Adv DSPSpring-2015

Lecture#10Spectrum Estimation

Introduction Spectrum Estimation: To estimate the power

spectral density (PSD) of WSS random process. PSD is the DTFT of the autocorrelation

sequence. Estimating the power spectrum is equivalent to

estimating the autocorrelation.

Introduction The autocorrelation is generally estimated by using

If x[n] is known for all ‘n’, then power spectrum estimation is straight forward

Determine autocorrelation sequence above and then compute its Fourier transform.

Two problems with this: The amount of data is never unlimited , in many cases very

small Data is often corrupted by noise

Introduction Spectrum estimation is the problem that involves

estimating Px(ejw) from a finite number of noisy measurement of x[n].

Spectrum estimation is a problem that is important in a variety of different fields and applications.

Applications: Optimum Filter design, Signal Detection & Tracking, Harmonic Analysis & Prediction, Spectral Smoothing, Beamforming & Direction Finding,

Techniques for Spectrum Estimation Classical or Non-parametric Approach

Estimating the autocorrelation sequence from given set of data.

Power Spectrum is then obtained by Fourier transforming the estimated autocorrelation.

Non-Classical or Parametric Approach Based on using a model for the process to estimate

the power spectrum. AR, MA or ARMA Models

Techniques for Spectrum Estimation

Power Spectral Density

Non-Parametric Methods Based on the idea of estimating the autocorrelation

sequence of a random process from a set of measured data.

Take the Fourier transform to obtain an estimate of the power spectrum.

Periodogram (1898) Easy to compute. Limited in its ability to produce an accurate estimate of the power

spectrum, particularly for short data records. Modification to Periodogram to improve its statistical

properties Modified periodogram method, Bartlett’s method, Welch’s

method, Blackman-Tukey method.

The Periodogram The power spectrum of WSS random process is the

Fourier transform of the autocorrelation sequence:

Spectrum estimation Autocorrelation estimation problem

Autocorrelation sequence for an unlimited amount of data can be estimated using the time average:

The Periodogram If x[n] is only measured over a finite interval, say n=0,1,….,N-1 ,

then we can estimate the autocorrelation sequence using:

In order to ensure that the values of x[n] that fall outside the interval [0,N-1] are excluded, we can write:

0)(ˆ ||

)(ˆ)(ˆ ,0 *

krNk

krkrk

x

xx

The Periodogram Taking the Fourier transform of the autocorrelation

sequence gave us the Periodogram.

Next we will represent the Periodogram directly in terms of the process x[n].

Let xN[n] be the finite length signal of length ‘N’ that is equal to x[n] over the interval [0,N-1] and is zero otherwise:

The Periodogram Thus, xN[n] is the product of x[n] with rectangular window

wR[n],

In term of xN[n] , the estimated autocorrelation sequence may be written as follows:

Taking the Fourier transform and using convolution theorem, the Periodogram becomes:

The Periodogram XN(ejw) is the DTFT of the N-point data

sequence xN[n],

Thus, the Periodogram is proportional to the squared magnitude of DTFT of xN[n], and may be easily computed using a DFT (FFT) as follows,

Example 8.2.1: Periodogram of White Noise

If x[n] is the white noise with variance σx2

Thus the power spectrum should be constant. Consider a sample realization of white noise of unite

variance and N=32

2

2

)(

)()(

xjw

x

xx

eP

kkr

Example 8.2.1: Periodogram of a White Noise

Sample realization

Estimated autocorrelation sequence

Periodogram

White Noise in Matlab rand( ) : Uniformly distributed white noise

with unit variance. randn( ): Gaussian Distributed white noise

with unit variance.

Example: The Periodogram

Matlab Codeclear all;close all;A=1;N=100;fs=0.3;

n=1:N;rn=randn(1,N);phi=2*pi*rand(1,N);x=A.*exp(j*2*pi*fs*n+phi)+rn;Px=abs(fft(x,1024)).^2/length(n);w=2.*pi.*[0:1023]./1024;plot(w./(2*pi),Px)xlabel('\bf Normalized Frequency')ylabel('\bf Periodogram')Title('\bf [A=1,N=100]')

Matlab Function can also be used directly: periodogram ( )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

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8x 10

5

Normalized Frequency

Per

iod

og

ram

[A=1,N=100]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

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7x 10

6

Normalized Frequency

Per

iod

og

ram

[A=1,N=1000]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

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7x 10

6

Normalized Frequency

Per

iod

og

ram

[A=1,N=10000]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

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Normalized Frequency

Per

iod

og

ram

[A=0.001,N=100]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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10

Normalized Frequency

Per

iod

og

ram

[A=0.001,N=1000]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Normalized Frequency

Per

iod

og

ram

[A=0.001,N=10000]

A=1

A=0.001

N=100 N=1000 N=10000

Modified Periodogram The Periodogram is proportional to the squared

magnitude of the Fourier transform of the windowed signal, i.e. xN[n]=x[n]wR[n]

Instead of applying a rectangular window , the above equation suggests the possibility of using other data windows.

Modified Periodogram Amount of smoothing in the Periodogram is

determined by the window that is applied to the data.

A rectangular window has a narrow main lobe compared to other windows, and therefore produces the least amount of spectral smoothing, it has relatively large side lobes that may lead to masking of weak narrowband components.

Rectangular Window DTFT amplitude‘M’ is the length

Example: Two sinusoid in White Noise Rectangular Window (Narrow main lobe High

Resolution, large side lobes) Hamming Window (Wide main lobe, Less

Resolution, Small side lobes)

Example: Two sinusoid in White Noise

Example: Two sinusoid in White Noise Rectangular window: Sinusoid at frequency w1 is

almost completely masked by the side lobes of the window at frequency w2.

Hamming window: The sinusoid at w1 is clearly visible. This is due to smaller side lobes of a Hamming window

Reduction in the side lobe amplitude comes at the expense of an increase in the width of main lobe, which in turn, effects the resolution.

Modified Periodogram The Periodogram of a process that is windowed

with a general window w[n] is called a “Modified Periodogram”.

“N” is the length of window“U” is the constant given by

Modified Periodogram

Matlab Function : periodogram ( ) has a window as an input parameterMatlab has functions to generate the above mentioned windows

Matlab Window Functions Rectangular: rectwin ( n) Hanning: hanning (n ) Hamming: hamming (n ) Blackman: blackman (n) Bartlett: bartlett(n )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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14x 10

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Normalized Frequency

Per

iod

og

ram

[A=1,N=100]

Rectangular

Hamming

HanningBartlett

Blackman

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38-0.5

0

0.5

1

1.5

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2.5

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x 104

Normalized Frequency

Per

iod

og

ram

[A=1,N=100]

Rectangular

Hamming

HanningBartlett

Blackman

Rectangular window has the best resolution