Coherence Function in Matlab

In Matlab and Octave, `cohere(x,y,M)` computes the coherence
function using successive DFTs of length with a Hanning
window and 50% overlap. (The window and overlap can be controlled
via additional optional arguments.) The following matlab illustrates
`cohere` on a simple example:

% Illustrate estimation of coherence function 'cohere' % in Matlab, or Octave + 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 8.14 shows a plot of `cxyM` for this example.
We see a coherence peak at frequency 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 FFT length.)

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

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