Spatial Correlation and Time Delay Estimation

To find the position of an acoustic source in a room, a set of relative delays among different microphone pairs has to be determined. The generalized cross-correlation method is the most popular to do so and is well explained in a landmark paper by Knapp and Carter. In this technical memorandum, we show how we can take advantage of the redundancy when more than two microphones are available. It is believed that the redundancy will help to better cope with noise and reverberation. The idea of cross-correlation coefficient between two random signals is generalized to the multichannel case by using the notion of spatial prediction. The multichannel spatial correlation matrix is then deduced and it is shown how it can be used for time delay estimation.