Multirate Digital Signal Processing1a

7/29/2019 Multirate Digital Signal Processing1a

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S-88.3105 Digital signal

processing systems Lectures on Wednesdays 14.15 15.30 Room S5

and Fridays 8.45 10.00 Room S3

Digital signal processing has become one of themost important methods to handle information.The rapid development of different types of largemarket products such as mobile phones, dvd:s, etccould not have been possible without modern

DSP methods.

Lecturer: Professor Iiro HartimoRoom G406, e-mail: [email protected]
mailto:[email protected]:[email protected]


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Exercises, homework and

course assistant Exercises: check the web

All information about the exercises and homeworkcan be found from the exercise page.

Assistant: Mobien Shoaib

Room G401a, e-mail: [email protected] time after the exercises.
http://localhost/var/www/apps/conversion/tmp/scratch_3/s88115/fall2001/exercise.htmlmailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/s88115/fall2001/exercise.html


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Course prerequisites

Basics of digital signal processing (eg., the

HUT course T-61.246) or equal knowledge.

Basic Matlab knowledge will help in

solving the Matlab-homework problems

It is mandatory to join the course via

its web pages!


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Course contents and goals

This course is intended to present

fundamental, as well as advanced, concepts

that are important to the understanding oftimely methods of digital signal processing

needed for modern communication systems.

Digital filter banks constitute the main

subject of the course.


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Multirate digital signal

processing The rapid development of multirate digital

signal processing is complemented by the

emergence of new applications. Theseinclude subband coding of speech, audio,and video signals, multicarrier datatransmission, fast transforms using digital

filter banks and discrete wavelet analysis ofall types of signals.


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Multirate algorithms

A key characteristic of multirate algorithms

is their high computational efficiency. In

many cases, these algorithms are the primereason that an application can now be

implemented economically using modern

digital signal processors


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1 Sampling Rate Conversion

Many different sampling rates in the

system

Reduction of computational complexity.

What happens when the sampling rate ischanged?


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1.1.1 Discrete sampling

Discrete signals are often described using the

complex number

This is one of the M different M-th roots of 1,

since

On the unit circle of the complex plane:

1WM

M

(1.1)1/M)(-j2exp MMW


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Figure 1.1 Definition of WM

Im

Re

2/M

2/M

ZWM

Z

0 1


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Figure 1.2 Sampling of a discrete

signal

1 2 4 515

0 3

x(n)

n

n

32 4 5 15n

x(n)w4(n)

w4(n)

a)

b)

c)1


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Figure 1.3 Sampling with a phase

offset of =3

3

a)

b)

c)

n

15

x(n)

1 2 4 50 3 n

1 2 4 5 15

x(n)w4(n- )

n

w4(n- )


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The sampling function

(1.4)otherwise0

intgermmM,nfor1

M

1)(n

1M

0

)(n

MM Ww

(1.3)M

1n)((n)

1M

0

n

MMM Www

(1.2)otherwise0

integermmM,nfor1W

M

1(n)W

1M

0

n

MM

With the phase offset :

Note:


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1.1.2 Polyphase Representation

It is possible to sample M different

signals out of x(n) each having every

Mth sample of the original .

Or in general,

3)(nx(n)2)(nx(n)1)(nx(n)(n)x(n)

(1.5)(n)(n)(n)(n)x(n)

wwwwxxxx

4444

(p)

3

(p)

2

(p)

1

(p)

0

(1.6))(nw(n)x(n)x(n) M

1M

0

1M

0

(p)

x


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The z-transform

Can likewise be partitioned into M sub-signals. For example, for the

finite signal x(n) in Fig. 1.4, we have

(1.7)n-(n)X(z)n

zx

(1.8)x(15)x(11)x(7)x(3)

x(14)x(10)x(6)x(2)

x(13)x(9)x(5)x(1)x(12)x(8)x(4)x(0)X(z)

15-11-7-3-

14-10-6-2-

13-9-5-1-

-12-8-4-0

zzzzzzzz

zzzz zzzz


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gives0.....3,,offactoroutTaking z

(1.9)x(15)x(11)x(7)x(3)

x(14)x(10)x(6)x(2)

x(13)x(9)x(5)x(1)

x(12)x(8)x(4)x(0))(

zzzzzzzzzz

zzzzz

zzzzz

12-8-4-0-3-

12-8-4-0-2-

12-8-4-0-1-

12-8-4-0-0-

zx

These are polynomials in z-4


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Figure 1.4Polyphase representation of adiscrete signal (next slide):


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150 1 2 4 53

x(n)

n

n

n

n

n

x 1(p)(n)

x 3(p)(n)

x 1(n)

x 2(p)(n)

x 0(p)(n)

=0

=1

=2

=3


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(1.10))().x(mMX(z)

Mmn

zXzz

M1M

0

(p)

1-M

0 -m

)(mM-

m

m M-M(p)

(1.11))x(mM)( zzX

Equation (1.10) is called the

polyphase representation of the z-transform X(z)

where


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(1.12))12()8()4()0()(

:polynomialFirst

3210(p)

0x zzzz xxxxz

One-to-one Mapping

z n

(1.13)1-M0,1,2....(n),)()()(-

z

xzxpMp

(1.14)T(z)(z),....,(z),(z):FormVector

XzXzXX(p)

1M

1)(M(p)

1

1(p)

0

(p)


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(1.17))1Mx(mM(m)

and

(1.16)(m).)(

Where

(1.15))(X(z)

:88bVai

tionrepresentapolyphase2typethetoleads-1-MbyReplacing

x

zxzX

zXz

(p2)

-m

mM(p2)

M(p2)

M1M

0

(p2)

)1(m


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:83Crotionrepresentastandardthefromobtainedbetion willrepresenta

polyphase3typethe-byReplacing

order.reversein the

doneisindexingOnly theidentical.ethereforare

(n)and1.4cFig.from(n)signalsThe

(1.18)1-M...0,1,2.....(z),(z)

byrelatedaretionsrepresenta2typeand1typeThe

xx

xx(p2)

2

(p)

1

(p)

1M

(p2)


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identicalthereforeare(n)and1.4cfigin(n)signalsThe

(1.23)1...M1,2,3.....(z),(z)and

(1.22)(z)(z)

byrelatedaretionsrepresenta3typeand1typeThe(1.21))x(mM(m)

and

(1.20)(m)Where

(1.19))((z)

xx

xzx

xx

x

zxzX

zXz

(p3)

3

(p)

1

(p)

M

1(p3)

(p)

0

(p3)

0

(p3)

mM(p3)

m

M(p3)

M(p3)

1-M

0

X


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1.1.3 Modulation Representation

M

k2byfrequencytheshiftingModulation

(1.25))()()(

Transform)Fourier(Z

(1.24)1-0,1,2....Mk),(z)(

byzargumentheMultiply t

k/M]2-j [k/Mj2-jj(m )

k

j

k

M

j(m )

k

kM

eXeeXeX

eWXeX

W


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In The Time Domain

kn/M)sin(2jx(n)kn/M)cos(2x(n)

(1.26)kn/M)exp(j2x(n)x(n))()X(z WW

kM

kM


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)(signalaof

tionrepresentaModulation

1.5Figure

eXj


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0

0 2

0 2

0 2

x(m )

1

x(m )

2

M/2

M/4

M/6

x(m )

3

)b

)c

2

XX(m )

0

/M

)a0k

1k

2k

3k)d


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m

n(p)

M(p)

-

T(m )

1M

(m )

1

(m )

0

(m )

kn

M

kn-

M

k-M

M

k

M

(1.29).)(

(1.28)(z)......(z)(z)(z)

:formMatrixin the

Transforms-zmodulatedofsetcompleteThe

(1.27)kn/M)cos(22x(n)

x(n)x(n))(z)(z

combinetohavewe

signalsDomainTimecomplexavoidTo

zXzXz

xxxx

WWWXWX


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1.1.4 Transformation of the

signal Components

(1.30)

M

1x(n)

)(nx(n)(n)

1M

0

n)(

M

M(p)

W

wx

From (1.13):


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substituting the expression in

(1.30) into equation (1.29) gives,

replacing by k,

(1.31)x(n)M

1

nx(n)M1

.M

1(n))(

-

W(zW

Wz

zWxzXz

k

M

1M

0k

n-

n

k

M

1M

0k n

n)k(

M

n

-n

1M

0k

n)k(

M

M(p)

(z))x(z XW(m)

k

k

M


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The relationship between the polyphase components

and the modulation components is thus

the following

1M

0k

k

M

(m)

k

M(p)

WXzXz (z)M1

)()32.1(

(1.33)(z).M1(z)

MatrixDFTUsing

xWx (m)M(p)


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Reducing the Sampling Rate

Decimation

ANTI-ALIASING

M


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1.2.1 Downsampling

The sampling rate of a discrete signal x(n) is

reduced by a factor M by taking only every

M-th value of the signal. The relationship between

the resulting signal y(m) and

the original signal x(n) is as follows:

y(m) = x(mM) (1.34)

Fig1.6 shows a signal flow representation of thisprocess:


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Figure 1.6 Downsampler

Mx(n) y(m)


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Can also be described using the polyphase

representation using discrete sampling

function and leaving out the zeroes.In the z-domain, we can use the z-transform of

the original signal,

)n(wM

(1.36)zYzY

zy(m)

zM)x(m

z

transform-zobtain theto0,with(1.11)and

(1.35)zx(n)(z)

M

m

m

M

mM

m

M(p)

0

n

n-

x

X


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Figure 1.7

steps in signalprocessing used to perform

downsampling


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15

x(n)

1 2 4 50 3n

1 4 m

T

31 2 4 5 15n

)(

x

(p)

0

n

)0(x )4(x )12(x

y(m)

)0(x )4(x )12(x

3

T4

a)

b)

c)


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(1.41)zz

thenisvariablestwoebetween thiprelationshthe(1.39),from

(1.40)z

variablenewadefinecanwesoand(1.39)MTT

nowissamplesbetweenspaceThe

(1.38)zplaneLaplaceusing

(1.37)y(m).)zY()Y(

explainedbecanvalueszeroout theLeaving

M

Ts

sT

M

e

e

z


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(1.45)zx(12)zx(8)zx(4)x(0))zY(

1.7c,fig.insignalddownsampletheoftransform-zthethusand

(1.44)x(12)x(8)x(4)x(0))Y(

polynomialget thewe,thisFrom

(1.43)x(12)x(8)x(4)x(0))(

is1.7bfig.in0withcomponentpolyphaseThe

(1.42)x(15)x(14)....x(2)x(1)x(0)X(z)bygivenis1.7afiginx(n)signaltheoftransform-zThe

:1.1Example

321

34

24

144

12844(p)

0

151421

zzzz

zzzzX

zzzz


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1.2.2 Spectrum of the

Downsampled Signal

(1.47))X(M

1)Y(

ansformFourier trtime-discretethederiveushelpsezonsubstitutithesignals,stableFor

(1.46))X(zM

1)Y(

obtainwe

,0with(1.32)From

.componentsmodulationusingexpressedbecan(1.36)in

)Y(signalsampleddiscretelytheoftransform-zThe

ee

Wz

z

k/Mj2j1-M

0k

jM

j

1-M

0k

k

M

M

M


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In the following, the magnitude of this

transform is referred to

as the magnitude spectrum,often shortened to just spectrum.

i 1 8 b i d i


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Fig1.8 spectra obtained using

downsampling

0 2

0

Xm

X)(

0

2

M/

must be bandlimited

0 2

Y

0 k 1 k 2 k 3 k 0 k

M/2 M/4 M/6

M2 2isRateSamplingNormalised


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(1.50))(XM

1)Y(

ofin termsSpectrum

Tspacingtimescorrespond

(1.49)MfrequencynormalizedThe(1.48)

js

M

2ratesamplingNew

MfactorbydecreasedisMagnitude

2isRateSamplingNormalised

ee

eeee

k]/M2j [1M

0k

j

jTjTjMjM


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(1.51)/(3M)T2T:isfrequency

normalisedingcorrespondThe/(6M).

frequencyat thespectrumtheofvaluethefindto

wishWe.Moffactorabyddownsamplethenand1/T

frequencyaatsampledbeenhaswhichsignalaConsider

f

ff

f

111

o1

o

:1.2Example


44/67

44

1.8b.fig.inshownalsoisThis

(1.53))X(

M

1)Y(

namely(1.50),into(1.52)fromngsubstitutiasresultsamethegives(1.47)into(1.51)fromthengSubstituti

(1.52)/3T2T

:frequencynormalizednewtherespect towith

frequencynormalizedthecalculatesimilarlycanWe

1-M

0k

k/Mj2-/3Mj/3j

/

1

1

11

/

1

ee

fw


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45

1.2.3 Aliasing Effects

(Vierastumisilmit)

h(n) Mu(n)x(n) y(m)

Fig 1.9 Decimator consisting of an anti-

aliasing filter h(n) and

a down-sampler M

Ua) limitedbandnot


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Fig 1.10 The effect of anti-aliasing

low-pass filter

Anti-aliasing

filter

Band-limited

Signal0 2M/

0 2M/

X

U

H

a)

b)

c)

0 2M/

M)/(to

limited-bandnot

Signal


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After Filtering:

k

k

kmMhku

knhkununx

(1.55)).()(y(m)

sampleddownbewill

(1.54)),()(h(n)*)()(


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48

In the z-transform Domain:

(1.57))(zU)H(zM

1)Y(

:thenis

signaldecimatedtheoftransform-zthe(1.46),From

(1.56)U(z)H(z)X(z)

WWzK

M

1M

0k

K

M

M


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1.2.4 Scaling of the Anti-Aliasing

FilterThe gain of the Anti-aliasing filter=?

Consider the sampled analog signal

(1.59)/T2

,)(T1)X(x(n)

thenis

x(n)signaldiscretetheofspectrumperiodicThe(1.58)).((n)x

0

0a

Tj

a

Xe

x

n

jnj

nT

i ld i t dhid ti il lW


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(1.61)/M)jn-(jMT

1

)Y(y(n)

isspectrumperiodicingcorrespondThe(1.60)(mMT)y(m)

:MTofspacingsampleawith

timebut this,(t)signalcontinuoussamethe

samplingbyobtainedbeenhaveto(1.34)iny(m)

signaldecimatedheconsider tsimilarlycanWe

noa

MTj

a

a

x

e

x

x


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51

M

1factorscaling

(1.63)),(MT

1)(

and

(1.62))(T1)(

:0issexpression

summetheofnindexthewherebaseband,in the

(1.61)and(1.59)spectratwothecomparingBy

Y

X

o

o

j

j

Xe

Xe

a

MTj

aTj


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1.2.5 Decimation of Band-pass

Signals

/M2ofshiftfrequencya

bySignalBasebandthefromformedis

(1.64)integer),X(z(z)signalpass-Band

WX M(m )

modulation

)(


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Figure 1.11 Spectra of decimated

band pass signals

0 2M/

1 k 2 k 3 k0 k

Xm

X)(

1

1 k

2M/2 M/4 M/6

)X(zsignalpass-bandthengDownsampli MW


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54

(1.66))X()Y(

and

(1.65))X()Y(

givesX(z),ofinstead,(1.47)and(1.46)

into)X(zsignalpass-bandthengsubstituti

X(z).signalbasebandtheofngdownsampliingcorresponda

asresultsamethegivesMfactoraby

)(gpgp

1-M

0k

k)/M(j2jjM

1-M

0k

k

MM

M

M

ee

WWz

W

M1

zM1

Ua)


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Figure 1.12 The effect of anti-

aliasing band-pass filtering

0 2

0 2

0 2

X

U

H

a)

b)

c)

M/2


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Anti-Aliasing Band-passFilter is needed

1byshiftedkindexwith the

(1.47)inspectrumthetoidenticalisIt

)2(periodinperiodicis)(z WMX


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1.2.6 Downsampling with

a phase offset

(1.67)1M...0,1,2),Mx(m(m)

:introducedbewilloffestPhase

y

x(n)


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58

Figure 1.13 Downsampling

with a phase offset

1501 2 4 53

x(n)

n

n

x 2(p)(n)

15

m

y2(m)

M/1 M/3 M/

componentpolyphase

2

(1 13)t f


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59(1.70))X(

:(n)componentspolyphasetheofspectrumthededuceto

ezsetcanwehere,From

(1.69))X()(

:(1.24)indefinedcomponentsthe

usingwrittenbecanthis(1.32),From

(1.68)(n))(

(1.13)transform-z

We

x

WzWzz

xzxz

k

M

1M

0k

k/Mj2-j

(p)

j

kM

1M

0k

kM

M(p)

-

(p)

M(p)

-

M

1

X

givenalreadyx(n)signaltheshows1 13afig


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60

(1.71))X()((m)

thereforeisspectrumIts

1.2.1).section(seelyrespective,/by

orby Zreplacingby1.13c,fig.in

shown(m)signalddownsamplethe

obtaincanwehere,From(n).components

polyphasethe1.13bfigand1.2a,figin the

givenalreadyx(n),signaltheshows1.13afig

1

0

k/]2j [

j

M

2

(p)2

We

eYy

z

y

x

M1

M

kM

Mk

M


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The nonzero values of the downsampled polyphase

components will not appear at integeral values

of m unless =0. This is shown in fig 1.13c. This

representation is occasionally used for

hypothetical filter prototypes.


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62

Example 1.3:

Consider the finite real signal x(n) in the Fig 1.14a,

whose spectrum X(exp j ) is shown in the Fig

1.15a, the non-causal signal is even, i.e. x(n)= x(-n).

The spectrum is thus real and has even symmetry.

x(n)


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63

Figure 1.14 Downsampled

polyphase components with M=2

n0 1 2

m0 1

m0

x~0

x~1

1/2 3/2 7/2

= 0

= 1


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64

(1.73),)X(

2

1

1.2.2).section(see2ofafactorbyratesampling

thereducedhavingand0offestphaseahavingwith

compatiblebeseen toisresultThis.respctivly2or

periodawithspectrumrealaformtocombine

termssumtwoThe1.15b.figinshownisspectrumThis

(1.72))X(2

1

:spectrumthe

obtainwe(1.70),into2MngSubstituti.0fori.e.

(m),componentddownsampletheshows1.14bFig

1

0k

K

2

k]j[(p)

1

1

0k

k]j[(p)

0

(p)

0

1)(k

Wex~

ex~

x~


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65

true.alwaysist thisevdent thaisIt

spectrum.theofperiodaffect thenotdoes0

offsetphaseagIntroducin.4or2periodahas(m)ofspectrumtheresult,aAs

.signsoppositehavetermssumtwothe1offest

phaseaithbut that wmagnitude,samethehavespectrathat tworeveals1.15cfigwith1.15bFig.or

(1.73)with(1.72)Comparing1.15c.FigindisplayedisThis

x~(p)

1

X


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66

Figure 1.15 spectra of

downsampled polyphase

components

0 2

20

20

42

20

a)

b)

c)

Zero Padding = insert zeroes


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Zero Padding insert zeroes

between existing samples

M=4M=5

Documents

Multirate Digital Signal Processing1a