiplies two input vectors a and b, and stores the result in output vec-tor c.

ALGORITHM

SYNOPSIS void vmul ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvmul.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami Bmj• m 0 1 2 …n, 1–, ,{ }==



vnabs ( a, i, c, k, n )

NAME Vector Negative Absolute Value

DESCRIPTION This function computes the negated absolute values of all elements in input vector a and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vnabs ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvnabs.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami–= m 0 1 2 …n 1–, , ,{ }=



vnand ( a, i, b, j, c, k, n )

NAME Vector Not And

DESCRIPTION This function performs a 32-bit logical AND of each element in input vector a with each element in input vector b. Next, the logical complement (NOT) of the AND is performed. The results are then stored in output vector c.

ALGORITHM

SYNOPSIS void vnand ( a, i, b, j, c, k, n )









ACCURACY 32-bits


NOTES The file tvnand.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami NAND Bmj( )= m 0 1 2…n 1 }–, ,{=



vnco ( f, amp, c, k, n )

NAME Vector Compute NCO Values

DESCRIPTION This function computes the nco ( numerically controlled oscillator ) values for a given bin, amplitude, and number of stages. The values are stored in output vector c[ ]. The nco values are computed for points. The cosine function is used to compute each nco function.

ALGORITHM Loop j = 0 to 2^n-1 nco = f * j nco = nco masked with 2^n-1 nco = nco * 2 * Pi / 2^n nco = amp * cos ( nco ) / 2^nEnd Loop

SYNOPSIS void vnco ( f, amp, c, k, n )

int f ; /* Input Bin Selection */

float dm amp ; /* Input Amplitude */

float dm *c ; /* Pointer to output vector C[ ] */

int k ; /* Vecotr C[ ] element stride */

int n ; /* Number of Stages (ie. 2^n) */

DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 49 + 29 * 2^N cycles

NOTES The file tvnco.c included in the distribution diskette provides an example of this func-tion’s use.

2n 1–



vncoarb ( table, tabsize, phase, acc, output, n )

NAME Vector Compute NCO Values

DESCRIPTION This function computes a series of output NCO (numerically controlled oscillator) values in output vector output based on a scalar input value tabsize, an input vector table, and a input scalar phase argument. The number of points to compute is based on the input parameter supplied in scalar n.

ALGORITHM

SYNOPSIS void vncoarb ( table, tabsize, phase, acc, output, n )

float dm *table ; /* Pointer to input nco vector table */

int tabsize ; /* Input Table Size */

int phase ; /* Input Phase Step */

int acc ; /* Input/Output Accumulator Index */

float dm *output ; /* Pointer to output vector */

int n ; /* Number of points to compute */

DOMAIN -3.4 x 1038 to +3.4 x 1038


output tabletabsize phase•=




NOTES The file tvncoarb.c included in the distribution diskette provides an example of this function’s use.

It is assumed that the input accumulator is initialized prior to this function being called. Otherwise, successive calls allows this function to generate a continuous nco table.

vncoarb( ) extracts the nco values for a given input table, phase, accumulator index

The accumulator index acc is a starting offset into the input nco vector table[ ]. The phase step phase is the added offset to the accumulator index up to n points. The table size tabsize is the size of the input nco vector table[ ]. It is used to wrap the accumula-tor index acc to the beginning of the input nco vector table[ ] if the accumulator index acc is greater than tabsize.

for i = 0 to n

if acc >= tabsize then

acc = acc - tabsize

end if

output[i] = table[acc]

acc = acc + phase

end loop

vncoarb ( table, tabsize, phase, acc, output, n )



vne ( a, i, b, j, c, k, n )

NAME Vector Not Equal

DESCRIPTION This function compares each element in input vectors a and b for non-equality. Either 1.0 or 0.0 is stored in output vector c depending on the results of the comparison.

ALGORITHM if then

else

SYNOPSIS void vne ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvne.c included in the distribution diskette provides an example of this func-tion’s use.

Ami Bmj≠

Cmk 0.0=

Cmk 1.0=

m 0 1 2 …n 1–, , ,{ }=



vneg ( a, i, c, k, n )

NAME Vector Negate

DESCRIPTION This function computes the negated values all elements in input vector a and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vneg ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvneg.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk A– mi= m 0 1 2 …n 1–, , ,{ }=



vnfix ( a, i, c, k, n )

NAME Vector Floating-Point To Integer Conversion for Negative Arguments

DESCRIPTION This function converts each real element in input vector a to the next smallest 32-bit integer and stores the results in integer output vector c. The algorithm assumes that only negative numbers are supplied in input vector a, and performs no error checking for positive numbers.

ALGORITHM

SYNOPSIS void vnfix ( a, i, c, k, n )






DOMAIN Floating-Point value of Ami must be within 0 to +2,147,483,648



NOTES The file tnfix.c included in the distribution diskette provides an example of this func-tion’s use. The routine is the opposite of vfloat( ). All conversions using vnfix( ) trun-cate the fractional portion of the negative number.

Cmk fix negative Ami( )= m 0 1 2 …n 1–, , ,{ }=



vnmerg ( a, i, b, j, c, k, d, l, n )

NAME Vector Negative Merge


ALGORITHM if then

else

SYNOPSIS void vnmerg ( a, i, b, j, c, k, d, l, n )










DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvnmerg.c included in the distribution diskette provides an example of this function’s use.

Cmk 0.0<

Dml Bmj=

Dml Ami=

m 0 1 2 …n 1–, , ,{ }=



vnor ( a, i, b, j, c, k, n )

NAME Vector Not Or

DESCRIPTION This function performs a 32-bit logical OR of each element in input vector a with each element in input vector b. Next the logical complement (NOT) of the OR is per-formed. The results are then stored in output vector c.

ALGORITHM

SYNOPSIS void vnor ( a, i, b, j, c, k, n )









ACCURACY 32-bits


NOTES The file tvnor.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk NOT Ami OR Bmj( )= m 0 1 2…n 1 }–, ,{=



vnot ( a, i, c, k, n )

NAME Vector Not (Complement)

DESCRIPTION This function performs a bitwise logical complement (NOT) on the elements of input vector a. The complemented results are then stored in output vector c.

ALGORITHM

SYNOPSIS void vnot ( a, i, c, k, n )







ACCURACY 32-bits


NOTES The file tvnot.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk NOT A= mi m 0 1 2…n 1 }–, ,{=



vnxor ( a, i, b, j, c, k, n )

NAME Vector Not Exclusive Bitwise Or

DESCRIPTION This function performs a bitwise exclusive OR of each element in input vector a with each element in input vector b. Next, the logical complement (NOT) of the bitwise OR is performed. The results are then stored in output vector c.

ALGORITHM

SYNOPSIS void vnxor ( a, i, b, j, c, k, n )









ACCURACY 32-bits


NOTES The file tvnxor.c included in the distribution diskette provides an example of this function’s use.

Cmk NOT Ami XOR Bmj( ) m 0 1 2…n 1 }–, ,{==



vor ( a, i, b, j, c, k, n )

NAME Vector Or

DESCRIPTION This function performs a 32-bit logical OR of each element in input vector a with each element in input vector b and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vor ( a, i, b, j, c, k, n )









ACCURACY 32-bits


NOTES The file tvor.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami OR Bmj( )= m 0 1 2…n 1 }–, ,{=



vpack ( a, i, b, j, fill, n )

NAME Pack a Floating-Point Vector of Ones and Zeros Into 32-Bit Words Arguments

DESCRIPTION This function reads an input vector a, which is assumed to contains either floating-point 1s or 0s. If a 1 is read, a bit is set in output vector b. Likewise, if a 0 is read a bit is cleared in output vector b. The next value is read and the next corresponding bit is set/cleared.

The order in which the bits are set/cleared is determined by the fill flag.

ALGORITHM

SYNOPSIS void vpack ( a, i, b, j, fill, n )



int *b ; /* Pointer to output vector c */

int j ; /* Address stride in words for output vector c */

int fill ; /* Fill direction flag ( Either 1 or -1 ) */


DOMAIN -3.4 x 1038 to +3.4 x 1038


EXECUTION TIME 43 +65*(Integer) (N/32) + 2*(M Mod 32) cycles

NOTES The file tvpack.c included in the distribution diskette provides an example of this function’s use.

32 elements are packed into a 32-bit word. For non-multiples of 32, the remaining ele-ments are packed into a 32-bit work and is zero filled.

The vector size of b should be at least (N/32). If N is not a multiple of 32 then the vec-tor size of b should be (N/32 + 1).

The order in which the bits are set/cleared is determined by the fill flag. If the fill flag is set = 1, the bits are packed from LSB to MSB. If the fill flag is set = -1, the bits are packed from MSB to LSB.

Bmj Ami= m 0 1 2 …n 1–, , ,{ }=



vpcki ( a, i, b, j, fill, n )

NAME Pack an Integer Vector of Ones and Zeros Into 32-Bit Words Arguments

DESCRIPTION This function reads an input vector a, which is assumed to contains either integer 1s or 0s. If a 1 is read, a bit is set in output vector b. Likewise, if a 0 is read a bit is cleared in output vector b. The next value is read and the next corresponding bit is set/cleared.

The order in which the bits are set/cleared is determined by the fill flag.

ALGORITHM

SYNOPSIS void vpcki ( a, i, b, j, fill, n )



int *b ; /* Pointer to output vector c */

int j ; /* Address stride in words for output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvpack.c included in the distribution diskette provides an example of this function’s use.



The order in which the bits are set/cleared is determined by the fill flag. If the fill flag is set = 1, the bits are packed from LSB to MSB. If the fill flag is set = -1, the bits are packed from MSB to LSB.

Bmj Ami= m 0 1 2 …n 1–, , ,{ }=



vpfix ( a, i, c, k, n )

NAME Vector Floating-Point To Integer Conversion for Positive Arguments

DESCRIPTION This function converts each real element in input vector a to the next smallest 32-bit integer and stores the results in integer output vector c.

The algorithm assumes that only positive numbers are supplied in input vector a, and performs no error checking for negative numbers.

ALGORITHM

SYNOPSIS void vpfix ( a, i, c, k, n )






DOMAIN Floating-Point value of Ami must be within 0 to +2,147,483,647



NOTES The file tvpfix.c included in the distribution diskette provides an example of this func-tion’s use. The routine is the opposite of vfloat( ). All conversions using vpfix( ) trun-cate the fractional portion of the positive number.

Cmk fix positive Ami( )= m 0 1 2 …n 1–, , ,{ }=



vphase ( re, i, im, j, ph, k, n )

NAME Vector Phase

DESCRIPTION This function computes the phase of the elements of real input vector re and imagi-nary input vector im, and stores the result in real output vector ph.

ALGORITHM Real Imaginary Ph [ ] Quadrant

>0 >0 Phi 1

<0 >0 PI - Phi 2

<0 <0 PI + Phi 3

>0 <0 2*PI - Phi 4

where: if ( |real| > |imaginary| )

Phi = atan ( |real| / |imaginary| )

else

Phi = PI/2 - atan ( |imaginary| / |real| )

SYNOPSIS void vphase ( re, i, im, j, ph, k, n )

float *re ; /* Pointer to complex input vector re */

int i ; /* Address stride in words for input vector re */

float *im ; /* Pointer to real input vector im */

int j ; /* Address stride in words for input vector im */

float *ph ; /* Pointer to real output vector ph */

int k ; /* Address stride in words for output vector ph */


DOMAIN -3.4 x 1038 to +3.4 x 1038, the real or imaginary component cannot equal zero or a divide by zero error will occur.



NOTES The file tvphase.c included in the distribution diskette provides an example of this function’s use.



vpimul ( a, i, c, k, n )

NAME Vector Multiply By

DESCRIPTION This function multiplies the contents of input vector a by the constant . The results are placed in output vector c.

ALGORITHM

SYNOPSIS void vpimul ( a, i, c, k, n )






DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvpimul.c included in the distribution tape provides an example of this func-tion’s use.

This routine computes as the sum of two number toachieve 40-bit precision. To use this routine as a scalar multiplier where the number of points computed is one (n = 1), set the stride of the input vector a to 0.

π

π

Cmk Ami π • m 0 1 2 …n 1–, , ,{ }==

π



vpmerg ( a, i, b, j, c, k, d, l, n )

NAME Vector Positive Merge


ALGORITHM if then

else

SYNOPSIS void vpmerg ( a, i, b, j, c, k, d, l, n )










DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvpmerg.c included in the distribution diskette provides an example of this function’s use.

Cmk 0.0≥

Dml Bmj=

Dml Ami=

m 0 1 2 …n, 1 }–, ,{=



vpoly ( a, as, b, bs, c, cs, n, p )

NAME Vector Polynomial Evaluation

DESCRIPTION Evaluates values of a p-order polynomial for vector b using coefficients in vector a. The number of coefficients in vector a must be p+1 since p+1 coefficients are required for a p-order polynomial.

ALGORITHM

SYNOPSIS void vpoly ( a, as, b, bs, c, cs, n, p )

float *a ; /* Pointer to coefficient vector a */

int as ; /* Element stride for vector a */


int bs ; /* Element stride for vector b */


int cs ; /* Element stride for vector c */


int p ; /* Polynomial order (degree) */

DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 46 + (5+2*(P-1))*N

NOTES The file tvpoly.c included in the distribution tape provides an example of this func-tion’s use.

Input vector a must have a minimum length of .Note that this is an index from zero to p.

Input vector b must have a minimum length of

Output vector c must have a minimum length of .

n must be a minimum of 1.

Ccs m• Aai m• Bbs m•p i– •

i 0=

p

∑=

m 0 1 2 … n, 1–, , ,{ }=

as p 1+( )•

bs n•

cs n•



vpow ( a, i, b, c, k, n )

NAME Vector Power

DESCRIPTION This function computes the base b exponential for each element in vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vpow ( a, i, b, c, k, n )



float b ; /* Value of the base b */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tpow.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk BAmi

m 0 1 2 …n 1–, , ,{ }==



vprn ( a, b, c, d, l, n )

NAME Vector PRN Generator

DESCRIPTION Generates a series of binary values representing the Pseudo-Random Number sequence for the polynomial specified. The sequence is stored in output vector d.

ALGORITHM

SYNOPSIS void vprn ( a, b, c, d, l, n )

int *a ; /* Pointer to input seed a */

int b ; /* Pointer to register length b */

int c ; /* Pointer to polynomial value */

int *d ; /* Pointer to output vector d */

int l ; /* Element stride for vector d */


DOMAIN 0 to 232 - 1 Unsigned (= 4,294,967,295)


EXECUTION TIME TBS

NOTES The file tvprn.c included in the distribution tape provides an example of this func-tion’s use.

Note the domain range on the input.

Bitm 1– of Ami Bitm n– Bit0⊕=

Ami Ami>>1=

Cmk LSB of Ami=

m 0 1 2 …n 1–, , ,{ }=



vpspec ( data, pspec, j, n )

NAME Vector Power Spectra

DESCRIPTION This function computes the power spectra from the input vector data [ ] and places the results in output vector pspec [ ]. The number of elements processed is n.

ALGORITHM

SYNOPSIS void vpspec ( data, pspec, j, n )

float dm *data ; /* Pointer to input vector data [ ] */

float dm *pspec ; /* Pointer to output vector pspec */

int j ; /* Element stride for vector pspec [ ] */


DOMAIN -3.4 x 1038 to +3.4 x 103


pspec 0[ ] data 0[ ]2=

pspec m[ ] data 2 m•[ ]2

data 2 m 1+•[ ] d– ata 2 m 1+•[ ]( )•–=

m 1 2 …n 2–, ,{ }=

pspec n 1–[ ] data 1[ ]2=




NOTES The file tpspec.c included in the distribution tape provides an example of this func-tion’s use.

The number of elements processed is n. This number represents 1/2 the rfft size (ie. for an rfft size of N=128, there are N/2 = 64 complex values).

This function assumes the input vector data is derived from the output of the rfftip( ) function call. This data is formatted as follows (F[ ] represents a complex values vec-tor):

Note: n = N/2 where N represents the rfft size

Re (F[0]) = data [0]

Re (F[k]) = data [2*k] for k = { 1,2, ... n-2}

Im (F[k]) = data [2*k+1] for k = { 1,2, ... n-2}

Re (F[n-1]) = data [1]

The power spectra is computed as follows:

pspec[0] = data [0]^2

pspec = data[2*m]^2 - data[2*m+1] * (-data[2*m+1] ) for m=1 to n-2

pspec [n-1] = data[1]^2

Note that vector yi is in pm memory. This function is tailored to use the output prod-ucts from a rfftip( ) function call.

vpspec ( data, pspec, j, n )



vpythag ( a, i, b, j, c, k, d, l, e, h, n )

NAME Vector Pythagoras

DESCRIPTION This function calculates the the Pythagorean formula for four input input vectors a, b, c, and d. The results are stored in output vector e.

ALGORITHM

SYNOPSIS void vpythag ( a, i, b, j, c, k, d, l, e, h, n )










int h ; /* Address stride in words for output vector h */


DOMAIN (a2-c2) +(b2-d2) should not exceed the domain of 0.0 to +3.4 x 1038



NOTES The file tvpythag.c included in the distribution diskette provides an example of this function’s use.

Emh A( mi Cmk– )2 B( mj Dml– )

2+=

m 0 1 2 …n 1–, , ,{ }=



vqint ( a, i, b, j, c, k, n )

NAME Vector Quadratic Interpolation

DESCRIPTION Performs a quadratic interpolation on the elements of input vector a using values spec-ified in input vector b. The results are stored in output vector c.

ALGORITHM

SYNOPSIS void vqint ( a, i, b, j, c, k, n )



float *b ; /* Pointer to coefficient vector b */





DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvqint.c included in the distribution tape provides an example of this func-tion’s use.

The values of input interpolation vector b must be greater than or equal to 0 after con-version to a truncated integer. Input interpolation vector b must also be less than orequal to m-2 after conversion to a truncated integer.

Cmk 0.5 Ax 1– y2

y–( )• 2.0 Ax•+• 1 y2

–( )• Ax 1+ y2

y+( )•+=

x max 1 int Bj,( )=

y Bj x–=

y2

y y•=

1 int Bj( ) m 2–≤≤

m 0 1 2 …n 1–, , ,{ }=



vqsort ( a, b, n )

NAME Vector Quick Sort

DESCRIPTION This function performs a quick sort on all elements within input vector a, and stores the result in output vector a. The input sort flag b determines if the results are returned in ascending or decending order.

ALGORITHM

SYNOPSIS void vqsort( a, b, n )

float dm *a ; /* Pointer to input/output vector a */

int *b ; /* Input sort flag b */


DOMAIN -3.4E+38 to 3.4E+38


If b = 0 then Output Am Input Am sorted largest to smallest (descending) =

If b = 1 then Output Am Input Am sorted smallest to largest (ascending) =

m 0 1 2 …n 1–, , ,{ }=



EXECUTION TIME 47 cycles overhead

While loop 43 cycles

Partition scans (left to right) 3

N * log N comparisons (Average Case)

N2 comparisons (Worst Case)

NOTES The file tvqsort.c included in the distribution diskette provides an example of this function’s use.

The sort flag b determines the sort order of the algorithm, as described in the equation system above. Note that vector a is both an input and an output vector.

This algorithm uses a non-recursive implementation. The crux of the algorithm is the partitioning scheme. Partitioning arranges the elements such that (for some random reference elements in the array):

• Elements less that the reference element a[i] are positioned before the ref-erence element (a[0] ... a[i-1].

• Elements greater that the reference element a[i] are positioned after the ref-erence element (a[i+1] ... a[n].

The reference element is choosen randomly, in this case the right most element of the array.

Partitioning returns the index of the left most element. This is then used to determine the left and right indices for the next round of partitioning until the array is sorted.

vqsort ( a, b, n )



vrad ( a, i, c, k, n )

NAME Vector Convert Radian to Degrees

DESCRIPTION This function converts each element in input vector a, assumed to be in radians to degrees, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vrad ( a, i, c, k, n )






DOMAIN -5.9 x 1036 to +5.9 x 1036


EXECUTION TIME 21 +2*(N-1) cycles

NOTES The file tvrad.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk 57.2957795 Ami• m 0 1 2 …n 1–, , ,{ }==



vramp ( a, b, c, k, n )

NAME Vector Build Ramp

DESCRIPTION Creates a ramp of ascending or descending elements and stores the result to output vector c. Input scalar value a is the offset of the ramp to be created. Input scalar value b is the slope of the ramp to be created.

ALGORITHM

SYNOPSIS void vramp ( a, b, c, k, n )

float a ; /* Value of ramp offset */

float b ; /* Value of ramp slope */




DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 10 + N

NOTES The file tvramp.c included in the distribution tape provides an example of this func-tion’s use.

Cmk A m B• m 0 1 2 … n, 1–, , ,{ }=+=



vrandn ( a, c, k, n )

NAME Vector Build Normal Distribution

DESCRIPTION Creates a Normally distributed group of elements and stores the result to vector c.

ALGORITHM

SYNOPSIS void vrandn ( a, c, k, n )

unsigned int *a ; /* Pointer to input/output seed a */




DOMAIN Seed (*a) is valid unsigned integer from 1 to 232 - 1 (= 4,294,967,295). Returns uni-form point [0 to 1].


Ck U3 2⁄3±

i 0=

9

∑=

U x± is uniform distribution in x, x–( )




NOTES The file tvrandn.c included in the distribution tape provides an example of this func-tion’s use.

vrandn( ) generates a vector of pseudo-random numbers (PRNs) having a normal dis-tribution.The distribution is said to be normal if the mean of the distribution is equal to 0 and the variance is equal to 1.

Uniformly-distributed PRNs are used to generate the normal PRNs. From calculus, recall that the Central Limit Theorem states that when samples are generated by sum-ming “large” numbers of uniform random numbers, the resulting distribution is nor-mal. In this case, ten uniform PRNs are summed to produce each normal point.

The uniform PRNs are generated using a linear-congruent method. This method multi-plies a “seed” by a large fixed number, resulting in overflow; the least-significant bits of the product are retained, and the integer value they represent is uniformly random in nature. This integer PRN forms the mantissa of a uniform floating-point PRN in the range [0,1]. The sum is shifted and scaled to produce a normal PRN.

The final seed is returned through argument a and can be used as the initial seed on subsequent calls to randn( ). For additional information, see the discussion of linear congruent generators in the vrandu( ) notes section.

vrandn ( a, c, k, n )



vrandu ( a, c, k, n )

NAME Vector Build Uniform Distribution

DESCRIPTION Creates a group of elements uniformly distributed in [0,1] and stores the result in out-put vector c.

ALGORITHM

SYNOPSIS void vrandu ( a, c, k, n )

unsigned int *a ; /* Pointer to input/output seed a */




DOMAIN Seed (*a) is valid unsigned integer from 1 to 232 - 1. Returns normal point [0 to 1].



NOTES The file tvrandu.c included in the distribution tape provides an example of this func-tion’s use.

This algorithm generates uniform pseudo-random numbers (PRNs) in [0,1] using a lin-ear-congruent method. The method used multiplies a “seed” value supplied by the user by a large fixed number, resulting in overflow. The least-significant bits of the product are retained, and the integer value they represent is uniformly random in nature. This integer PRN forms the mantissa of a uniform floating-point PRN in the range [0,1].

The linear congruent method is pseudo-random: the sequence of pseudo-random num-bers will repeat itself within 232 iterations. For an excellent discussion of the methods used see pages 275-277 in Numerical Recipes in C by Press, Teukolsky, Vetterling & Flannery, Second Edition, Cambridge University Press, New York, New York, 1992.

The final seed is returned through argument a and can be used as the initial seed on subsequent calls to vrandu( ).

Ck float Ik( ) 232⁄( )=

where I0 *a=

Ik 110351524 Ik 1– 12345+•[ ]% 232

=

k 0 1 2 … n, 1–, , ,{ }=



vreal ( a, i, c, k, n )

NAME Vector Copy Real Parts

DESCRIPTION This function creates a real output vector c based on the real portion of complex input vector a.

ALGORITHM

SYNOPSIS void vreal ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvreal.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Re Ami{ } m 0 1 2 …n 1–, , ,{ }==



vrecip ( a, i, c, k, n )

NAME Vector Reciprocal

DESCRIPTION This function computes the reciprocal values all elements of input vector a and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vrecip ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038, except 0.0



NOTES The file tvrecip.c included in the distribution diskette provides an example of this function’s use.

Should one of the elements of input vector a be set to zero, this routine does not con-tain any logic to check for any attempted divide by zero. If this is a concern, use the vrecipz( ) routine, which executes slower but checks for attempted divides by zero.

Cmk1

A----

mi=

m 0 1 2 …n 1–, , ,{ }=



vrecipz ( a, i, b, c, k, n )

NAME Vector Reciprocal with Zero Domain Check

DESCRIPTION This function computes the reciprocal values all elements of input vector a and stores the results in output vector c. If element a is found to be zero, scalar b is stored to out-put vector c.

ALGORITHM

SYNOPSIS void vrecipz ( a, i, b, c, k, n )



float *b ; /* Value of input scalar b */




DOMAIN -3.4 x 1038 to +3.4 x 1038, except 0.0



NOTES The file tvrecipz.c included in the distribution diskette provides an example of this function’s use.

if Ami 0.0 then C= mk B m 0 1 2 …n 1–, , ,{ }==

else Cmk1

Amk-----------=



vreim ( a, i, b, j, c, k, n )

NAME Vector Copy Real and Imaginary Parts

DESCRIPTION This function extracts the real and imaginary components from a complex input vector a and saves the results into a real output vector b and a real ouput vector c respectively.

ALGORITHM

SYNOPSIS void vreim ( a, i, b, j, c, k, n )



float *b ; /* Pointer to real output vector b */

int j ; /* Address stride in words for output vector b */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvreims.c included in the distribution diskette provides an example of this function’s use.

Bmj Re Ami{ } m 0 1 2 …n 1–, , ,{ }==

Cmk Im Ami{ } m 0 1 2 …n 1–, , ,{ }==



vreim_pdp ( a, i, b, j, c, k, n )

NAME Vector Copy Real and Imaginary Parts, Vector B[ ] in Data Memory

DESCRIPTION This function extracts the real and imaginary components from a complex input vector a[ ] and saves the results into a real output vector b[ ] and a real ouput vector c[ ] respectively. Vectors a[ ] and c[ ] are in pm memory while vector b[ ] is in dm mem-ory.

ALGORITHM

SYNOPSIS void vreim_pdp ( a, i, b, j, c, k, n )



float dm *b ; /* Pointer to real output vector b */


float pm *c ; /* Pointer to real output vector c */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvreims_pdp.c included in the distribution diskette provides an example of this function’s use.

Bmj Re Ami{ } m 0 1 2 …n 1–, , ,{ }==

Cmk Im Ami{ } m 0 1 2 …n 1–, , ,{ }==



vrfftip ( x, real, imag, n )

NAME Sort Ouptut Data from rfftip( ) Real, In-Place FFT Function

DESCRIPTION This function sorts the output data in x[ ] that was generated by the rfftip( ) function into its real and imaginary components. The components outputted are placed in vec-tors real[ ] and imag[ ] respectively. This function is intended to condition the output from rfftip( ) for use in computing the inverse real FFT with the rffti( ) function.

ALGORITHM

SYNOPSIS void vrfftip ( x, real, imag, n )

float dm *x ; /* Pointer to output data from rfftip() */

float dm *real ; /* Pointer to Real output vector */

float pm *imag ; /* Pointer to imaginary output vector */


DOMAIN No Restrictions



NOTES The file tvrfftip.c included in the distribution diskette provides an example of this function’s use.

The value of n should be 1/2 the size of the original FFT.

The vector sizes for real[ ] and imag[ ] are n/2+1; where real[0] = real[n/2] and imag[0] = imag[n/2].

For a detailed review of the interelationships between the various FFT functions and how to use them with one another, see the final section of Chapter 4.

Re F 0[ ] ( ) data 0[ ]=

Re F 1[ ] ( ) data 2 k•[ ] k= 0 1 2 … n 2⁄, 1–, , ,{ }=

Im F k[ ] ( ) data 2 k 1+•[ ]k= 0 1 2 … n 2⁄, 1–, , ,{ }=

Re F n 2⁄[ ] ( ) data 1[ ]=



vrffts ( x, real, imag, n )

NAME Sort Ouptut Data from rffts( ) Real, In-Place FFT Function

DESCRIPTION This function sorts the output data in x[ ] that was generated by the rffts( ) function into its real and imaginary components. The components outputted are placed in vec-tors real[ ] and imag[ ] respectively. This function is intended to condition the output from rffts( ) for use in computing the inverse real FFT with the rffti( ) function.

ALGORITHM

SYNOPSIS void vrffts ( x, real, imag, n )

float dm *x ; /* Pointer to output data from rfftip() */

float dm *real ; /* Pointer to Real output vector */

float pm *imag ; /* Pointer to imaginary output vector */


DOMAIN No Restrictions


Re F 0[ ] ( ) x 0[ ]=

Re F 1[ ] ( ) x 1[ ]=

. .

Re F n 2⁄[ ] ( ) x n 2⁄[ ]=

Im F n 2 1–⁄[ ] ( ) x n 2 1+⁄[ ]=

Im F n 2 2–⁄[ ] ( ) x n 2 2+⁄[ ]=

. .

Im F 1[ ] ( ) x n 1–[ ]=




NOTES The file tvrffts.c included in the distribution diskette provides an example of this func-tion’s use.

The value of n should be 1/2 the size of the original FFT.

The vector sizes for real[ ] and imag[ ] are n/2+1; where real[0] = real[n/2] and imag[0] = imag[n/2].

For a detailed review of the interelationships between the various FFT functions and how to use them with one another, see the final section of Chapter 4.

vrffts ( x, real, imag, n )



vrol ( a, i, c, k, n )

NAME Vector Rotate Left

DESCRIPTION This function performs a 1-bit left-rotation on each element in input vector a. The results are stored in output vector c.

ALGORITHM

SYNOPSIS void vrol ( a, i, c, k, n )

unsigned int *a ; /* Pointer to input vector a */


unsigned int *c ; /* Pointer to output vector c */




ACCURACY 32-bits


NOTES The file tvrol.c included in the distribution diskette provides an example of this func-tion’s use.

The carry bit is not rotated in this routine. For left-rotations greater then 5 use the vrot( ) routine.

Cmk bit0[ ] Ami bit31[ ]=

Cmk bitn[ ] Ami bitn 1–[ ]=

m 0 1 2 …n 1–, , ,{ }=



vror ( a, i, c, k, n )

NAME Vector Rotate Right

DESCRIPTION This function performs a 1-bit right-rotation on each element in input vector a. The results are stored in output vector c.

ALGORITHM

SYNOPSIS void vror ( a, i, c, k, n )







ACCURACY 32-bits


NOTES The file tvror.c included in the distribution diskette provides an example of this func-tion’s use. The carry bit is not rotated in this routine. For right-rotations greater then 5 use the vrot( ) routine.

Cmk bit31[ ] Ami bit0[ ]= Cmk bitn[ ] Ami bitn 1+[ ]=

m 0 1 2…n 1 }–, ,{=



vrot ( a, i, c, k, n, h )

NAME Vector Rotate N Bits

DESCRIPTION This function performs multiple h-bit right or left-rotations on each 32-bit element in input vector a. The results are then stored in output vector c. If h is negative the bits are rotated-right with the least significant bit (LSB) being rotated into the most signif-icant bit (MSB). If h is positive the bits are rotated-left with the MSB being rotated into the LSB.

ALGORITHM

SYNOPSISvoid vrot ( a, i, c, k, n, h )






int h ; /* Length And direction of rotation */

DOMAIN h must be an unsigned integer between -65,536 and +65,536. The argument is later range reduced within the routine to an argument between 0 to -31.

ACCURACY 32-bits


NOTES The file tvrot.c included in the distribution diskette provides an example of this func-tion’s use.

The carry-bit is not rotated in this routine. For bit-rotation less then 5 use the vror( ) or vrol( ) routines. For bit-rotations greater then 5 use this routine.

if h 0<( )then (rotate right h-bits)

c bit31[ ] a bit0[ ] a bitn[ ] a bitn 1+[ ]=,=

else h 0>( )then (rotate left h-bits)

c bit0[ ] a bit31[ ] a bitn 1+[ ] a bitn[ ]=,=

m 0 1 2…n 1 }–, ,{=



vround ( a, i, c, k, n )

NAME Vector Round

DESCRIPTION This function computes the nearest whole number of each element in input vector a and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vround ( a, i, c, k, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvround.c included in the distribution diskette provides an example of this function’s use.

This function is the opposite of the vfloor( ) function, described in the preceding page.

Cmk floor Ami 0.5+( )= m 0 1 2 …n 1–, , ,{ }=



vrsqrt ( a, i, c, k, n )

NAME Vector Reciprocal Square Root

DESCRIPTION This function computes the reciprocal square root for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vrsqrt ( a, i, c, k, n )






DOMAIN 0.0 to +3.4 x 1038,



NOTES The file tvrsqrt.c included in the distribution diskette provides an example of this function’s use.

Since division by zero is an illegal operation which will result in an error, input vector a must be greater then zero. This routine will execute faster than vsqrtz( ), as it does not contain logic to check for domain entires less than or equal to zero.

Cmk1.0

Ami

-------------- m 0 1 2 …n, 1–, ,{ }==

Amj 0>



vrsqrtz ( a, i, b, c, k, n )

NAME Vector Reciprocal Square Root with Domain Check

DESCRIPTION This function computes the reciprocal square root for each element in vector a, and stores the result in output vector c.

If input vector a is less or equal to zero, the value of scalar b is stored to output vector c.

ALGORITHM

SYNOPSIS void vrsqrtz ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvrsqrtz.c included in the distribution diskette provides an example of this function’s use.

This routine is slower than vrsqrt( ), but checks for the presence of the less than or equal to 0.0 condition in the denominator.

if Ami 0.0 then m 0 1 2 …n 1–, , ,{ }=>

Cmk1.0Ami

-------------- else=

Cmk B=



vrsum ( a, i, b, c, k, n )

NAME Vector Running Sum

DESCRIPTION Computes a running sum integration on the elements in input vector a scaled by input scalar b. The results are placed in input/output vector c.

ALGORITHM

SYNOPSIS void vrsum ( a, i, b, c, k, n )



float b ; /* Weighting Factor */




DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvrsum.c included in the distribution tape provides an example of this func-tion’s use.

Vector c is both an input and output vector. If you desire to clear the vector, use the vclr( ) function to initialize the value of vector c to zero.

is not used in the sum to provide an output format consistent with vtrap( ) and vsimp( ).

Cmk C m 1–( )k Ami B•+=

C0 0.0=

m 0 1 2 … n, 1–, , ,{ }=

Ami



vrvrs ( c, k, n )

NAME Vector Reverse Order

DESCRIPTION This function reverse orders the elements of real input/output vector c. This operation is performed in-place with the output saved in input/output vector c.

ALGORITHM

SYNOPSIS void vrvrs ( c, k, n )




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvrvrs.c included in the distribution diskette provides an example of this func-tion’s use.

Note that this process is performed in-place, and is therefore destructive to the original elements in vector c.

Temp Cmk=

Cmk C n 1– m–[ ]k=

C n 1– m–[ ]k Temp=

m 0 1 2…n 1 }–, ,{=



vsasm ( a, i, b, c, d, l, n )

NAME Vector Scalar Add and Scalar Multiply

DESCRIPTION This function adds input vector a with input scalar b, multiplies the sum by input sca-lar c, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vsasm ( a, i, b, c, d, l, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsasm.c included in the distribution diskette provides an example of this function’s use.

Dml Ami B+( ) C• m 0 1 2 …n, 1–, ,{ }==



vsassb ( a, i, b, c, d, l, n )

NAME Vector Scalar Add and Scalar Subtract

DESCRIPTION This function adds input vector a with input scalar b, subtracts input scalar c from the sum, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vsassb ( a, i, b, c, d, l, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsassb.c included in the distribution diskette provides an example of this function’s use.

Dml Ami B+( ) C– m 0 1 2 …n, 1–, ,{ }==



vscadd ( a, i, b, c, k, n )

NAME Vector Scalar Add

DESCRIPTION This function adds input scalar b to each element of input vector a. The results are stored in output vector c.

ALGORITHM

SYNOPSIS void vscadd ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvscadd.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami B+ m 0 1 2 …n, 1–, ,{ }==



vscatr ( a, b, j, c, n )

NAME Vector Scatter

DESCRIPTION This function copies the elements of input vector a to output vector c using integer input vector b as the indices to be copied into vector c.

ALGORITHM

SYNOPSIS void vscatr ( a, b, j, c, n )






DOMAIN -3.4 x 1038 to +3.4 x 1038, Bmj must be within the valid range of integers.



NOTES The file tvscatr.c included in the distribution diskette provides an example of this function’s use.

This function is the opposite of the vgathr( ) and vindex( ) functions, vgathr( ). This function performs the same operation as vgathr( ) execpt that vgathr( ) expects input vector b to be type real. Note that only vectors a and b may stride.

CBmj

Am= m 0 1 2 … n 1–, , , ,{ }=



vscdiv ( a, i, b, c, k, n )

NAME Vector Scalar Divide

DESCRIPTION This function divides input vector a by input scalar b, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vscdiv ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvscdiv.c included in the distribution diskette provides an example of this function’s use.

This routine does not contain logic to check for an attempted divide by zero. If scalar b is set to zero an attempt to divide by zero will occur.

Cmk

Ami

B---------- m 0 1 2 …n, 1–, ,{ }==



vscma ( a, i, b, j, c, d, l, n )

NAME Vector Scalar Multiply and Vector Add

DESCRIPTION This function multiplies input vector a times input scalar c, then adds input vector b to the product, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vscma ( a, i, b, j, c, d, l, n )






float *d ; /* Pointer to output vector b */



DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvscma.c included in the distribution diskette provides an example of this function’s use.

Dml Ami C•( ) Bmj+ m 0 1 2 …n, 1–, ,{ }==



vscmsb ( a, i, b, j, c, d, l, n )

NAME Vector Scalar Multiply and Vector Subtract

DESCRIPTION This function multiplies input vector a by scalar c, subtracts input vector b from the product, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vscmsb ( a, i, b, j, c, d, l, n )









DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvscmsb.c included in the distribution diskette provides an example of this function’s use.

Dml Ami C•( ) Bmj– m 0 1 2 …n, 1–, ,{ }==



vscmul ( a, i, b, c, k, n )

NAME Vector Scalar Multiply

DESCRIPTION This function multiplies input vector a by input scalar b and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vscmul ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvscmul.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami B• m 0 1 2 …n 1–, , ,{ }==



vscsub ( a, i, b, c, k, n )

NAME Vector Scalar Subtract

DESCRIPTION This function subtracts input scalar b from input vector a, and stores the result in out-put vector c.

ALGORITHM

SYNOPSIS void vscsub ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvscsub.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami B– m 0 1 2 …n 1–, , ,{ }==



vsec ( a, i, c, k, n )

NAME Vector Secant

DESCRIPTION This function computes the secant for each element in input vector a, and stores the result in output vector c. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vsec ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvsec.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amisec m 0 1 2 …n 1–, , ,{ }==



vsech ( a, i, c, k, n )

NAME Vector Hyperbolic Secant

DESCRIPTION This function computes the hyperbolic secant for each element in input vector a, and stores the result in output vector c. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vsech ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvsech.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amisech m 0 1 2 …n 1–, , ,{ }==



vsfloat ( a, c, sign, n )

NAME Vector Integer To Floating-Point Conversion of Short Word

DESCRIPTION This function successively applies the float instruction to the short word in integer vector a and stores the results into output vector c. The lower 16-bit word or LSW of the element in vector a is extracted and then converted. The same is done for the upper 16-bit word or MSW.

The number of elements in n is defined by the size of vector a. The sign extension flag determines whether the 16-bit word is sign extended or not before conversion ( 0 = unsigned, 1 = signed ).

ALGORITHM

SYNOPSIS void vsfloat ( a, c, sign, n )

int dm *a ; /* Pointer to input vector a */


int sign ; /* Sign Extension Flag */

int n ; /* Element count ( size of a ) */

DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tsvfloat.c included in the distribution diskette provides an example of this function’s use.

Cmk float Ami( )= m 0 1 2 …n 1–, , ,{ }=



vsimps ( a, i, b, c, n )

NAME Vector Simpson’s Rule Integration

DESCRIPTION Integrates input vector a scaled by input scalar b according to Simpson’s rule, and stores the result to output vector c.

ALGORITHM

SYNOPSIS void vsimps ( a, i, b, c, n )




float *c ; /* Pointer to accumulator output vector c */


DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvsimp.c included in the distribution tape provides an example of this func-tion’s use.

The number of points n must be an odd number great than or equal to 3.

C0 0.0=

C1 0.5 B A0 B1+( )••=

Ci Ci 2– 0.333333 B Ai 2– 4 Ai 1– Ai+•+( )••+=



vsin ( a, i, c, k, n )

NAME Vector Sine

DESCRIPTION This function computes the sine for each element in input vector a, and stores the result in output vector c. All angles in input vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vsin ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvsin.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=sin=



vsinh ( a, i, c, k, n )

NAME Vector Hyperbolic Sine

DESCRIPTION This function computes the hyperbolic sine for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vsinh ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvsinh.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Amisinh m 0 1 2 …n 1–, , ,{ }==



vsm ( a, i, b, j, c, k, d, l, n )

NAME Vector Subtract and Multiply

DESCRIPTION This function subtracts input vector b from input vector a, multiplies the difference by input vector c, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vsm ( a, i, b, j, c, d, l, n )



float *b ; /* Pointer to input scalar b */


float *c ; /* Pointer to input scalar c */





DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsm.c included in the distribution diskette provides an example of this func-tion’s use.

Dml Ami Bmj–( ) Cmk• m 0 1 2 …n 1–, , ,{ }==



vsmsad ( a, i, b, c, d, l, n )

NAME Vector Scalar Multiply and Scalar Add

DESCRIPTION This function multiplies input vector a by input scalar b, adds input scalar c to the product, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vsmsad ( a, i, b, c, d, l, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsmsad.c included in the distribution diskette provides an example of this function’s use.

Dml Ami B•( ) C m 0 1 2 …n 1–, , ,{ }=+=



vsmssb ( a, i, b, c, d, l, n )

NAME Vector Scalar Multiply and Scalar Subtract

DESCRIPTION This function multiplies input vector a by input scalar b, and then subtracts input sca-lar c from the product. The result are stored in output vector d.

ALGORITHM

SYNOPSIS void vsmssb ( a, i, b, j, c, d, l, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsmssb.c included in the distribution diskette provides an example of this function’s use.

Dml Ami B•( ) C – m 0 1 2 …n 1–, , ,{ }==



vsmvsma ( a, i, b, j, c, k, d, e, n )

NAME Vector Muliply By Scalar, Add to Vector Multiplied By Scalar

DESCRIPTION This function multiplies input vector a by input scalar d and adds the results to the multiplication of input vector b times scalar e, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vsmvsma ( a, i, b, j, c, k, d, e, n )







float *d ; /* Pointer to input scalar d */

float *e ; /* Pointer to input scalar e */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsmvsma.c included in the distribution diskette provides an example of this function’s use.

Cml Ami D•( ) Bmj E•( ) m 0 1 2 …n 1–, , ,{ }=+=



vsq ( a, i, c, k, n )

NAME Vector Square

DESCRIPTION This function computes the square for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vsq ( a, i, c, k, n )






DOMAIN -1.8 x 1019 to +1.8 x 1019



NOTES The file tvsq.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami2

m 0 1 2 …n 1–, , ,{ }==



vsqrt ( a, i, c, k, n )

NAME Vector Square Root

DESCRIPTION This function computes the square root for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vsqrt( a, i, c, k, n )






DOMAIN 0.0 to +3.4 x 1038, Ami must be > 0.0



NOTES The file tvsqrt.c included in the distribution diskette provides an example of this func-tion’s use.

This routine does not check for the presence of a number less than zero in Ami.

Cmk Ami m 0 1 2 …n 1–, , ,{ }==



vsqrtz ( a, i, b, c, k, n )

NAME Vector Square Root With Zero Domain Check

DESCRIPTION This function computes the square root for each element in input vector a, and stores the result in output vector c. If input vector a is found to be less than zero, the value of input scalar b is stored to output vector c.

ALGORITHM

SYNOPSIS void vsqrtz( a, i, b, c, k, n )



float *b ; /* Value of input scalar b */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsqrtz.c included in the distribution diskette provides an example of this function’s use.

if Ami 0.0 then m 0 1 2 …n 1–, , ,{ }=<

Cmk B else=

Cmk Ami=



vssbsa ( a, i, b, c, d, l, n )

NAME Vector Scalar Subtract and Scalar Add

DESCRIPTION This function subtracts input scalar b from input vector a, adds the difference to input scalar c, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vssbsa ( a, i, b, c, d, l, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvssbsa.c included in the distribution diskette provides an example of this function’s use.

Dml Ami B–( ) C+ m 0 1 2 …n 1–, , ,{ }==



vssbsm ( a, i, b, c, d, l, n )

NAME Vector Scalar Subtract and Scalar Multiply

DESCRIPTION This function subtracts input scalar b from input vector a, multiplies the difference by input scalar c, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vssbsm ( a, i, b, c, d, l, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvssbsm.c included in the distribution diskette provides an example of this function’s use.

Dml Ami B–( ) C• m 0 1 2 …n 1–, , ,{ }==



vssca ( a, i, b, j, c, d, l, n )

NAME Vector Subtract and Scalar Add

DESCRIPTION This function subtracts input vector b from input vector a, adds input scalar c to the difference, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vssca ( a, i, b, j, c, d, l, n )









DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvssca.c included in the distribution diskette provides an example of this func-tion’s use.

Dml Ami Bmj–( ) C+ m 0 1 2 …n 1–, , ,{ }==



vsscm ( a, i, b, j, c, d, l, n )

NAME Vector Subtract and Scalar Multiply

DESCRIPTION This function subtracts input vector b from input vector a, multiplies the difference by input scalar c, and stores the result in output vector d.

ALGORITHM

SYNOPSIS void vsscm ( a, i, b, j, c, d, l, n )









DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsscm.c included in the distribution diskette provides an example of this function’s use.

Dml Ami Bmj–( ) C• m 0 1 2 …n 1–, , ,{ }==



vssm ( a, i, b, j, c, k, d, l, e, h, n )

NAME Vector Subtract, Subtract & Multiply

DESCRIPTION This function subtracts two input vectors, a minus b, subtracts two other input vectors c minus d, then multiplies the two intermediate results, and stores the result in output vector e.

ALGORITHM

SYNOPSIS void vssm ( a, i, b, j, c, k, d, l, e, h, n )












DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvssm.c included in the distribution diskette provides an example of this func-tion’s use.

Emh Ami Bmj–( ) Cmk Dml–( )•=

m 0 1 2 …n 1–, , ,{ }=



vssort ( a, b, c, n )

NAME Selection Sort

DESCRIPTION This function performs a selection sort on input/output vector a according to the num-ber of elements defined in input scalar c. The sort order (ascending or descending) is defined by input scalar b, while the number of elements in input/output vector a is defined in input scalar n.

ALGORITHM for i = 0 to c

min = 1

for j = i+1 to n

if a[j] < a[min] then

min = j

end if

swap a[min] and a[i]

end loop

end loop

SYNOPSIS void vssort ( a, b, c, n )

float dm *a ; /* Pointer to input/output vector a */

int b ; /* Input sort flag */

int c ; /* Number of elements to sort */

int n ; /* Count of floating point elements */

DOMAIN -3.4 x 1038 to +3.4 x 1038




EXECUTION TIME 22 + (10 + 5*N) cycles

NOTES The file tvssort.c included in the distribution diskette provides an example of this function’s use.

vssort ( ) performs a selection sort on input vector a. A selection sort searches the vec-tor for a value (largest or smallest) and swaps it with a vector position (either from the left or the right). This process is repeated for the next value and position until the sort is finished. The type of sorting is determined by flags explained as follows:

The sort order is determined by the sort flag b.

For b =1, the sort order is ascending order (smallest to largest).

For b =0, the sort order is descending order (largest to smallest to largest).

The number of elements that are sorted is determined by input scalar c. The size of the input/output vector is defined by input scalar a.

The remaining non-sorted elements in the vector are left in the respective swapped positions.

Example:

float a[8] = { 1.2, 10.5, 3.0, 4.0, 12.2, 7.8, 9.0, 0.5 };

vssort ( a, 0, 3, 8 ) ;

a = { 12.2, 10.5, 9.0, 4.0, 1.2, 7.8, 3.0, 0.5 } ;

vssort ( a, 1, 3, 8 ) ;

a = { 0.5, 1.2, 3.0, 4.0, 10.5, 7.8, 9.0, 12.2 } ;

The number of elements to sort must be less than or equal to the number of elements in a. ( c <= n ).

If the order of the sorted elements is out of order (i.e., want the lowest values in descending order) use the function vrvrs( ) to reverse the order . Likewise, if the desired sorted elements are out of position in the vector ( i.e., want lowest three ele-ments in the middle of the vector ) use the function vswap( ) to switch the order.

vssort ( a, b, c, n )



vssq ( a, i, c, k, n )

NAME Vector Signed Square

DESCRIPTION This function computes the signed square for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vssq ( a, i, c, k, n )






DOMAIN -1.8 x 1019 to +1.8 x 1019



NOTES The file tvssq.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami Ami• m 0 1 2 …n 1–, , ,{ }==



vsub ( a, i, b, j, c, k, n )

NAME Vector Subtract

DESCRIPTION This function subtracts input vector b from input vector a, and stores the result in out-put vector c.

ALGORITHM

SYNOPSIS void vsub ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvsub.c included in the distribution diskette provides an example of this func-tion’s use.

Cm Ami Bmj– m 0 1 2 …n 1–, , ,{ }==



vswap ( a, i, c, k, n )

NAME Vector Swap

DESCRIPTION This function swaps each element of input vector a with each element of output vector c.

ALGORITHM

SYNOPSIS void vswap ( a, i, c, k, n )

float a ; /* Pointer to input vector a */

int i ; /* Address stride in words for input/output vector a */


int k ; /* Address stride in words for input/output vector c */


DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvswap.c included in the distribution diskette provides an example of this function’s use.

Temp Bmj C= mk m 0 1 2 …n 1–, , ,{ }=

Cmk Ami=

Ami Temp Bmj=



vtan ( a, i, c, k, n )

NAME Vector Tangent

DESCRIPTION This function computes the tangent for each element in vector a, and stores the result in output vector c. All angles in vector a are assumed to be in radians.

ALGORITHM

SYNOPSIS void vtan ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvtan.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=tan=



vtanh ( a, i, c, k, n )

NAME Vector Hyperbolic Tangent

DESCRIPTION This function computes the hyperbolic tangent for each element in input vector a, and stores the result in output vector c.

ALGORITHM

SYNOPSIS void vtanh ( a, i, c, k, n )






DOMAIN -1.0 x 109 to +1.0 x 109



NOTES The file tvtanh.c included in the distribution diskette provides an example of this function’s use.

Cmk Ami m 0 1 2 …n 1–, , ,{ }=tanh=



vthr ( a, i, b, c, k, n )

NAME Vector Threshold


ALGORITHM

SYNOPSIS void vthr ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvthr.c included in the distribution diskette provides an example of this func-tion’s use.

This routine is also an alias for the vlthr( ) routine.

if Ami B then Cmk B m 0 1 2 …n 1–, , ,{ }==<

else Cmk Ami=



vthres ( a, i, b, c, k, n )

NAME Vector Threshold With Zero Fill

DESCRIPTION This function compares input vector a to input scalar b. For each element in input vec-tor a greater than or equal to the threshold input scalar b, the value of the element in vector a is stored in the corresponding element of output vector c. Otherwise, the ele-ment in output vector c is set equal to 0.0.

ALGORITHM

SYNOPSIS void vthres ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvthres.c included in the distribution diskette provides an example of this function’s use.

if Ami B then Cmk Ami m 0 1 2 …n 1–, , ,{ }==≥

else Cmk 0.0=



vthrsc ( a, i, b, c, d, l, n )

NAME Vector Threshold With Signed Constant

DESCRIPTION This function compares input vector a to input scalar b. For each element in input vec-tor a greater than or equal to the threshold input scalar b, the value of input scalar c is stored in the corresponding element of output vector d. Otherwise, the output element in vector d is set equal to negative input scalar c.

ALGORITHM

SYNOPSIS void vthrsc ( a, i, b, c, d, l, n )




float c ; /* Value of fill scalar c */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvthrsc.c included in the distribution diskette provides an example of this function’s use.

if Ami B then Dml C m 0 1 2 …n 1–, , ,{ }==≥

else Dml C–=



vtmerg ( a, i, b, j, c, k, n )

NAME Vector Tapered Merge

DESCRIPTION This function performs a tapered merge. A comparison of two input vectors a and b determines whether vectors a or b are copied into output vector c.

ALGORITHM if then

else

SYNOPSIS void vtmerg ( a, i, b, j, c, k, n )








DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvtmerg.c included in the distribution diskette provides an example of this function’s use.

Ami Bmj<

Cmk Bmj=

Cmk Amj=

m 0 1 2 … n, 1–, , ,{ }=



vtrans ( a, i, b, j, c, k, n )

NAME Transfer Function

DESCRIPTION Computes the transfer function for the auto-spectrum provided by complex input vec-tor a and the cross-spectrum provided by input vector b. The result is stored in com-plex output vector c.

ALGORITHM

SYNOPSIS void vtrans ( a, i, b, j, c, k, n )

complex *a ; /* Pointer to input vector a */




complex *c ; /* Pointer to accum output scalar c */


int n ; /* Element count to compute */

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvtrans.c included in the distribution tape provides an example of this func-tion’s use.

Vector a is assumed to be a cross-spectrum.

Vector b is assumed to be an autospectrum.

Re Cmk{ }Re Bmj{ }

Ami------------------------= Im Cmk{ }

Im Bmj{ }

Ami------------------------=

m 0 1 2 … n, 1–, , ,{ }=



vtrapz ( a, i, b, c, k, n )

NAME Vector Trapezoidal Integration

DESCRIPTION Performs a trapezoid-rule integration on input vector a placing the results in outputvector c. B is a scalar value which typically represents the abcissa increment betweenelements of vector a (the step size), hence allowing scaling.

ALGORITHM

SYNOPSIS void vtrapz ( a, i, b, c, k, n )



float b ; /* Scaling/Factor (Step Size) */




DOMAIN -3.4E+38 to 3.4E+38



NOTES The file tvtrapz.c included in the distribution tape provides an example of this func-tion’s use.

C0 0.0=

Cmk C m 1–( )kB Ami A m 1–( )i+[ ]•

2------------------------------------------------------+=

m 0 1 2 … n, 1–, , ,{ }=



vunpack ( a, i, b, j, fill, n )

NAME Unpack a Input Vector of 32-Bit Words Into a Floating Point Vector of Ones and Zeros

DESCRIPTION This function reads an input vector a, bit by bit. If a 1 is read, a bit is set in output vec-tor b corresponding to floating-point 1. Likewise, if a 0 is read a bit is set in output vector b to floating-point 0. The next value is read and the next corresponding bit is set.

The order in which the bits are set/cleared is determined by the fill flag. If the fill=1, the bits are read from LSB to MSB. If fill=-1, the bits are read from MSB to LSB.

ALGORITHM

SYNOPSIS void vunpack ( a, i, b, j, fill, n )



int dm *b ; /* Pointer to output vector b */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvunpack.c included in the distribution diskette provides an example of this function’s use.



Bmj Ami= m 0 1 2 …n 1–, , ,{ }=



vunpcki ( a, i, b, j, fill, n )

NAME Unpack an Input Vector of Ones and Zeros Into Integer 32-Bit Words Arguments

DESCRIPTION This function reads an input vector a bit by bit. . If a 1 is read, a bit is set in output vec-tor b to integer 1. Likewise, if a 0 is read a bit is in output vector b is set to integer 0. The next value is read and the next corresponding bit is set.

The order in which the bits are set is determined by the fill flag. If fill =1, then the bits are read from LSB to MSB. If fill = -1, the bits are read from MSB to LSB.

ALGORITHM

SYNOPSIS void vunpcki ( a, i, b, j, fill, n )



int dm *b ; /* Pointer to output vector b */




DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvunpack.c included in the distribution diskette provides an example of this function’s use.



Bmj Ami= m 0 1 2 …n 1–, , ,{ }=



vuthr ( a, i, b, c, k, n )

NAME Vector Upper Threshold


ALGORITHM

SYNOPSIS void vuthr ( a, i, b, c, k, n )







DOMAIN -3.4 x 1038 to +3.4 x 1038



NOTES The file tvuthr.c included in the distribution diskette provides an example of this func-tion’s use.

if Ami B then Cmk Ami =<

else Cmk B=

m 0 1 2 …n 1–, , ,{ }=



vxcs ( a, i, b, c, d, l, n )

NAME Vector Multiply by Sine and Cosine

DESCRIPTION The function vxcs( ) computes the real and imaginary components from the values held within vector a [ ] by multiplying by the sine and cosine of the input frequency b and phase (in radians) c. The results are stored in output vector d.

ALGORITHM

SYNOPSIS void vxcs ( a, i, b, c, d, l, n )


int i ; /* Element stride for input vector a */

float *b ; /* Pointer to Freq in radians per element*/

float *c ; /* Pointer to input/output phase in radians*/

complex *d ; /* Pointer to output vector d */

int l ; /* Element stride for output vector d */


DOMAIN -3.4E+38 to 3.4E+38


Re Dml{ } Ami B m•( ) C+( )cos•=

Im Dml{ } Ami B m•( ) C+( )sin•=

C n B C+•=

m 0 1 2 … n, 1–, , ,{ }=



EXECUTION TIME 66 + 48 * N

NOTES The file tvxcs.c included in the distribution tape provides an example of this function’s use.

vxcs ( ) computes the real and imaginary components from the values held within vec-tor a by multiplying by the sine and cosine of the input frequency b and phase c. The return value is stored in output vector d.

This routine reduces the argument to a modulo fraction before using a polynomial approximation from Cody and Waite to estimate the sine value. The polynomial used is appropriate for 32-bit computations. In addition, the innner loop of coefficient is unrolled to eliminate setting up a loop each time.

All arguments are assumed to be in radians.

The phase c is also computed for the next frame such that succesive calls to vxcs ( ) will produce a continuous signal.

should be normalized such that .

vxcs ( a, i, b, c, d, l, n )

Re Dmk{ } Ami Bm C+( )cos•=

Im Dmk{ } Ami Bm C+( )sin•=

m 0 1 2 … n, 1–, , ,{ }=

π

C n B C+•= π– c π< <



vxor ( a, i, b, j, c, k, n )

NAME Vector Exclusive Or

DESCRIPTION This function performs a bitwise exclusive OR of each element in input vector a with each element in input vector b and stores the results in output vector c.

ALGORITHM

SYNOPSIS void vxor ( a, i, b, j, c, k, n )









ACCURACY 32-bits


NOTES The file txor.c included in the distribution diskette provides an example of this func-tion’s use.

Cmk Ami XOR Bmj( )=

m 0 1 2…n 1 }–, ,{=



welch ( a, i, c, k, n )

NAME Welch Window Multiply

DESCRIPTION Computes a Welch window on the values held in real input vector a. The results are stored in real output vector c.

ALGORITHM

SYNOPSIS void welch ( a, i, c, k, n )



float *c ; /* Pointer to resultant output vector c */


int n ; /* Element count for ooutput vector c */

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file twelch.c included in the distribution tape provides an example of this func-tion’s use.

welch( ) is a vector function. You may therefore use the stride arguments i and k todecimate both input and output for data congruence with other routines.

Cm Ami 1m 0.5 n 1–( )–[ ]

0.5 n 1+( )---------------------------------------

2–• =

m 0 1 2 … n, 1–, , ,{ }=



winwts ( wr, i, n )

NAME Window Cosine Weights Array

DESCRIPTION The routine computes the cosine weights array for a windowing computation such as the hamm( ) and hann( ) functions.

ALGORITHM

SYNOPSIS void winwts ( wr, i, n )

float pm *wr ; /* Pointer to output weights table wr[ ] */

int i ; /* Vector wr[ ] element stride */

int n ; /* Count of floating point elements */

DOMAIN -3.4E+38 to 3.4E+38



NOTES The file twinwts.c included in the distribution tape provides an example of this func-tion’s use.

This function is introduced to correct a discrepancy with the hamm( ) and hann( ) window function, whose weights were originally calculated using the formula

. This was incorrect as this required the fft function to allocate twice the amount of memory to contain the weights. To correct the problem, a new function was introduced called the winwts( ) (shown above) which uses the formula to perform the weights calculation while the fftwts( ) function was modified to use the formula for the fft weights calculation.

wrm 2 πmn----••

for m 0 1 2 … n, 1–, , ,{ }=cos=

2 π m•• n⁄

2 π m•• n⁄

π m• n⁄



wtmean ( a, i, b, j, &c, n )

NAME Weighted Mean

DESCRIPTION This routine calculates the mean of the elements within input vector a weighted by the elements within input vector b, and places the results in scalar c.

ALGORITHM

SYNOPSIS void wtmean ( a, i, b, j, &c, n)





float *c ; /* Pointer to resultant scalar c */


DOMAIN -3.4E+38 to 3.4E+38



NOTES The file twtmean.c included in the distribution tape provides an example of this func-tion’s use.

C

Ai Bj•

i 0 j 0=,=

n 1–

∑

Bj

j 0=

n 1–

∑

----------------------------------------- m 0 1 2 … n, 1–, , ,{ }==



zeroxng ( a, i, n )

NAME Zero Crossing Detector

DESCRIPTION This routine detects the number of times the elements within input vector a cross the zero axis (ie. in connecting the points, how many times the connection crosses the zero axis). The result is returned from the function.

ALGORITHM

SYNOPSIS void zeroxng ( a, i, n )




DOMAIN -3.4E+38 to 3.4E+38


EXECUTION TIME 45 + N * ( 7+H-1 )

NOTES The file tzeroxng.c included in the distribution tape provides an example of this func-tion’s use.

if Ami A m 1+( )i•( ) 0<( )

count count 1+=

m 0 1 2 … n, 1–, , ,{ }=





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Chap5 - ADSP 21K Manual