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Multi-rate Digital

Signal Processing

Dr Suprava Patnaik

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Pending Topics

Multirate Signal Processing

- Decimation

- Interpolation

Multistage decimation and interpolation Poly-phase Filtering Filter banks and wavelet transform 23rd and 25th Tutorial (Assignment & class

performance evaluation- 25 marks) Quiz test of 20 marks. Midtest(30)+ End test(50)

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Why Multi-rate Processing

Multirate signal processing deals with a change insampling rate for discrete signals.

Multirate signal processing is used for the practicalapplications in signal processing to save costs,

processing time, for compression, feature extraction,device compatibility and many other practical reasons.

Basic Sampling Rate Alteration DevicesBasic Sampling Rate Alteration Devices

Up-samplerUp-sampler - Used to increase the sampling rate by an

integer factor Down-samplerDown-sampler - Used to decrease the sampling rate byan integer factor

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Up-SamplerUp-SamplerTime-Domain CharacterizationTime-Domain Characterization

An up-sampler with an up-sampling factorup-sampling factorL, where L is apositive integer, develops an output sequence witha sampling rate that is L times larger than that of theinput sequencex[n].

Up-sampling operation is implemented by inserting

equidistant zero-valued samples between twoconsecutive samples ofx[n]

Input-output relation

Block-diagram representation

][nxu

Lx[n] ][nxu

==otherwise,0

,2,,0],/[][

LLnLnxnxu

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0 10 20 30 40 50-1

-0.5

0

0.5

1Input Sequence

Timeindexn

Amplitude

0 10 20 30 40 50-1

-0.5

0

0.5

1Output sequenceup-sampledby3

Timeindexn

Amplitude

Figure above demonstrates up-sampling by factor 3.

In practice, the zero-valued samples inserted by the up-sampler

are replaced with appropriate nonzero values using some type of

filtering process.

This process is called interpolationinterpolation

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Example of Up-Sampling

X(n)=8 8 4 -5 -6

Y(n)=8 0 0 8 0 0 4 0 0 -5 0 0 -6 0 0, L=3

If T=original sampling rate, TL=T/L

fsL=Lfs ( Folding frequency will increase by a factor L) After up-sampling, the spectral replicas originally centered

at fs, 2fs,.are included in the frequency range 0Hz to the

new Nyquist limit Lfs/2 Hz.

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Equivalent z-domain equations

1 1

1 NZ

z

+ 1 1

1N

Z

z

Time Domain figure

Is inserting zero equivalent to inserting some other value

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Up-SamplerUp-Sampler

Frequency-Domain CharacterizationFrequency-Domain Characterization Consider first a factor-of-2 up-sampler whose input-output

relation in the time-domain is given by

In terms of the z-transform, the input-output relation is then

given by

=

=

==

even

]/[][)(

nn

n

n

nuu znxznxzX 2

== otherwise,,,,],/[][ 0 4202

nnxnxu

2 2

[ ] ( )

m

mx m z X z

== =

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Up-SamplerUp-Sampler

In a similar manner, we can show that for a factor-of-factor-of-LL up-up-

samplersampler

On the unit circle, for , the input-output relation isgiven by

In the case of a factor-of-L sampling rate expansion, there

will be L-1 additional images of the input spectrum inthe baseband

Lowpass filtering of removes the L-1 images and in

effect fills in the zero-valued samples in with

interpolated sample values

][nxu

)()(L

u zXzX =

j

ez=)()(

Ljju eXeX

=

][nxu

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Up-Sampling is responsible for spectrum compression and

presence of more than one image spectrum below folding

frequency.

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Imaging( Removal requires

Interpolation/ anti-image filter)

As can be seen, a factor-of-2 sampling rate expansion leadsto a compression of by a factor of2 and a 2-fold repetition in

the baseband [0, 2].

This process is called imagingimagingas we get an additional

image of the input spectrum

To remove those extra spectral replicas, an interpolation filter

with a stop frequency edge fs/2 must follow the up-sampler.

Normalized stop frequency edge is:2

2

sstop

f Tradians

L L

= =

4000 25003250

2

12 3250 0.2708

24000

cut off

c

f

= =

= =

2 1000 2 2500( ) 5sin cos

8000 8000

n nx n

= +

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Down-SamplerDown-SamplerTime-Domain CharacterizationTime-Domain Characterization

An down-sampler with a down-sampling factordown-sampling factorM,where Mis a positive integer, develops an output

sequence y[n] with a sampling rate that is (1/M)-th of

that of the input sequencex[n]

Down-sampling operation is implemented by keepingevery M-th sample ofx[n] and removing M-1

samples in-between samples to generatey[n]

Input-output relation

y[n] =x[nM] Block-diagram representation

Mx[n] y[n]

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0 10 20 30 40 50-1

-0.5

0

0.5

1Input Sequence

Timeindexn

Amplitu

de

0 10 20 30 40 50-1

-0.5

0

0.5

1Outputsequencedown-sampledby3

Amplitude

Timeindexn

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

Input sampling frequency

TFT 1=

Output sampling frequency

'1'TM

FF TT ==x[n]=8 7 4 8 9 6 4 2 -2 -5 -7 -7 -6 -4 with T=0.1 , fs=10

y[n]=8 8 4 -5 -6 T=3 x 0.1=0.3 , fsM=3.3

Precaution has to be taken to avoid aliasing due to reducedsam lin rate.

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After down-sampling new folding frequency

reduces by factor M.

If the original signal has frequency componentslarger than the new folding frequency aliasing

noise will be introduced into down-sampled data.

To overcome this the original signal has to be

processed by a LPF before down-sampling,

which to stop frequency components above fs/

(2M) Hz.

Normalized cut-off frequency is22

sstop

f Tradians

M M

= =

max

2

sff

M

M the system will not introduce aliasing

If M>L ,x[n] must be band limited to the new nyquist rate either intrinsically

or by a filter.

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Changing sampling rate by L/M

Interpolation and anti-aliasing filtersappear in cascade.

Since both operate at same rate, we can

select one of them. We choose the one with the lower stop

frequency edge and choose the most

demanding requirement for pass-bandgain and stop-band attenuation for filterdesign.

=ML

s

,min

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Cascade EquivalencesCascade Equivalences Two other cascade equivalences are shown below

L][nx ][2 ny)(LzH

L][nx ][2 ny)(zH

M][nx ][1 ny)(zH

M][nx )(MzH ][1 ny

Cascade equivalence #1Cascade equivalence #1

Cascade equivalence #2Cascade equivalence #2

1

1 11

0

( ) ( ) ( )( )

1( ) ( )

M

Mk k MM M

k

Y z X z H z downsampler

X z W H z WM

=

=

=

1

1

1( ) ( ) ( )

kMY z X z W H z M

=

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Why FIR Filter?

Linear Phase

FIR requires less computation

=

=1

0

N

m

mnxmhnv ][][][

If the decimation filterH(z) is an FIR filter of length Nimplemented in a

direct form, then

Now, the down-sampler keeps only every M-th sample of v[n] at its output.

Hence, it is sufficient to compute v[n] only for values ofn that are multiples ofM

and skip the computations of in-between samples. This leads to a factor of M

savings in the computational complexity

Now assume H(z) to be an IIR filter of orderKwith a transfer function

nK

n nzpzP

==

0

)( nK

n nzdzD

=+=

11)(

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Its direct form implementation is given by

Since v[n] is being down-sampled, it is sufficient

to compute v[n] only for values ofn that are

integer multiples ofM

However, the intermediate signal w[n] must be

computed for all values ofn

As a result, the savings in the computation inthis case is going to be less than a factor ofM

= ][][][ 21 21 nwdnwdnw ][][ nxKnwdK +

][][][][ Knwpnwpnwpnv K +++= 110

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Multistage Decimation

On Board

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Thank You