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 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: Sample coherence function.

Figure 8.14 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 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 cxyN, because no averaging is performed.