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Lecture 5

Spectrum Estimation

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Periodogram Based

.Periodogram

.Modified Per.

.Bartlett

.Welch

Classical Methods(Fourier Transform

Based)

ACF Based

.Blackmen Tukey

Filterbank Approach.Minimum Variance

Nonclassical Methods

(Non Fourier Based)

Family of Non Parametric Methods

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In practice we have one set of finite duration data.This approach has 2 practical problems

1. Cant do the expected value2. Cant do the limit

The periodogram is a method that ignores them both

Based on the definition of PSD:

In practice we compute it using the discrete Fourier transform (DFT)(possibly with zero padding) which computes the discrete time

Fourier transform (DTFT) at discrete frequency points

Periodogram Definition (recalled)

2

1( ) lim ( )

2 1

M

jwnxM

n M

S w E x n eM

21

0

1 ( ) ( )N

jwnPER

n

S w x n eN

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Periodogram-Viewed as a filterbank

Although the Periodogram is Always implemented using

the DFT, a filterbank interpretation is also very helpful.

Defining the impulse response of an FIR filter as

1, 0 >1 and ww1, then WR(w-w1)0

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function Px = minvar(x,p)%MINVAR Spectrum estimation using the minimum variance method.% The spectrum of a process x is estimated using the minimum% variance method (sometimes called the maximum likelihood method).% x : Input sequence% p : Order of the minimum variance estimate - for short% sequences, p is typically about length(x)/3% The spectrum estimate is returned in Px using a dB scale.%x = x(:);

R = covar(x,p);[v,d]=eig(R);U = diag(inv(abs(d)+eps));V = abs(fft(v,1024)).^2;Px = 10*log10(p)-10*log10(V*U);end;

The Minimum Variance Spectral Estimation:MATLAB code

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function R=covar(x,p)%COVAR Generates a covariance matrix/

%USAGE R=covar(x,p)%% Generates a p x p covariance matrix for the sequence x.

%---------------------------------------------------------------

x = x(:);m = length(x);x = x - ones(m,1)*(sum(x)/m);R = convm(x,p)'*convm(x,p)/(m-1);end;

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