Multi Rate DSP.ppt

8/14/2019 Multi Rate DSP.ppt

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Multirate Digital SignalProcessing


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Multirate Digital Signal

Processing

What is multirate signal processing?

Processingof digital signal withdifferent sampling rates in the system.

Sampling Rate Conversion


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Processing

Up-sampler- Used to increase

the sampling rate by an integer

factor

Down-sampler- Used to decrease

the sampling rate by an integer

factor

Basic Sampling Rate Alteration Devices


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Why sample rate conversion? (I)

Compat ib i l i ty : convert sample frequencies ofdifferent stds.

Eff ic iency: easier data processing(computationally more efficient), less storage,

lower transmission speed, All-digi tal: Change sample frequency in an

efficient manner

Cost : Avoid need for expensive analogue anti-

aliasing filters


Processing


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Up-Sampler

Time-Domain Characterization An up-sampler with an up-sampling

factorL, where Lis a positive integer,

develops an output sequence witha sampling rate that is Ltimes larger

than that of the input sequencex[n]

Block-diagram representation

][nxu

Lx[n] ][nxu


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Up-Sampler

Up-sampling operation is implemented by

inserting equidistant zero-valued

samples between two consecutive

samples ofx[n]

Input-output relation

1L

otherwise,0

,2,,0],/[][

LLnLnxnxu


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Up-Sampler

In practice, the zero-valued samples

inserted by the up-sampler are replaced

with appropriate nonzero values using

some type of filtering process

Process is called interpolationand will be

discussed later


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Down-Sampler

Time-Domain Characterization An down-sampler with a down-sampling

factorM, where Mis a positive integer,

develops an output sequence y[n]with asampling rate that is (1/M)-th of that of

the input sequencex[n]

Block-diagram representation

Mx[n] y[n]


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Down-Sampler

Down-sampling operation is implemented

by keeping every M-th sample ofx[n]and

removing in-between samples to

generatey[n]

Input-output relation

y[n] =x[nM]

1


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Down-Sampler

Figure below shows explicitly the time-

dimensions for the down-sampler

M )(][ nMTxny a)(][ nTxnx a

Input sampling frequency

TFT

1

Output sampling frequency

'1'

TMFF TT


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Up-Sampler

Figure below shows explicitly the time-

dimensions for the up-sampler

Input sampling frequency

TFT

1

otherwise0

,2,,0),/( LLnLnTxa

L)(][ nTxnx a y[n]

Output sampling frequency

'

1'T

LFF TT


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Basic Sampling Rate Alteration Devices

The up-samplerand the down-samplerare

linearbut time-varying discrete-time systems

Consider a factor-of-Mdown-sampler defined

by

Its output for an input is

then given by

From the input-output relation of the down-

sampler we obtain

y[n] =x[nM]

][1 ny ][][ 01 nnxnx

][][][ 011 nMnxMnxny

)]([][ 00 nnMxnny ][][ 10 nyMnMnx


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Up-Sampler

Frequency-Domain Characterization

Consider first a factor-of-2up-sampler

whose input-output relation in the time-domain is given by

otherwise,

,,,],/[][

0

4202 nnxnx

u


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Up-Sampler

In terms of the z-transform, the input-

output relation is then given by

even

]/[][)(

nn

n

n

nuu znxznxzX 2

2 2[ ] ( )m

m

x m z X z


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Up-Sampler

In a similar manner, we can show that

for a factor-of-Lup-sampler

On the unit circle, for , the input-

output relation is given by

)()( L

u zXzX jez

)()( Ljju eXeX


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Up-Sampler

Figure below shows the relation betweenand for L= 2in the

case of a typical sequencex[n])( jeX )(

ju eX


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Up-Sampler

As can be seen, a factor-of-2sampling

rate expansion leads to a compression

of by a factor of 2and a 2-foldrepetition in the baseband[0, 2p]

This process is called imagingas we

get an additional image of the inputspectrum

)( j

eX


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Up-Sampler

Similarly in the case of a factor-of-L

sampling rate expansion, there will be

additional images of the input spectrum in

the baseband

Lowpass filtering of removes the

images and in effect fills in the zero-

valued samples in with interpolatedsample values

1L

1L

][nxu

][nxu


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Down-Sampler

Frequency-Domain Characterization

Applying the z-transform to the input-output

relation of a factor-of-Mdown-sampler

we get

The expression on the right-hand side cannot

be directly expressed in terms ofX(z)

n

nzMnxzY ][)(

][][ Mnxny


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Down-Sampler

To get around this problem, define a

new sequence :

Then

otherwise, ,,,],[][int 0 20

MMnnxnx

][int nx

n

n

n

n

zMnxzMnxzY ][][)( int

)(][ /int/

intM

k

Mk zXzkx 1


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Down-Sampler

Now, can be formally related tox[n]through

where periodic train c[n]

A convenient representation of c[n]is givenby

where

][int nx

][][][int nxncnx

otherwise,

,,,,][

0

201 MMnnc

1

0

1 M

k

knMW

Mnc ][

Mj

M eW

/p2


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Down-Sampler

Taking the z-transform of

and making use of

we arrive at

][][][int nxncnx

1

0

1 M

k

kn

MWMnc ][

n

n

M

k

knM

n

n

znxWMznxnczX

][][][)(int

1

0

1

1

0

1

0

11 M

k

kM

M

k n

nknM WzX

MzWnx

M][


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Down-Sampler

Consider a factor-of-2down-samplerwith an inputx[n]whose spectrum is asshown below

The DTFTs of the output and the inputsequences of this down-sampler arethen related as

)}()({

2

1)( 2/2/ jjj eXeXeY


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Down-Sampler

Now implyingthat the second term in the

previous equation is simply obtained by

shifting the first term to the rightby an amount 2pas shown below

)()( 2/)2(2/ p jj eXeX)( 2/ jeX

)( 2/jeX


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Down-Sampler

The plots of the two terms have an overlap,

and hence, in general, the original shape

of is lost whenx[n]is down-sampled

as indicated below

)( jeX


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Down-Sampler

This overlap causes the aliasingthat takes

place due to under-sampling

There is no overlap, i.e., no aliasing, only if

Note: is indeed periodic with a

period2p, even though the stretched

version of is periodic with a period

4p

2/0)( p forjeX

)( jeX

)( jeY


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Down-Sampler

For the general case, the relation between

the DTFTs of the output and the input of a

factor-of-Mdown-sampler is given by

is a sum of Muniformly

shifted and stretched versions of

and scaled by a factor of1/M

p 1

0

/)2( )(1

)(M

k

Mkjj eXM

eY

)( j

eY

)( jeX


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Down-Sampler

Aliasing is absent if and only if

as shown below for M= 2

2/for0)( pj

eX

MforeX j /0)( p


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Filters in Sampling Rate

Alteration Systems The bandwidth of a critically sampled

signal must be reduced by lowpass

filteringbefore its sampling rate is

reduced by a down-sampler to avoid

aliasing

Likewise, the zero-valued samples

introduced by an up-sampler must beinterpolated by lowpass filteringto more

appropriate values for an effective

sampling rate increase


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Filter Specifications

Since up-sampling causes periodic

repetition of the basic spectrum, the

unwanted images in the spectra of the up-

sampled signal must be removed byusing a lowpass filter H(z), called the

interpolat ion f i l ter, as indicated below

The above system is called aninterpolator

][nxu

L][nx ][ny)(zH][nxu


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Filter Specifications

On the other hand, prior to down-

sampling, the signal v[n]should be

bandlimited to by means

of a lowpass filter, called the decimationfilter, as indicated below to avoid aliasing

caused by down-sampling

The above system is called adecimator

M/p

M][nx )(zH ][ny


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Interpolation Filter

Specifications If we passx[n]through a factor-of-Lup-sampler generating , the relation

between the Fourier transforms of x[n]and

are given by

It therefore follows that if is passedthrough an ideal lowpass filter H(z)with a

cutoff at p/Land a gain of L, the output of

the filter will be precisely y[n]

][nxu

][nxu

)()( Ljju eXeX

][nxu


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Interpolation Filter

Specifications If is the highest frequency that needs

to be preserved inx[n], then

Summarizing the specifications of the

lowpass interpolation filter are thus given

by

c

Lcp /

pp

L

LLeH cj

/,

/,)(

0


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Decimation Filter Specifications

In a similar manner, we can develop thespecifications for the lowpass decimationfilter that are given by

The design of the filter H(z) is a standardIIR or FIR lowpass filter designproblem

pp

M

MeH cj

/,

/,)(

0

1


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The FIR filter is realized using direct form

To avoid unnecessary calculations the decimator

is replaced with efficient transversal structure.

For the polyphase structure

][*][)(1

0

nxnpny mm

M

m

Polyphase Decomposition

][][)(1

0

nxmhny m

N

m


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Decomposition ofH(z)=hmz -m in blocks ofM:

H(z) = ...+h(M)zM+h(M+ 1)zM1+ ... +h(1)z1

+h(0)z0+h(1)z1+ ... +h(M 1)z(M1)

+h(M)zM+h(M+ 1)z(M+1)+ ... +h(2M 1)z(2M1)

+h(2M)z2M+h(2M+ 1)z(2M+1)+ ... +h(3M 1)z(3M1)+ ...

=z0[... +h(0)z0+h(M)zM+ ...]+z1[... +h(1) +h(M+ 1)zM+ ...]

+z2[... +h(2) +h(M+ 2)zM+ ...] + ...+z(M1)[... +h(M 1) +h(2M 1)zM+ ...]

H(z)=z Pizi

i=0

M1

( )Mwhere Pi(z)=

n=

z h(nM+i)n+


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Implementation of Decimation

Using noble identity:

Operations performed at Operations at low rate

high rate more efficient


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Using commutator:

Implementation of Decimation

one input perDpulses;

counter-clockwise rotation


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THANK YOU

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Multi Rate DSP.ppt