matlab - Generalized Cross Correlation

What is the definition of Generalized Cross Correlation?

I need it for estimating the delay (time offset) between two audio signals. The time offset between the two signals might be as large as 500ms.

Answer

Cross-correlation implies that you have 2 signals, and there is a delay between them. A 'basic' frequency domain cross-correlator does an FFT on each signal, conjugates one of them, multiplies one versus the other, then inverse transforms. A peak in the result indicates a time delay.

There are a lot of problems with this simple approach. For instance: you want a linear cross-correlation instead of a circular one; you may have strong sinusoidal interference which will generates multiple peaks; you may have very low SNR; the noise may not be what you expect it to be, etc.

A 'generalized' cross-correlation adds a windowing (or filtering) function prior to the inverse transform. It's purpose is to improve the estimation of the time delay, depending on the specific characteristics of the signals and noise (broadband/narrowband/interference signals, Gaussian noise, etc.). Since there are many different types of signals and noise, there are many different window functions (eg.: SCOT, Ekhart, etc.) Each one is designed for specific problems. Understanding these differences is not trivial, nor is proper calculation of the window function. They are typically dealt with in graduate-level time delay estimation or sonar/radar courses.

Some references for generalized cross-correlation are:

J. C. Hassab, R. E. Boucher, “Optimum Estimation of Time Delay by a Generalized Correlator,” IEEE T-ASSP, vol. 27, no. 4, Aug. 1979, pp. 373-380 (discusses various window types).

J. C. Hassab, R. E. Boucher, “Performance of the Generalized Cross Correlator in the Presence of a Strong Spectral Peak in the Signal,” IEEE T-ASSP, vol. 29, no. 3, June 1981, pp. 549-555.

J. C. Hassab, R. E. Boucher, “An Experimental Comparison of Optimum and Sub-Optimum Filters’ Effectiveness in the Generalized Correlator,” J. Sound and Vibration, 1981, pp. 4+ (12 pages total)(Fig. 1 contains a block diagram of a generalized cross-correlator).

There are many others.