LRA Multi-Rate DSPsite.iugaza.edu.ps/zshaban/files/2010/10/P3_MultiRateAppl.pdf · LRA DSP LRA DSP 6'(Sigma-Delta) Converters Analogue (digital) integrator 1-bit ADC Mth band Digital

LRA

DSP

LRA

DSP MultiMulti--Rate DSP Rate DSP

Applications:

Oversampling, Undersampling,

Quadrature Mirror Filters

Professor L R Arnaut © 1

LRA

DSP

LRA


Oversampling


LRA

DSP

LRA

DSP


Optimal Sampling vs. Optimal Sampling vs.

OversamplingOversampling

Sampling at Nyquist rate Fs =2FB

Allows perfect reconstruction in principle, but…

Pre-sampling anti-aliasing filter must have very steep roll-off:

High-order analogue filter: expensive, difficult, imprecise, large phase distortion, …

Sampling at Fs >>2FB & decimation to 2FB

Larger separation between images easierfiltering of aliases (lower-order filter)

Cheaper analogue component; easier digital than analogue VLSI filters; greater digital complexity

LRA

DSP

LRA

DSP Oversampling Noise ReductionOversampling Noise Reduction

Quantisation step (b-bit ADC, range R):

Noise power density (per unit sampling bandwidth):

Total in-band noise power:

Low Pin for high Fs:

b

RQ

2

]W/Hz[62/

12/

2/

222

sss

NN

F

Q

F

Q

Fp

FsFB FB’ Fs

’

pN

N2

pN’

Pin’

Pin

2'2' NinBs PFF

2/12

)2/(

6d)(

22

0s

Bb

s

BFNin

F

FR

F

FQffpP B


LRA

DSP

LRA

DSP

Oversampling:Oversampling:

Effective ResolutionEffective Resolution

Equivalent -bit ADC operating at Fs=2FB

giving same noise power as b-bit ADC operating at Fs>>2FB over same range R:

i.e.,

M = oversampling ratio, -b = resolution increase

2/212

)2/(

2/12

)2/( 22

B

B

s

Bb

F

FR

F

FR

2

)(log

2

)2/(log 22 Mb

FFb Bs


LRA

DSP

LRA

DSP Oversampling RatioOversampling Ratio

Example:

M=106: =b+10, i.e., standard 16-bit ADC @ Fs=2FB has equivalent resolution w.r.t. noise power as an oversampling6-bit ADC @ Fs=2 106FB

or as oversampling 11-bit ADC @ Fs=2000FB

Thus, 0.5 bit length reduction of ADC per doubling of MProfessor L R Arnaut © 6

LRA

DSP

LRA

DSP (Sigma-Delta) Converters

(digital)Analogue

integrator

1-bit

ADC

Mth band

Digital LPF

M

1-bit

DAC

-

++(analogue)

(2MFB) (2MFB) (2MFB)

1 b

2FB

quantiser M downsampler

(b-bit multipliers)

-High-rate oversampling allows for differential encoding: only 1 bit needed to quantify change between input and delayed output of 1-bit ADC, for closely spaced consecutive samples in output of quantiser (bitmap of sample increments)

-Mth order alias digital LPF eliminates out-of-band quantisation noise (sharp cut-off); heavy comp.; remedy: perform multiplications at later stage (down-rate 2FB) instead

-Long word length of digital o/p determines overall oversampling rate

Analogue

integrator

z -1

+x(n)-y(n-1)w(n)


LRA

DSP

LRA

DSP


ConvertersConverters

Accumulator (integrator) & quantiser outputs:

noise transfer function:

Output psdf (one-sided spectrum):

z -1

+w(n)

z -1

-

++x(n)

quantiser

y(n)+

eADC(n)

eDAC(n)

)1()1()1( nenwny

)1()1()()( nwnynxnw

)]1()([)()()()( nenenxnenwny

11)( zzH

)(2

sin4)()( 22

fpT

fpeHfp Ns

NTj

ys

zshaban

Note

e(n-1)= x(n)-w(n)

zshaban

Note

at zero input [ x(n) = 0 ]

zshaban

Note

| H( exp(jwT) |^2 = H( exp(jwT) ) . H( exp( -jwT ) ) = ( 1-exp(jwT) ) ( 1 - exp(-jwT) ) = 2 - [ exp (jwT) + exp(-jwT) ] = 2 - [cos(wT) + j sin(wT) + cos(wT)- j sin(wT)] = 2 - 2 cos(wT) = 2 ( 1 - cos (wT) ) = 4 sin^2(wT/2) because: cos( A )= 1 - 2 sin^2(A/2)

LRA

DSP

LRA

DSP OversamplingOversampling

Precise psdf of output noise depends on pdf & spectral characteristics of {x(n)}

Assume: {e(n)} is random,

uncorrelated, white:

psdf of output noise:


z -1

+w(n)

z -1

-

++x(n)

quantiser

y(n)+

eADC(n)

eDAC(n)

2/

12/

2/

22

ss

NN

F

Q

Fp

s

sy

F

QTffp

22

3

)(sin2)(

LRA

DSP

LRA

DSP

Oversampling:Oversampling:

Performance ComparisonPerformance Comparison

For efficient oversampling :

Output noise power for modulator:

Improvement over standard oversampling:


),1( sFfM

3

22222

3

2

3

)/(2)(

ss

sy

F

fQ

F

QFffp

3

322

09

2d)(

s

BFyy

F

FQffpP B

dB)](log2017.5[3

log10log10 102

2

1010 MM

P

P

y

in

zshaban

Note

Here and under these conditions, sin^2(A) ~ A^2

zshaban

Note

Pin= PN*FB

LRA

DSP

LRA


Undersampling


LRA

DSP

LRA

DSP NyquistNyquist ConditionCondition

Alias-free (sub)sampling of (discrete) function

Spectrum (Fourier transformation): convolution

Thus,

No spectral overlap (aliasing, stroboscopy) iffProfessor L R Arnaut © 16

Bs FF 2

kss

ksTn kTtkTxkTttxttxtx

s

)()()()()()()(

ks

s

ks

s

n

dkFfXT

dkFT

fXfX

)()(1

1)()(

ks

s

n kFfXT

fX )(1

)(

LRA

DSP

LRA

DSP


UndersamplingUndersampling: Aliasing: Aliasing

LRA

DSP

LRA

DSP UndersamplingUndersampling: Baseband: Baseband

Shannon sampling theorem:

Applies to baseband signals (DC-coupled): FB islargest frequency component in signal

Motivation: avoid spectral overlap of baseband frequency responses that are periodically continued due to sampling operation

For bandpass (narrowband modulated) signals (e.g., radio- and optical communications, IF filters, etc.): condition is too conservative: large spectral gaps occur because Fc >> |FB - Fc |


Bs FF 2

LRA

DSP

LRA

DSP UndersamplingUndersampling:: BandpassBandpass

Aliasing of bandpass signal is avoided if baseband can be folded periodically around carrier frequency without causing overlap

Range of permissible sample frequencies:

i.e.,

Yields additional permissible lower samplingrates for narrowband signals without aliasing

Practically useful (slower computations)Professor L R Arnaut © 19

1

22

n

FF

n

F ls

u

lu

u

FF

Fn1

LRA

DSP

LRA

DSP UndersamplingUndersampling: Applications: Applications

58.7

(n=4)(n=3)

Fu

=108f (MHz)Fl

=88

20 54 2 Fu

=216

2 Fl

=176

…(n=1)(n=2)

72


Example 1: digitization of analogue FM audio

Fl = 88 MHz, Fu = 108 MHz (nonzero gap)

•n=1: classical ( Nyquist rate)

•n=2: between modulated (Iu=1) signal and doubled (Il=2) signal

•n=3: Du=2/3, Il=1

•n=5:

51 n

MHz44MHz2.43 sF MHz1105.2...2MHz4.86 sF

LRA

DSP

LRA


Quadrature Mirror Filters

for Subband Coding


LRA

DSP

LRA

DSP Subband CodingSubband Coding

Problem statement:

Efficient transmission of realistic speech or video signals

Contain most energy at relative low frequencies (time/space)

Coding scheme to be tailored to assign more bits to LF band

Solution:

Subband coding:divide total frequency band in unequal subbands;

narrowest subband for interval with highest energy (equalization ofpower across band)

each subband is encoded separately

is alternative to companding (pre-distortion) Professor L R Arnaut © 22

LRA

DSP

LRA

DSP

Example:

(approximate equalisation)

- Multi-rate conversion by factor I/D after each frequency subdivision (LPF/HPF)

- Reduced bitrate of digitized signal (bandwidth compression) due to nonuniform coding (variable number of bits per sample)


Subband CodingSubband Coding

S(f)

fmfm/29fm/329fm/960

1 bit/sample5 3 2

(D=2)(I/D=9/16)(I/D=1/3)

LRA

DSP

LRA

DSP

Implementation:

Brickwall Filter: Quadrature Mirror Filter (QMF):Physically aliasing for decimated subbands can be removed

unrealizable by judicious choice of H0( ) and H1( )


Subband CodingSubband Coding

LRA

DSP

LRA

DSP

Implementation (analyzer / synthesizer):


TwoTwo--Channel QMFChannel QMF

(for I/D=1/2)

LRA

DSP

LRA

DSP

QMF Analyzer:

TwoTwo--Channel QMF: AnalysisChannel QMF: Analysis

2D,2

exp2

exp1 1

0

11

00

D

k

DD zkD

jXzk

D

jH

DzX

22222

1000 XHXHX

22222

1111 XHXHX


LRA

DSP

LRA

DSP

QMF Synthesizer:

Cascaded QMF analyzer-synthesizer:


TwoTwo--Channel QMF: SynthesisChannel QMF: Synthesis

2I,)( 00

IzYzV

22 1100 YGYGY

1100 , XYXY

XGHGHXGHGHY 110011002

1

2

1

Aliasing (k=1)

LRA

DSP

LRA

DSP

Elimination of aliasing for any input signal:

results in time-invariant filter

example: alias-free symmetric subband coding


QMF AntiQMF Anti--AliasingAliasing

e.g.

01100 GHGH

0110 , HGHG

0100 , HGHG

LRA

DSP

LRA

DSP

Distortion-free & alias-free reconstruction:

Example: symmetric subband

i.e., independent of (all-passfilter), but may exhibit phase distortion!

It can be shown: linear-phase FIR QMF causes amplitude distortion


QMF Perfect ReconstructionQMF Perfect Reconstruction

2),exp(1100 DjkDGHGH

)exp(0110 jkDHHHH

)exp(20

20 jkDHH

2

0

2

0 HH

LRA

DSP

LRA

DSP

M branches; M in analyzer, M in synthesizer

Output kth analyzer branch (BPF+D):

Output synthesizer (I+BPF):


MM--Channel QMF BankChannel QMF Bank

1

0

M

k

M

kk zYzGzY

)(,2

exp2

exp1 1

0

/1/1 DMM

mjzX

M

mjzH

MzX

M

m

MMkk

1

0

1

0

2exp

2exp

1 M

m

k

M

k

kM

mjzX

M

mjzH

MzGzY

,2

exp1

0

M

m

mM

mjzXzL

1

0

)(2

exp1 M

k

kkm zGM

mjzH

MzL

LRA

DSP

LRA

DSP

Alias-free QMF:

Distortion-free & alias-free QMF:

independent of (all-pass filters)


MM--Channel QMFChannel QMF

XLY )()( 0)(,0

2exp

1

1

zXM

mjzXzL

M

m

miff

i.e. 11,0 MmzLm

0L

Documents

LRA Multi-Rate DSPsite.iugaza.edu.ps/zshaban/files/2010/10/P3_MultiRateAppl.pdf · LRA DSP LRA DSP 6'(Sigma-Delta) Converters Analogue (digital) integrator 1-bit ADC Mth band Digital