On the statistical distributions of active intensity directions

Pressure and pressure differences measured using closely-spaced microphone pairs enable the calculation of active intensity directions. These directions are exploited in various applications such as acoustical analysis and modelling, multichannel audio systems, source localization and real-time blind source separation. Despite their widespread usage, the distribution of the active intensity directions in the presence of interference has not been investigated before, except for the diffuse sound field case for which the distribution becomes uniform. In this paper, the interference problem has been formulated using the pressure and pressure differences. Algebra of random variables, fractional linear transformations and Brownian motion have been used for the exposition of the model distributions. It is shown that the distribution is wrapped Cauchy for a single sound source in the presence of independent interference in an anechoic chamber. In a direct plus diffuse sound field, the distribution is offset Cauchy for a nonstationary target source and von Mises for the special case of a stationary target source, related to the Brownian motion. In the presence of multiple independent sound sources, the distribution can be modeled as mixtures of circular Cauchy distributions and a uniform distribution.