dsp_lect_99.L1

Created:June 1999

Last update:©AFMH DSP_L1_T1Created:June 1999

Last update:July 2010

Discrete-Time Signals: Review

©AFMH DSP_L1_T1

Develop the notion of a discrete-time (d-t) signal and a d-t system.

Define a d-t signal and its properties,and perform operations on these signals.

☺ Tutorial:Solve problems related to the above.MATLAB Exercises

Chap. 1, ‘’Digital Signal Processing, Sanjit K. Mitra.

LEARNING OBJECTIVES

x[n]

n-4 -3 -2 -1 0 1 2 3 5

... ...

-Nx[n] x[n-N] x[-n]x[n]

x[2n]x[n]2

x[2n]x[n]2 b[n]

a[n] +/- y[n] x[n] xy[n]

b[n] or scaling by c

Created:June 1999




©AFMH DSP_L1_T2

Discrete-Time Signals

• A discrete-time signal is an indexed sequence of real or complex numbers. Thus, it is function of an integer-valued variable, n and is denoted by x[n].

• x[n] is generally referred to as a function of time. A d-t signal is undefined for noninteger values of n.

• A real-valued signal x[n] will be represented graphically in the form of a lollipop plot as shown.

x[n]

n-4 -3 -2 -1 0 1 2 3 5

... ...

↑

= ...)522023342(...][nx

Created:June 1999




©AFMH DSP_L1_T3

• Discrete-time signals or sequences are often derived by sampling a continuous-time signal, such as speech, with an analog-to-digital (A/D) converter.

• For example, a c-t signal xa(t) that is sampled at a rate of fs=1/Ts samples per second produces the sampled signal x[n], which is related to xa(t) as follows:

x[n] = xa(nTs)• Not all d-t signals are obtained in this

manner. Eg. Daily stock market prices, population statistics, etc.

Generating a D-T Signal or Sequence

x[n]

n-4 -3 -2 -1 0 1 2 3 5

Samplinginterval, Ts

xa(t)

xa(t)

t

A/D

Created:June 1999




©AFMH DSP_L1_T4

Complex Sequences

• In general, a d-t signal may be complex-valued. In digital communications, complex signals arise naturally. A general complex signal x(t) may be expressed either in terms of its real and imaginary parts,

z[n] = Re{z[n]} + j Im{z[n]}or in polar form in terms of its magnitude and phase,

z[n] = |z[n]| exp[j arg{z[n]}]

• The magnitude may be derived as follows:|z[n]|2 = Re2{z[n]} + Im2{z[n]}

• The phase may be found usingarg{z[n]} = tan-1[Im{z[n]}/Re{z[n]}]

• If z[n] is a complex sequence, the complex conjugate denoted by z*[n] is formed by changing the sign on the imaginary part of z[n]:

z*[n] = Re{z[n]} - j Im{z[n]} = |z[n]| exp[-j arg{z[n]}]

z = reiφ

z = a + jb

b

a Re

Im

tan-1 b/a

-b

z[n]

z*[n]

,...),(...,][ jdcjbanz ++=

Created:June 1999




©AFMH DSP_L1_T5

Fundamental Sequences:Unit Impulse Sequence

• The unit impulse (or unit sample) sequence δ [n], is defined as

• The delayed/shifted unit impulse/sample sequence δ[n-k] is defined by

⎩⎨⎧

≠=

=−

⎩⎨⎧

≠=

=

knkn

kn

nn

n

01

][

0001

][

δ

δ

δ[n]

1

n-2-1 0 1 2 3

δ[n-k]

1

n-2

...

-1 0 1 2 k

][nδ

Created:June 1999




©AFMH DSP_L1_T6

• The unit step sequence u[n], is defined as

• Value of u[n] at n=0 is defined and equals to unity (unlike c-t step function, u(t)).

• The shifted unit step sequence u[n-k] is defined as

Fundamental Sequences:Unit Step Sequence

⎩⎨⎧

<≥

=−

⎩⎨⎧

<≥

=

knkn

knu

nn

nu

01

][

0001

][u[n]

1

n-2

...

-1 0 1 2 3 4 5

u[n-k]

1

n-2

...

-1 0 1 2 k

][nu

Created:June 1999




©AFMH DSP_L1_T7

Fundamental Sequences:Unit Impulse Sequence

• From the definitions ofδ[n] and δ[n-k], it isreadily seen that

• Note that δ[n] and u[n]are related by

• Any sequence x[n] canbe expressed as(see DSP_L1_T24) ∑

∑

∞

−∞=

−∞=

−=

−−=

=

−=−=

k

n

k

knkxnx

nunun

knu

knkxknnxnxnnx

][][][

]1[][][

,][][

][][][][][]0[][][

δ

δ

δ

δδδδ

Created:June 1999




©AFMH DSP_L1_T8

Fundamental Sequences:Complex Exponential Sequences

An exponential sequence is given as

The complex exponential sequence is of the form

).exp()exp(

,][

φωσαα

α

jAAjA

Anx

oo

n

=+=

=

and,i.e. numberscomplex or realmay and

)sin()cos(

][ )(

φωφω σσ

φωσ

+++=

= ++

neAjneA

eAnx

on

on

njn

oo

oo

Created:June 1999




©AFMH DSP_L1_T9

Fundamental Sequences:Complex Exponential Sequences

The real sinusoidal sequence with a constant amplitude is of the form

For x[n] (with σo=0 for complex seq) to be periodic with period N, ω0must satisfy the following condition

The smallest N yields the fundamental period of sequence,N0

)cos(][ φω += nAnx o

amplitude angular frequency initial phase

integer positive== mNm

πω2

0

000

12f

N =⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωπ

Created:June 1999




©AFMH DSP_L1_T10

Fundamental Sequences:Complex-valued Exponential Sequences

0 5 10 15 20 25 30 35 40-2

-1

0

1

2

Time index n

Ampl

itude

Real part

0 5 10 15 20 25 30 35 40-1

0

1

2

Time index n

Ampl

itude

Imaginary part

% Generation of a complex exp. sequence

clf;c = -(1/12)+(pi/6)*i;K = 2;n = 0:40;x = K*exp(c*n);subplot(2,1,1);stem(n,real(x));xlabel('Time index n');ylabel('Amplitude');title('Real part');subplot(2,1,2);stem(n,imag(x));xlabel('Time index n');ylabel('Amplitude');title('Imaginary part');

njnx )exp(2][ 6121 π+−=

122cos2 12

1 nne π−

122sin2 12

1 nne π−

Ref: Program P1_2, DSP Lab using Matlab, Mitra

Created:June 1999




©AFMH DSP_L1_T11

Fundamental Sequences:Period Complex-valued Exponential Sequences

% Generate periodic complex sequence%n=[0:40];x=exp((j*pi/10)*n);subplot(2,2,1);stem(n, real(x)) % real partsubplot(2,2,2);stem(n, imag(x)) % imaginary partsubplot(2,2,3);stem(n, abs(x)) % magnitudesubplot(2,2,4);stem(n, angle(x)*(180/pi))

% phase in degrees

njnx )exp(][ 10π=

0 10 20 30 40-1

-0.5

0

0.5

1

0 10 20 30 40-1

-0.5

0

0.5

1Real part Imaginary part

0 10 20 30 400

0.2

0.4

0.6

0.8

1

0 10 20 30 40-200

-100

0

100

200Magnitude Phase

Ref: Program P1_2new

Created:June 1999




©AFMH DSP_L1_T12

Fundamental Sequences:Real-valued Exponential Sequences

0 5 10 15 20 25 30 350

20

40

60

80

100

120

Time index n

Ampl

itude

% Generation of a real exp.l sequenceclf;n = 0:35; a = 1.2; K = 0.2;x = K*a.^n;stem(n,x);xlabel('Time index n');ylabel('Amplitude');


Created:June 1999




©AFMH DSP_L1_T13

Fundamental Sequences:Sinusoidal Sequences

• A sinusoidal sequence can be expressed as

If n is dimensionless, then both ω0 and θ have units of radians.

Sequence above is periodic with fundamental period 12.

)cos(][ 0 θω += nAnx

-12 -9-6

-3 0 36

9 12

x[n]=cos(πn/6)

n

Created:June 1999




©AFMH DSP_L1_T14

Fundamental Sequences:Sinusoidal Sequences

0 5 10 15 20 25 30 35 40-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Sinusoidal Sequence

Time index n

Ampl

itude

% Generation of a sinusoidal sequencen = 0:40;f = 0.1;phase = 0;A = 1.5;arg = 2*pi*f*n - phase; x = A*cos(arg);clf; % Clear old graphstem(n,x);% Plot the generated sequenceaxis([0 40 -2 2]);grid; title('Sinusoidal Sequence');xlabel('Time index n');ylabel('Amplitude');axis;


Created:June 1999




©AFMH DSP_L1_T15

• A real-valued sequence x[n] is said to be even if, for all n

» x[n] = x[-n]• whereas it is said to be odd if, for all n

» x[n] = -x[-n]

• Any signal x[n] can be decomposed into a sum of its even part xe[n], and its odd part xo[n], as follows:

x[n] = xe[n] + xo[n],where

xe[n] = 1/2 {x[n]+x[-n]},xo[n] = 1/2 {x[n]-x[-n]}.

Symmetric Sequences

n

x[n]

0

n

x[n]

0

Example:x[n]=(3 6 2 10 1 8 5)x[-n]=(5 8 1 10 2 6 3)

xe[n]=(4 7 1.5 10 1.5 7 4)xo[n]=(-1 -1 0.5 0 0.5 1 1)

Created:June 1999




©AFMH DSP_L1_T16

• A complex sequence x[n] is said to be conjugate symmetric (see note below) if, for all n

» x[n] = x*[-n]

• whereas it is said to be conjugate antisymmetric if, for all n

» x[n] = -x*[-n]

Symmetric Sequences (Complex)

Note: A sequence that is conjugate symmetric is sometimes said to be hermitian. See Example 2.5 in Mitra for conjugate -symmetric sequence

Re

Im

Re

Im

See Example 2.5 in Mitra for conjugate -symmetric sequence

jba +

jba −

jba +jba +−

Created:June 1999




©AFMH DSP_L1_T17

Periodic and Aperiodic Sequences

• A sequence (discrete-time signal) x[n] is said to be periodic with period N if for some positive real integer N for which

x[n] = x[n+N] for all n

• It follows that x[n+mN] = x[n] for all n and any integer m

• The fundamental period N0 of x[n] is the smallest positive integer for which the above equation is satisfied.

• If the equation is not satisfied for any integer N, x[n] is said to be aperiodic.

-N N0 n

x[n]

Sequence repeats itself every N samplesSee TP DSP_L1-T8

Sequence is also periodic with period 2N, period 3N, and all other integer multiples of N

Created:June 1999




©AFMH DSP_L1_T18

Exercises

Determine whether or not the following signals are periodic and, for each sequence that is periodic, determine the fundamental period.

)17/cos(][(d)

)2.0sin(][(c)}Im{}Re{][(b)

)125.0cos(][(a)

16

18/12/

π

π

π

π

ππ

nenx

nnxeenx

nnx

nj

jnjn

=

+=+=

=

Created:June 1999




©AFMH DSP_L1_T19

Solutions

Because 0.125π = π/8,

From TP DSP_L1_T8, we have ω0 = π/8, and

For x[n] to be periodic with period N, ω0 must satisfy the following condition ω0/2π = m/N (=1/16)

Therefore, cos(π/8 n) = cos(π/8 (n+16))

x[n] is periodic with fundamental period N0 = 16.

)125.0cos(][(a) nnx π=

168

22

00 === π

πωπN

Try plotting - adapt Matlab code on DSP_L1_T13

Created:June 1999




©AFMH DSP_L1_T20

Solutions

Here we have the sum of two periodic signals

)18/sin()12/cos(][

}Im{}Re{][(b) 18/12/

ππ

ππ

nnnx

eenx jnjn

+=

+=

),gcd( 21

21

NNNNN =

Note: If x1[n] is a sequence with period N1, and x2[n] is another sequence with period N2, the sum x[n]=x1[n]+x2[n] will always be periodic with a fundamental period

where gcd(N1, N2) means greatest common divisor of N1 and N2.

The same is true for x[n]=x1[n]x2[n]; however, the fundamental period may be smaller

72

36,2418/,12/

12)36)(24(

)36,24gcd()36)(24(

18/2

212/2

1

0201

===

====

==

N

NN ππ

ππ

πωπωTherefore, the period of the sum is

Created:June 1999




©AFMH DSP_L1_T21

Solutions

In order for this sequence to be periodic, we must be able to find a value for N such that

The sine function is periodic with a period 2π.Therefore, 0.2N must be an integer multiple of 2π. However, because π is an irrational number, no integer value of N exists that will make the equality true.Thus, this sequence is aperiodic.

))(2.0sin()2.0sin(

)2.0sin(][(c)

Nnn

nnx

++=+

+=

ππ

π

Created:June 1999




©AFMH DSP_L1_T22

Solutions

Here we have the product of two periodic signals,Therefore, the period of the sum is

)17/cos(][(d) 16 ππ

nenxnj

=

544

34,3217/,16/

2)34)(32(

)34,32gcd()34)(32(

17/2

216/2

1

0201

===

====

==

N

NN ππ

ππ

πωπω

Created:June 1999




©AFMH DSP_L1_T23

• Note: The operations below are order-dependent.• Shifting

– If y[n]=x[n-n0], x[n] is shifted to the right by n0 samples (delay), given n0 positive

• Reversal– Given y[n]=x[-n] (simply involves “flipping”

the sequence x[n] w.r.t. to index n)

• Time scaling– Given y[n]=x[Mn] or y[n]=x[n/N] where M

and N are positive integers.

Signal Manipulation:Shifting, Reversal, Time scaling

x[n]

n-2-1 0 1 2 3 4 5 6x[n-2]

n-2-1 0 1 2 3 4 5 6x[-n]

n-8-7 -6-5-4-3-2-10

x[2n]

n-2-1 0 1 2 3 4 5 6

Down-sampling by a factor of 2

x[n/2]

n-2 -1 0 1 2 3 4 5 6 7 8 9 10 11

Up-sampling by a factor of 2

Created:June 1999




©AFMH DSP_L1_T24

• Addition– The sum of 2 sequences,

y[n]=a[n]+b[n] is formed by the pointwise addition of the two sequences.

• Multiplication (or modulation)– The product of 2 sequences,

y[n]=a[n]b[n] is formed by the pointwise product of the two sequences.

• Scaling (or scalar multiplication)– Amplitude scaling by a constant c,

y[n] = cx[n] is accomplished by multiplying every sample value by c.

Signal Manipulation:Addition, Multiplication, Scaling

n -1 0 1 2 3 4

a[n] 2 -1 0 4 7 3

b[n] 3 5 -2 7 4 -5

Addition a[b]+b[n]

5 4 -2 11 11 8

Multiplicationa[n]b[n]

6 -5 0 28 28 -15

Scaling 3 a[n]

6 -3 0 12 21 9

Created:June 1999




©AFMH DSP_L1_T25

• The unit sequence may be used to decompose an arbitrary sequencex[n] into a sum of weighted and shifted unit samples as follows:

• This decomposition may be written concisely as:

Signal Manipulation:Shifting, Reversal, Time scaling

...)5(.1)4(.2)3(.3)2(.2)1(.1)(.0)1(.1......)2()2()1()1()()0()1()1(...)(

+−+−+−+−+−++++=+−+−+++−+=

nnnnnnnnxnxnxnxnx

δδδδδδδδδδδ

x[n]

n-1 0 1 2 3 4 5

∑∞

−∞=

−=k

knkxnx )()()( δ

↑

= ...)1232101(...][nx

Created:June 1999




©AFMH DSP_L1_T26

Exercises

Express the sequence

as a sum of scaled and shifted unit steps.

⎪⎪⎩

⎪⎪⎨

⎧

===

=

elsennn

nx

0231201

)(

Created:June 1999




©AFMH DSP_L1_T27

Solution

There are several ways to derive the signal decomposition.(a) Express as a sum of weighted and shifted unit samples

Use the fact:Therefore

(b) Derive directly as follows: Decomposition should begin with a unit step which generates a value of 1 at index n=0. Because x(n) increases to a value of 2 at n=1, we must add a delayed unit step u(n-1). At n=2, x(n) again increases in amplitude by 1, so we add the delayed unit step u(n-2). We then bring the sequence back to zero for n>=3 by subtracting the delayed unit step 3u(n-3).

)2(3)1(2)()( −+−+= nnnnx δδδ)1()()( −−= nununδ

)3(3)2()1()()]3()2([3)]2()1([2)1()()(

−−−+−+=−−−+−−−+−−=

nununununununununununx

Created:June 1999




©AFMH DSP_L1_T28

Application Example #1 -Signal Smoothing

A common DSP application is the removal of noise from a signal corrupted by additive noise. A simple 3-point moving average algorithm is given by:

])1[][]1[(31][ +++−= nxnxnxny

0 5 10 15 20 25 30 35 40 45 50-2

0

2

4

6

8

Time index nAm

plitu

de

d[n]s[n]x[n]

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

Time index n

Ampl

itude

y[n]s[n]

% Signal Smoothing by Averagingclf;% Generate random noise, d[n]R = 51; d = 0.8*(rand(R,1) - 0.5); % Generate uncorrupted signal, s[n]m = 0:R-1; s = 2*m.*(0.9.^m); % Generate noise corrupted signal, x[n]x = s + d'; subplot(2,1,1);plot(m,d','r-',m,s,'g--',m,x,'b-.');xlabel('Time index n');ylabel('Amplitude');legend('d[n] ','s[n] ','x[n] ');% do smoothingx1 = [0 0 x];x2 = [0 x 0];x3 = [x 0 0];y = (x1 + x2 + x3)/3;subplot(2,1,2);plot(m,y(2:R+1),'r-',m,s,'g--');legend( 'y[n] ','s[n] ');xlabel('Time index n');ylabel('Amplitude');


Created:June 1999




©AFMH DSP_L1_T29

Application Example #2 -Amplitude Modulation

An amplitude modulated signal can be generated by modulating a high-frequency sinusoidal signal, with a low-frequency sinusoidal signal, . The resulting signal is of the form:

where m is the modulation index.

)cos(][ nnx HH ω=)cos(][ nnx LL ω=

)cos())cos(.1(][])[.1(][ nnmAnxnxmAny HLHL ωω+=+=

0 10 20 30 40 50 60 70 80 90 100-1.5

-1

-0.5

0

0.5

1

1.5

Time index n

Ampl

itude

% Generation of amplitude modulated seqclf;n = 0:100;m = 0.4;fH = 0.1; fL = 0.01;

xH = sin(2*pi*fH*n);xL = sin(2*pi*fL*n);y = (1+m*xL).*xH;stem(n,y);grid;xlabel('Time index n');ylabel('Amplitude');


Created:June 1999




©AFMH DSP_L1_T30

Application Example #2 -Amplitude Modulation

)cos())cos(.1(][])[.1(][ nnmAnxnxmAny HLHL ωω+=+=

0 10 20 30 40 50 60 70 80 90 100-1.5

-1

-0.5

0

0.5

1

1.5

Time index n

Ampl

itude

Created:June 1999




©AFMH DSP_L1_T31

Application Example #3 -Swept Frequency Sinusoidal Sequence

Note: To generate a swept-frequency sinusoidal signal whose frequency increases linearly with time, the argument of the sinusoidal signal must be a quadratic function of time. Assume the argument is of the form an2 + bn (i.e. the angular frequency is 2an + b since the frequency is the derivative of its phase w.r.t. time).Solve for values of a and b from the given conditions (minimum & maximum angular frequencies).

Ref: Program P1_7new, DSP Lab using Matlab, Mitra

% Generation of a swept freq sinusoidal seqn = 0:100; a = pi/2/400; b = 0;arg = a*n.*n + b*n;x = cos(arg);clf;stem(n, x);axis([0,100,-1.5,1.5]);title('Swept-Frequency Sinusoidal Signal');xlabel('Time index n'); ylabel('Amplitude');grid; axis;

0 10 20 30 40 50 60 70 80 90 100-1.5

-1

-0.5

0

0.5

1

1.5Swept-Frequency Sinusoidal Signal

Time index n

Ampl

itude

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dsp_lect_99.L1