Statistical models for pole‐zero distributions in the complex frequency plan—Reverberant amplitudes and reverberant phase

Abstract

Statistical energy analysis (SEA) is a way of estimating transfer function magnitudes in complex, multi‐degree of freedom (dof) systems. The system model is a Poisson distribution of poles along the pole line, with residue statistics determined by the mean and mean‐square values of the mode shape functions. The model gives the average and variance of the response amplitudes, and the effect of modal overlap on the statistics. On the other hand, the phase of transfer functions is determined by both the poles and zeros of the transfer function, and is important for determining group delay and distortions of waveforms. Since the zeros lie throughout the complex frequency plane, their distribution is two dimensional. The distribution of zeros normal to the pole line is the Cauchy distribution. This modeling is part of “statistical phase analysis” (SPA). The concept of “reverberant phase,” which has many corrolaries to the idea of reverberant amplitudes, arises quite naturally from the model. This paper will ex...