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Coherence Function in Matlab

In Matlab and Octave, cohere(x,y,M) computes the coherence function $ C_{xy}$ using successive DFTs of length $ M$ with a Hanning window and 50% overlap. (The window and overlap can be controlled via additional optional arguments.) The matlab listing in Fig.8.14 illustrates cohere on a simple example. Figure 8.15 shows a plot of cxyM for this example. We see a coherence peak at frequency $ 0.25$ cycles/sample, as expected, but there are also two rather large coherence samples on either side of the main peak. These are expected as well, since the true cross-spectrum for this case is a critically sampled Hanning window transform. (A window transform is critically sampled whenever the window length equals the DFT length.)

Figure 8.14: Coherence measurement example in matlab.

 
% Illustrate estimation of coherence function 'cohere' 
% in the Matlab Signal Processing Toolbox
% or Octave with Octave Forge:
N = 1024;           % number of samples
x=randn(1,N);       % Gaussian noise
y=randn(1,N);       % Uncorrelated noise
f0 = 1/4;           % Frequency of high coherence
nT = [0:N-1];       % Time axis
w0 = 2*pi*f0;
x = x + cos(w0*nT); % Let something be correlated
p = 2*pi*rand(1,1); % Phase is irrelevant
y = y + cos(w0*nT+p); 
M = round(sqrt(N)); % Typical window length
[cxyM,w] = cohere(x,y,M); % Do the work
figure(1); clf; 
stem(w/2,cxyM,'*'); % w goes from 0 to 1 (odd convention)
legend('');         % needed in Octave
grid on;
ylabel('Coherence');
xlabel('Normalized Frequency (cycles/sample)');
axis([0 1/2 0 1]);
replot;  % Needed in Octave
saveplot('../eps/coherex.eps'); % compatibility utility

Figure 8.15: Sample coherence function.
\includegraphics[width=\twidth]{eps/coherex}

Note that more than one frame must be averaged to obtain a coherence less than one. For example, changing the cohere call in the above example to ``cxyN = cohere(x,y,N);'' produces all ones in cxyN, because no averaging is performed.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA