unit 4 random signal processing

8/8/2019 Unit 4 Random Signal Processing

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Unit 4 Random Signal

Processing

MULTIRATE SIGNAL

PROCESSING-LAST PART OFUNIT 4


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Introduction to probability function, joint probability,conditional probability estimation parameters jointdistribution function, probability density function,ensemble average mean squared value, variance,

standard deviation, moments, correlation, covariance,orthogonality, auto-covariance, auto-correlation, cross-covariance and cross-correlation stationarity ergodic

white noise energy density spectrum powerdensity spectrum estimation periodogram directmethod, indirect method, Bartlett method Welch

method. Decimator (down sampling) frequency-domain analysis of decimator interpolation (upsampling) frequency-domain analysis ofinterpolator

BASICS OF RANDOM SIGNAL PROCESSING


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HOD WILL HANDLE:

power density spectrum estimation periodogram direct method, indirect method, Bartlett method Welch method

ProF.J.Valarmathi:VIT Introduction to probability function, joint probability,

conditional probability estimation parameters jointdistribution function, probability density function,ensemble average mean squared value, variance,

standard deviation, moments, correlation, covariance,orthogonality, auto-covariance, auto-correlation, cross-covariance and cross-correlation stationarity ergodic

white noise energy density spectrum


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Definition of Multirate Signal

Processing The systems that uses single sampling

rate from A/D converter to D/A converter

are known as single rate systems. When the sampling rate of the signals is

unequal at various parts of the system-

Those discrete time systems that process

data at more than one sampling rate are

known as multirate systems.


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Where Multirate signal processing

is used: 1.In high quality data acquisition and

storage systems.

2.In audio signal processing. For examplea CD is sampled at 44.1KHz but DAT issampled at 48 KHz. Conversion betweenDAT and CD use multirate signal

processing technique. 3.Narrow band filtering for fetal ECG and

EEG


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4.In video PAL and NTSC run at different

sampling rates. Therefore to watch an

American program in Europe one need asampling rate converter.

5.In speech processing to reduce the

storage space or the transmitting rate of

the speech data.

6.In transmultiplexers.


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Basic operations: Interpolation and

decimation.Basic Sampling Rate Alteration DevicesBasic Sampling Rate Alteration Devices

UpUp--samplersampler- Used to increase the

sampling rate by an integer factor DownDown--samplersampler- Used to decrease the

sampling rate by an integer factor.


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UpUp--SamplerSampler

Up-sampling operation is implemented by

inserting equidistant zero-valued

samples between two consecutive

samples ofx[n]

Input-output relation

1L

ss!!

otherwise,0

,2,,0],/[][

.LLnLnxnxu


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0 10 20 30 40 50-1

-0.5

0

0.5

1

Input Sequence

Time index n

Amplitude

0 10 20 30 40 50-1

-0.5

0

0.5

1Output sequence up-sampled by 3

Time index n

Amplitude


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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 interpolationinterpolation and will be

discussed later


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DownDown--SamplerSamplerTimeTime--Domain CharacterizationDomain Characterization

An down-sampler with a downdown--samplingsampling

factorfactorM, where Mis a positive integer,

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

the input sequence x[n]

Block-diagram representation

Mx[n] y[n]


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DownDown--SamplerSampler

Down-sampling operation is implemented

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

removing in-between samples to

generate y[n]

Input-output relation

y[n] = x[nM]

1M


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0 10 20 30 40 50-1

-0.5

0

0.5

1Input Sequence

Time index n

Amplitude

0 10 20 30 40 50-1

-0.5

0

0.5

1Output sequence do n-sampledby 3

Ampl

itude

Time index n


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Basic Sampling Rate AlterationBasic Sampling Rate Alteration

DevicesDevices Sampling periods have not been explicitly

shown in the block-diagram

representations of the up-sampler and the

down-sampler

This is for simplicity and the fact that the

mathematical theory of multirate systemsmathematical theory of multirate systems

can be understood without bringing thesampling period Tor the sampling

frequency into the pictureTF


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DownDown--SamplerSampler 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'TM

FF

T

T!!


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Figure below shows explicitly the time-

dimensions for the up-sampler

Input sampling frequency

TFT

1!

!!

other ise0

,2,,0),/( -LLnLnTxa

L)(][ nTxnx a! y[n]

Output sampling frequency

'

1'T

LFFTT!!


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Basic Sampling RateBasic Sampling Rate

Alteration DevicesAlteration Devices

The upup--samplersamplerand the downdown--samplersampler

are linearlinearbut timetime--varying discretevarying discrete--timetimesystemssystems

We illustrate the time-varying property

of a down-sampler The time-varying property of an up-

sampler can be proved in a similar

manner


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Basic Sampling RateBasic Sampling Rate

Alteration DevicesAlteration Devices Consider a factor-of-Mdown-sampler

defined by

Its output for an inputis then given by

From the input-output relation of thedown-sampler we obtain

y[n] = x[nM]

][1 ny ][][ 01 nnxnx !

][][][ 011 nMnxMnxny !!

)]([][ 00 nnMxnny !][][

10nyMnMnx {!


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FrequencyFrequency--Domain CharacterizationDomain 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)

g

g!

!n

nzMnxzY ][)(

][][ Mnxny !


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To get around this problem, define a

new sequence :

Then

ss!! otherwise,,,,],[][int 0

20 -MMnnxnx

][int nx

g

g!

g

g!

!!n

nn

n zMnxzMnxzY ][][)( int

)(][ /int/

intM

k

Mk zXzkx 1!! g

g!


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Now, can be formally related to

x[n] through

where

A convenient representation ofc[n] is

given by

where

][int nx

][][][int nxncnx !

!!

other ise,

,,,,][

0

201 -MMnnc

!!

1

0

1 M

k

knMW

Mnc ][

Mj

MeW /T2!


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Taking the z-transform of

and making use of

we arrive at

][][][int nxncnx !

!!

1

0

1 M

k

knMWMnc ][

n

n

M

k

knM

n

nznxWMznxncz

g

g!

!

g

g!

!! ][][][)(int1

0

1

!

!

g

g!

!

!

1

0

1

0

11 M

k

k

M

M

k n

nknM WzX

MzWnx

M][


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Consider a factor-of-2 down-samplerwith an input x[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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Now

implying that the second term

in the previous equation is simply

obtained by shifting the first termto the right by an amount 2T as shown

below

)()( 2/)2(2/ T[[ ! jj ee)( 2/[ je

)( 2/[je


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The plots of the two terms have an

overlap, and hence, in general, the original

shape of is lost when x[n] is down-

sampled as indicated below

)( [je


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This overlap causes the aliasingaliasingthat takes

place due to under-sampling

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

Note: is indeed periodic with a

period2T

, even though the stretchedversion of is periodic with a period

4T

2/0)( Tu[![ forje

)( [je

)( [jeY


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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 ofMuniformlyshifted and stretched versions of

and scaled by a factor of1/M

!

T[[ !1

0

/)2( )(1

)(M

k

Mkjj eXM

eY

)( [jeY

)( [je


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Aliasing is absent if and only if

as shown below forM= 2

2/for0)( Tu[![je

Mforej /0)( Tu[![


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Filter SpecificationsFilter 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

decimation filterdecimation filter, as indicated below to

avoid aliasing caused by down-

sampling

The above system is called a decimatordecimator

M/T[

M][nx )(H ][ny


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FrequencyFrequency--Domain CharacterizationDomain Characterization

Consider first a factor-of-2 up-sampler

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

ss!!otherwise,

,,,],/[][

0

4202 -nnxnx

u


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In terms of the z-transform, the input-

output relation is then given by

g

g!

g

g!

!!

even

]/[][)(

nn

n

n

nuu znxznxzX 2

2 2[ ] ( )m

m

x m z X zg

!g

! !


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In a similar manner, we can show that

for a factorfactor--ofof--LL upup--samplersampler

On the unit circle, for , the input-

output relation is given by

)()(L

u zXzX ![jez !

)()( Ljju eXeX[[

!


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Figure below shows the relation between

and forL = 2 in the

case of a typical sequence x[n])( [jeX )(

[ju e


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

rate expansion leads to a compression

of by a factor of2 and a 2-foldrepetition in the baseband [0, 2T]

This process is called imagingimagingas we

get an additional image of the inputspectrum

)( [jeX


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Similarly in the case of a factor-of-L

sampling rate expansion, there will be

additional images of the input spectrum in

the baseband

Low pass 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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Filter SpecificationsFilter Specifications

Since up-sampling causes periodicrepetition of the basic spectrum, the

unwanted images in the spectra of the

up-sampled signal must be

removed by using a lowpass filterH(z),

called the interpolation filterinterpolation filter, as

indicated below

The above system is called an

interpolatorinterpolator

][nxu

L][nx ][ny][nxu


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Cascade EquivalencesCascade Equivalences

A complex multirate systemmultirate system is formed

by an interconnection of the up-sampler,

the down-sampler, and the components

of an LTI digital filter

In many applications these devices

appear in a cascade form

An interchange of the positions of thebranches in a cascade often can lead to

a computationally efficient realization


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Interpolation Filter SpecificationsInterpolation Filter Specifications

On the other hand, if we pass x[n]through a factor-of-L up-sampler

generating , the relation between

the Fourier transforms of x[n] andare given by

It therefore follows that if is

passed through an ideal lowpass filter

H(z) with a cutoff at T/L and a gain ofL,

the output of the filter will be precisely

y[n]

][nxu][nxu

)()( Ljju eXeX[[ !

][nxu


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unit 4 random signal processing

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