A function related to cross-correlation is the coherence function, defined in terms of power spectral densities and the cross-spectral density by
The coherence is a real function between zero and one which gives a measure of correlation between and at each frequency . For example, imagine that is produced from via an LTI filtering operation:
so that the coherence function becomes
A common use for the coherence function is in the validation of input/output data collected in an acoustics experiment for purposes of system identification. For example, might be a known signal which is input to an unknown system, such as a reverberant room, say, and is the recorded response of the room. Ideally, the coherence should be at all frequencies. However, if the microphone is situated at a null in the room response for some frequency, it may record mostly noise at that frequency. This is indicated in the measured coherence by a significant dip below 1. An example is shown in Book III  for the case of a measured guitar-bridge admittance. A more elementary example is given in the next section.
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 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 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.)
% 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
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
Recommended Further Reading
Power Spectral Density Estimation