Multirate DSP Part 1 [Read-Only]

Multi rate Digital Signal

ProcessingProcessing

1

Syllabus

– Basics of Multirate systems and its application

– Up-sampling and Down-Sampling

– Fractional Sampling rate converter

– Poly-phase decomposition

– Efficient realization of Multirate systems– Efficient realization of Multirate systems

– Uniform filter banks

• Its implementation using poly-phase decomposition

– Two channel Quadrature Mirror Filter Banks

– Perfect Reconstruction

– M-channel PR QMFB

2

Basics of Multirate systems and its application

3

Introduction

– Sampling rate conversion � Changing the

sampling rate of a signal

• Required in many practical applications of DSP

• Change in sampling rate can be an increase or decrease

– Example on need for Sampling rate conversion

• Communication systems may transmit and receive

different types of signals � facsimile, speech, video etc

– Each of these signal is to be processed at a different rate

corresponding to its band width

4

Introduction

– Multi-rate Digital signal processing systems

• Systems that employ multiple sampling rates in the

processing of digital signals

• Example: Playing CD music in a studio

– CD music has a data rate of 44.1 kHz

– To play this in a studio that handles data at 48 KHz rate, CD

data rate must be increased to 48 KHz

5

Advantages of Sampling rate conversion

• Computational efficiency with sampling rate reduction

– Number of computations is less at lower sampling rate

– Relevant only when sampling rate reduction does not cause

loss of any desired information

• Achieve compatibility with another system

– Playing CD music in a studio

6

Operations in Multi-rate DSP

• Decimation that reduces sampling rate � Compresses

data to retain only desired information

• Interpolation that increases sampling rate

7

Concept of sampling rate conversion

– Resampling after reconstruction

• Pass the signal through a DAC and LPF

• Sample the analog signal at the desired rate

• Advantage of this method

– New sampling rate can be arbitrarily selected

– Do not need any special relationship to the old sampling rate

• Disadvantage is added Signal distortion

– In DAC (signal reconstruction) and

– In ADC (quantization)

8

Concept of sampling rate conversion

– Reason for the disadvantage

• In the conceptual method, signal is reconstructed

• Noise gets added in signal reconstruction

– The disadvantage is overcome if sampling rate – The disadvantage is overcome if sampling rate

conversion is performed in the digital domain

• Anti aliasing can be done in digital domain with linear

phase and sharper magnitude response

• Convert the signal to analog form only when it is

essential

9

Applications in Industry

1. Reduce cost of analog anti aliasing filters

• We can sample the analog signal at a rate much higher

than that specified by sampling theorem

– This calls for a much simpler anti aliasing filter

– Once in digital form, the signal can be reduced to the desired – Once in digital form, the signal can be reduced to the desired

rate using multi rate approach

2. Efficient implementation of DSP functions

• Example: Narrow band digital FIR filters

– Direct implementation requires a large number of coefficients

to meet their frequency response specifications

– Multi-rate techniques can reduce the Number of coefficients

10

Applications in Industry

3. Dispense with expensive high resolution ADCs

• Advantage of High resolution ADCs is lesser

quantization noise

• With multi-rate techniques, we can achieve similar

performance using cheap ADCsperformance using cheap ADCs

– Oversampling spread the Quantization noise over a wider

frequency range

– Signal bandwidth is reduced with Sampling rate reduction,

simultaneously reducing the in-band noise too

11

Theory of sampling rate conversion

– Mathematical formulation of the theory:

• Developed around the concept of resampling after

reconstruction

– An aid to understand the sampling rate conversion process in

the digital domainthe digital domain

2)( ofBandwidth

1);( signal timediscrete Generates

)(

x

xxx

x

Ftx

FTnTx

FrateNyquistatsampledistxsignaltimeContinuous

=

=⇒

12


{)(

sin

)(

,'

2,

,0

FFT

OtherwiseFx FG

t

T

t

tg

isfunctionionInterpolatIdealtheoremsShannonFrom

xx

=→←=

=

≤

π

π

summation infinite needsit since practical,not istion reconstruc Ideal

)()()(*)()(

)()()( )( samples theingInterpolat

xn

xx

x

x

nTtgnTxtgnTxty

txtytsreconstructgwithnTx

T

t

−==

=

∑∞

−∞=

π

13


sequence desired theof samples )(

)()()(

instants at time )( evaluating is conversion rate Sampling

⇒

−=

=

∑∞

−∞=

y

xyn

xy

y

mTy

nTmTgnTxmTy

mTtty

samplesinput andoutput ofinstant Time &

andfunction tion Reconstruc )(

sequenceinput theof Samples )(

need we,)( compute To

sequence desired theof samples )(

→→→

⇒

xy

x

y

y

nTmT

tg

nTx

mTy

mTy

14


≥==

>

⇒<

yxyy

y

xy

FFFXifonlymTxmTy

Ftx

FF

2

,2

minFfor 0)(),()(

2/)( of components

frequency out theFilter if occurserror Aliasing

( ){ }

−=∆

=

−∆+=

−=−=

∑

∑∑

∞

−∞=

∞

−∞=

∞

−∞=

x

y

x

ym

x

ym

mmxn

x

x

yx

nxxy

nxy

T

mT

T

mT

T

mTkwhere

nkTgnTx

nT

mTTgnTxnTmTgnTxmTy

&

)(

)()()()(

15


( ){ } ( ){ }

{ } )()()()(

)()(

fromsummation ofindex Changing

kxmxmxxmmy

m

nTgnTxTngnTx

TkkxTkTgTkymTy

nkkton

∗=∆+∗=

−∆+=∆+=

−=

∑∞

−∞=

• This is the fundamental equation for the discrete time

implementation of sampling rate conversion

• Process in this Discrete time sampling rate conversion

is demonstrated in the figure below

{ } )()()()( xmxxmx nTgnTxTngnTx ∗=∆+∗=

16


17


• mTy = (km + ∆m)Tx

�For each m, the fractional interval ∆mTx determines the

Impulse response coefficients

�km specifies the input samples to compute y(mTy)

• This sampling rate conversion needs different impulse

responses gm(nTx) for each output sample y(mTy)

– Thus we have a Discrete time linear and time varying system

• The conversion process becomes inefficient when the

number of required values of gm(nTx) is large

18


19


number rational a ,

when simplified gets process conversion rate sampling The

mDmDmTmTmThen

I

D

T

T

yy

x

y

−=

−=∆

=

( )

( )

valuesof setsdistinct only can take )(,

,1

,.....2

,1

,0 valuesonly the can take

11

InTgSoI

I

IIm

mDI

II

mDmD

I

I

mD

I

mD

T

mT

T

mTmThen

xm

I

x

y

x

y

−∆

=

−=

−=

−=∆

20


system DT varyingly timeperiodical andlinear a have We

,...2,1,0),()(

period with periodic is)(

±±== + rnTgnTg

InTg

xrImxm

xm

• This system is much simpler than the continuously

time varying DT system

• The two cases of sampling rate conversion are

– Decimation or Down-sampling

– Interpolation or Up-sampling

21

Downsampling by an integer factor D

nTmTgnTxmTy

DTTI

I

D

T

T

xyxy

xy

x

y

)()()(

& 1Then

D integer,an by ngDownsampliFor

−=

==

==

∑∞

mallfornTgI

DTT

Ttgandtx

TnmDgnTxmDTyT

nTmTgnTxmTy

xmx

xy

x

xn

xxy

xyn

xy

)( oneonly have weand0 1, Since

of incrementsat shifted is response Impulse

period with sampled are)()(

))(()()(,for ngSubstituti

)()()(

=∆=

=

−=

−=

∑

∑

∞

−∞=

−∞=

22

Downsampling by an integer factor

23

Process of Downsampling

Upsampling by an integer factor I

)()(get Then we

So 1,D integer,an by lingFor Upsamp

−=

=

==

∑∞

−∞=nT

I

TmgnTx

I

TmymTy

I

TT

xx

nx

xy

xy

1 .., 1, 0, mfor )( responses Impulse of sets

)( of resampling requires shifting fractionalEach

of incrementsat shifted is response Impulse

period with sampled are)()(

−=⇒

=

InTgI

tg

I

TT

Ttgandtx

xm

xy

x

24

Upsampling by an Integer factor I

x(t)

Two impulse responses

required – for odd

numbered outputs and

even numbered outputsy(t)

(a)

25

x(t)

y(t)

Only one impulse

response required

(b)

(c)

Upsampling by an Integer factor I

– Upsampling can also be achieved in the method

shown in (c) above

• Determine an impulse response g(nTy) as shown

• Create a new sequence v(nTy) by inserting (I-1) zero

valued samples between successive samples of x(nT )valued samples between successive samples of x(nTx)

• Compute y(nTy) as the convolution of the sequences

g(nTy) and v(nTy)

26

Decimation by an integer factor D

Decimation

FFinzerononbeXLet

sequencesignalthebeXnxLet

x

F

πωω

ω

≤≤≤−

→←

reduction rate sampling is2

;0)(

)()(

DD

FF

nxD

Fnx

nx

x

x

πω ==

⇒

maxmax

th

or 2

to)( ofBW thereducemust wealiasing, avoid To2

at frequency foldingwith ),( of version aliasedan get We

)( of sample Devery Select

27


• Decimation process consists of a digital anti aliasing

filter h(k) and a Down sampler

– Down sampling is a data compression operation, where (D-1)

samples are discarded for every D samples of signal v(n)

28


{

range in the )( of spectrum theeliminatesIt

)()(

sticscharacteri ideal thehas LPF

,1

,0

DX

Hnh DOtherwiseD

F

<<

=→←≤

πωπω

ωπω

)()()()(

r,Downsample ofOutput

)()()( LPF ofOutput

range in the )( of spectrum theeliminatesIt

0

0

mykmDxkhmDv

knxkhnv

DX

k

k

=−=

−=⇒

<<

∑

∑

∞

=

∞

=

πωω

29


Overall Decimation operation is not Time invariant

• Low pass Filtering is a Linear Time Invariant operation

• The Down sampler is not time invariant

– x(n) produces y(m) does not imply that x(n-n0) � y(m-n0),

except when n is a multiple of Dexcept when n0 is a multiple of D

• Thus overall Decimation operation is not Time invariant

30


)()()(

)(

)( sequenceoutput of sticscharacteridomain Frequency

,...2,,0),(

,0

npnvnv

nvsquencetheLet

my

DDnnv

Otherwise

== ±±=

)()()()(

1 )(DFS, of In terms

D period with impulses of train periodic a is )(

)()()(

1

0

2

mDpmDvmDvmy

eD

np

np

npnvnv

D

k

D

knj

==

=

=

∑−

=

π

31

∑

∑∑

∞

−∞=

−

∞

−∞=

−∞

−∞=

−

=

==

m

D

m

m

m

m

m

zmpmv

zmDpmDvzmyzY

)()(

)()()()(

32

=

∞

−∞=

−−

=∑ ∑

m

D

mD

k

D

kmj

zeD

mvzY

mp

1

0

21

)()(

),(for ngSubstituti

π

=

=

=

−−−

=

−−

=

−−−

=

∞

−∞=

∑

∑∑ ∑

DD

kj

DD

kjD

kD

DD

kjD

k

m

DD

kjD

k m

zeXzeHD

zeVD

zemvD

12121

0

121

0

121

0

1

1)(

1

ππ

ππ

33

( )( ) πωπωω

ω

−

−=

⇒=

∑−

=

yD

k

yDy

y

D

kX

D

kH

D

zYmy

aliasing eliminatecan filter suitableA

221Y

circleunit on the )( Y )( of Spectrum

1

0

( )

πωπω

ωππω

ωωωω

≤≤≤≤

===

=

=

yx

xxy

y

yyyDy

toD

ORDF

FD

F

F

DX

DDX

DH

D

00 stretches ngDownsampli

22

11YThen

aliasing eliminatecan filter suitableA

34

Decimation in detail

35

Decimation by factor D = 3

36

Decimation by factor D

37

Decimation by factor D

38

Documents

Multirate DSP Part 1 [Read-Only]