Generalized Cross-Correlation Function with Comprehensive Weighting Functions

The Generalized Cross-Correlation (GCC) function is a crucial tool in signal processing, primarily used for estimating time delays between two signals. Its core concept involves optimizing cross-correlation calculations by introducing different weighting functions, thereby improving the accuracy of time delay estimation. The implementation typically involves taking the Fourier transform of input signals, applying frequency-domain weighting, and performing inverse Fourier transformation to obtain time-domain correlation results.

Building upon classical cross-correlation, the GCC function incorporates frequency-domain weighting stages. Common weighting functions include PHAT (Phase Transform), Roth, and SCOT, with PHAT weighting demonstrating superior time-delay estimation performance in most applications due to its specialized phase information processing. In code implementation, these weighting functions can be applied as frequency-domain filters after computing the cross-spectral density of the input signals.

The advantage of PHAT weighting lies in its insensitivity to signal amplitude, utilizing only phase information for weighting, which effectively suppresses the impact of noise and multipath effects on time delay estimation. This makes it widely applicable in fields such as sound source localization and radar ranging. The PHAT weighting function can be implemented in MATLAB as a normalization of the cross-spectrum, where the magnitude information is discarded while preserving phase relationships.