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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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UpUp--SamplerSampler
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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UpUp--SamplerSampler
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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DownDown--SamplerSampler
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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UpUp--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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DownDown--SamplerSampler
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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UpUp--SamplerSampler
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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UpUp--SamplerSampler
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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UpUp--SamplerSampler
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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UpUp--SamplerSampler
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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UpUp--SamplerSampler
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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UpUp--SamplerSampler
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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