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CHAPTER-4

4/19/2012 1Prof.H.T.Patil,CCOEW,PUNE

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WHAT IS THE POWER DENSITYSPECTRUM AND THEIR ESTIMATION

The signal processing methods which characterises the

frequency content of signal is known as spectrum analysis.

The signals which are analysed in any communication system

are either purely random or will have noise component also.

If the signal is random ,then only an estimate of the signal can

be obtained.

This is possible only if the statistical attributes of the random

signals are known.


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AttributesConsider the signal x(t), which is deterministic,

complex and finite energy.

The signal energy is given by Parsevals relation-:

The density of the energy of x(t) w.r.t. frequency is

represented by |X(f)|2

Sxx

(f) = |X(f)|

2

Sxx(f)=Energy Spectral Density (ESD)

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Let Rxx() be the autocorrelation function of the

signal x(t) where Rxx() is given by

Rxx() =

Energy Spectral Density (ESD), Sxx(f), of the

signal is calculated as the Fourier Transform of

autocorrelation function of the signalSxx(f) = F(Rxx()) =|X(f)|2

4

Attributes


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Consider the signal x(t), which is random

stationary , complex and infinite energy.The infinite energy signal do not have

Fourier Transform.The infinite energy signal have finite

average power.

For such signals, spectral analysis id

done with Power Spectral Function.

5

Attributes


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Consider the signal x(t), which is random stationary ,

complex and infinite energy.

Let xx() be the statistical autocorrelation functionof the signal x(t) , given by

xx() = E[x(t) x(t+ )]Power Spectral Density (PSD), xx(f), of the signal is

calculated as the Fourier Transform of the statisticalautocorrelation function xx() of the signal x(t),

xx(f) = F(xx() )

6

Attributes


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Statistical Autocorrelation function of a

random process is time average autocorrelationthat will use to characterizing random signal in

the time domain .

Fourier transform of that autocorrelationfunction is called Power Density Spectrum.

Power Spectral Estimation method is to

obtain an approximate estimation of the power

spectral density of a given real random

process.

Power Spectral Estimation


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WHY WE USE POWER SPECTRUMESTIMATION ?

To estimate the spectral characteristics ofsignal characterized as random processes.

To estimation of spectra in frequency domain

when signals are random in nature.


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Spectral Estimation

ParametricNon Parametric

Ex: Periodogram

and Welch methodSubspace Based(high-resolution)AR, ARMA based

Ex: MUSIC

and ESPRIT

Model fitting based

Ex: Least Squares

AR: Autoregressive (all-pole IIR)ARMA: Autoregressive Moving Average (IIR)

MUSIC: MUltiple SIgnal Classification

ESPRIT: Estimation of Signal Parameters using Rotational Invariance Techniques

Spectral Estimation Techniques


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Estimators

Estimation error: difference between theestimator and the parameter.

Bias: expected value of the error.

Mean Square Error: mean square value oferror.

=

)(e

== ][][)(

EeEbias

)()var(][)( 22

biaseEMSE +==


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Estimator comparison

Bias: lower the bias the better.

Variance: smaller variance implies less

deviation hence better.

MSE: in most cases more convenient.measure ofefficiency.

Consistency: if bias and variance both tend

to zero as the number of observationsbecome larger.


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NON-PARAMETRIC METHODS

Non-parametric: PSE does NOT assume anydata-generating process or model i.e noassumption about how data were generated.

Methods that rely on the direct use of the givenfinite duration signal to compute the

autocorrelation to the maximum allowable length(beyond which it is assumed zero), are calledNon-parametric methods.

Non-Parametric Methods:: PSD is estimated

directly from the signal itself.


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Periodogram

Bartlett method

Welch method

Blackman-Tukey method

Types of Nonparametric methods


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The Periodogram

=

+=

kN

n

x nxknxN

kr1

0

)(*)(1

)(

=

=k

jk

x

jk

per ekreP

)()(

Estimated autocorrelation:

Estimated power spectrum or periodogram:


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Discrete-time Fourier transform of the samples

of the process and then the magnitude squaredof the result.

where,

Lf

fXfP

s

L

xx

2)(

)( =

=

=

1

0

/2][)(

L

n

fjfn

LLsenxfX

Lf

fXfP

s

kL

xx

2)(

)( =

N

kff sk =

1......,2,1,0=

Nk

The Periodogram


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Performance of the Periodogram (SpectralLeakage)

xL[n] can be interpreted as the result ofmultiplying an infinite signal, x[n], by a finite-

length rectangular window, wR[n]

xL[n] = x[n].wR[n]

Multiplication in the time domain corresponds

to convolution in the frequency domain.


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The side lobes account forthe effect known asspectral leakage.

While the infinite-length

signal has its powerconcentrated exactly at thediscrete frequencypoints fk, the windowed

signal has a continuum ofpower "leaked" around thediscrete frequencypoints fk.

frequency response of a rectangular

window

Performance of the Periodogram (SpectralLeakage)


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Performance of the Periodogram

(resolution)

Ability to discriminate spectral features.

In order to resolve two sinusoids, thedifference between the two frequenciesshould be greater than the width of themainlobe of the leaked spectra for eitherone of these sinusoids.

Mainlobe width is defined as 3 dB widthand is approximately equal to fs /L

L

fff s )( 21


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Performance of the Periodogram

(bias and variance)

A biased estimator. Its expected value can be

shown to be

The estimates correspond to a leaky PSD rather

than the true PSD.

Inconsistent estimator, its variance is


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Modified PeriodogramThe sidelobes can be interpreted as spurious

frequencies introduced into the signal by the

abrupt truncation that occurs when a rectangularwindow is used.

The time-domain signal is windowed prior to

computing the FFT in order to smooth the edgesof the signal.

A window function is a function that is zero-valued outside of some chosen interval.

The height of the sidelobes or spectral leakage isreduced.


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For nonrectangular windows, the end points of the

truncated signal are attenuated smoothly, andhence the spurious frequencies introduced are

much less severe.

But, nonrectangular windows also broaden themainlobe, which results in a net reduction of

resolution.

Nonrectangular windowing affects the averagepower due to attenuation of the time samples

Modified Periodogram


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To compensate for this, we normalize the windowto have an average power of unity.

The modified periodogram estimate of the PSD is

where Uis the window normalization constant

Choice of window does not affect the averagepower of the signal after introduction ofU.

Modified Periodogram


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For reducing the variance in the periodogram

involves three steps .

The N-point sequence is subdivided into Knonoverlapping segments where each segment

has length L .This results in K data segments.

For each segment, Compute the periodogram .

Averaging the periodogram for the K segmentto obtain the Bartlett power spectrum estimate.

Bartletts method


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The Bartlett Method: Averaging

PeriodogramsIt consists of Three steps

Step 1: The N-pt sequence is subdivided into K nonoverlappingsegments, where each segment has length M. This results in the K data

segments

( ) ( ); 0,1, 2,..., 1

0,1, 2, ..., 1

ix n x n iM i K

n M

= + =

=

Step 3: Finally, we average the periodograms for the K segments to obtain

the Bartlett PSD estimate.

21

( ) 2

0

1( ) ( ) ; 0,1,2,..., 1

Mi i fn

xx i

n

P f x n e i K M

=

= =

Step 2: For each segment, we compute the periodogram

1( )

0

1( ) ( )

KB i

xx xx

i

P f P f K

=

= 4/19/2012 24Prof.H.T.Patil,CCOEW,PUNE

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Mean:1

( ) ( )

0

1( ) ( ) = ( )

KB i i

xx xx xx

i

E P f E P f E P fK

=

=

[ ] [ ]

( )

( ) { ( )} { ( ) }| |

{ ( ). ( )} 1 . ( )

( ) ( )

i

xx xx xx

B xx xx

B xx

E P f E DTFT r m DTFT E r mm

DTFT w m m DTFT mM

W f f

= =

= =

=

2

1 sin( )( )

sin( )B

fMW f

M f

=

where

1 22

( )

12

1 sin( ( ) )( ) ( ) ( ) ( )

sin( ( ))

i

xx B xx xx

f ME P f W f f d

M f

= =

The Bartlett Method: Averaging

Periodograms


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Variance:

1

( ) ( )2

0

2

2

1 1( ) var ( ) = var ( )

1 sin(2 )( ) 1

sin(2 )

K

B i ixx xx xx

i

xx

var P f P f P f K K

fMf

K M f

=

=

= +

The effect of reducing the length of the data from N points to

M=N/K results in a window whose spectral width has been

increased by a factor of K. Consequently, the frequency resolution

has been reduced by a factor K.

In return for this reduction in resolution, we have reduced the

variance. The variance of the Bartlett estimate is

Therefore, the variance of the Bartlett PSD estimate has been

reduced by the factor K.

The Bartlett Method: Averaging Periodograms


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Bias:Bias:Bias:Bias:

VarianceVarianceVarianceVariance:

21 1

0 0

1 ( ) ( )

k L

j jnB

i n

P e x n iL eN

= =

= +

)()(2

1)}({

j

B

j

x

j

B eWePePE =

)(1

)}({ 2 jxj

per ePk

ePVar

Bartletts method


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Welch's Method The method consists of dividing the time series data into

(possibly overlapping) segments, computing a modified

periodogram of each segment, and then averaging the PSDestimates

The averaging of modified periodograms tends to decrease

the variance of the estimate relative to a single

periodogram estimate of the entire data record

Overlap between segments tends to introduce redundant

information, this effect is diminished by the use of a

nonrectangular window, which reduces the importance orweightgiven to the end samples of segments (the samples

that overlap).


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The combined use of short data records and

nonrectangular windows results in reducedresolution of the estimator. ie, there is a tradeoff

between variance reduction and resolution.

One can manipulate the parameters in Welch'smethod to obtain improved estimates relative to

the periodogram.

Welch's Method


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The Welch Method: Averaging Modified

PeriodogramsWelch made two modifications to the Bartlett method.

Step 1: The data segments are allowed to overlap. Thus the data segmentscan be represented as

( ) ( ); 0,1,2,..., 1

0,1, 2, ..., 1

ix n x n iD i L

n M

= + =

=

Step 2: Apply window to the data segments prior to computing the

periodogram. The result is a modified Periodogram

Where U is the normalization factor for the power in the window function

12

0

1( )

N

n

U w n

M

=

=

21

( )2

0

1( ) ( ) ( ) ; 0,1, 2,..., 1

Mi

i fnxx

in

P f x n w n e i LMU

=

= =

Where iD is the starting point for the ith sequence. If D=M/2, there is 50%

overlap between successive data segments and L=2K.


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The Welch PSD estimate is the average of these modified periodograms, thatis

{ }1 1

( )* 2 ( )

0 0

1 12 ( )

0 0

1( ) ( ) ( ) ( ) ( )

1

( ) ( ) ( )

M Mi

j f n mxx i i

n m

M Mj f n m

xxn m

E P f w n w m E x n x m eMU

w n w m n m eMU

= =

= =

=

=

Since

1/22

1/2

( ) ( ) j fnxx xxn e d

=

1( )

0

1( ) ( )

LiW

xx

xx i

P f P f L

=

=

Mean:1

( ) ( )

0

1( ) ( ) = ( )

Li iW

xx xxxx

i

E P f E P f E P fL

=

=

The Welch Method: Averaging Modified

Periodograms


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1 11/2( ) 2 ( )( )

1/20 0

1/2

1/2

1( ) ( ) ( ) ( )

( ) ( )

M Mij f n m

xx xx

n m

xx

E P f w n w m e dMU

W f d

= =

=

=

where 21

2

0

1( ) ( )

Mi fn

n

W f w n eMU

=

=

Variance:

{ }1 1 2( ) ( )

20 0

1( ) ( ) ( ) ( )

L Li j

W Wxx xxxx xx

i j

var P f E P f P f E P f L

= =

= Case(1): No overlap (L=K)

( )

2

1 1( ) var ( ) ( )

iW

xxxx xxvar P f P f f L L =

Case(2): 50% overlap (L=2K)

29( ) ( )

8

W

xx xxvar P f f

L

The Welch Method: Averaging Modified Periodograms


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A biased estimator of the PSD, the expected valueis

For a fixed length data record, the bias of Welch'sestimate is larger than that of the periodogrambecause Ls < L.

The variance of Welch's estimator is difficult tocompute because it depends on both the windowused and the amount of overlap between segments.

The variance is inversely proportional to the

number of segments whose modifiedperiodograms are being averaged.

Welch's Method


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The Blackman-Tukey Method: Smoothing the

PeriodogramThe sample autocorrelation sequence is windowed first and then Fourier

transformed to yield the estimate of PSD.

where

Where Pxx(f) is Periodogram. The effect of windowing the autocorrelation is to

smooth the periodogram estimate, thus decreasing its variance.

Where the window function w(n) has length 2M-1 and zero for |m|>(M-1).

12

( 1)( ) ( ) ( )

MBT i fm

xx xxm MP f r m w m e

=

=

1/2

1/2

( ) ( ) ( )BT

xx xxP f P W f d

=

Mean:

[ ]

1/2

1/2( ) ( ) ( )BT

xx xxE P f E P W f d =

[ ]1/2

1/2( ) ( ) ( )xx xx BE P W d

= 4/19/2012 34Prof.H.T.Patil,CCOEW,PUNE

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We should select the window length for w(n) such that M

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PARAMETRIC METHODS

Parametric methods:PSD is estimated from a signal that is assumed to

be output of a linear system driven by whitenoise.

Parametric methods:

ARMA ModellingYule-Walker autoregressive (AR) method,Burg method.


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