Stochastic Modeling of Impulse Responses in Reverberating Environments

Propagation of waves within media supporting reverberation is usually regarded as a direct extension of the case of multi-path propagation, where a set of independent paths, equivalent to plane-wave contributions, can be drawn between a transmitter and a receiver. This paper adopts an alternative approach, based on modal theory, in order to derive models of the stochastic behavior of impulse responses (IRs) measured within such media. IRs can be represented as stationary Gaussian random processes whose amplitude is modulated by a decay function that converges to an exponential only if the time constants of each mode are similar, otherwise displaying a decay rate slowing with time as the modal time constants become more diverse. The asymptotic convergence to a Gaussian process is controlled by the number of available modes, which modal theory predicts to increase linearly with the bandwidth, but quadratically with the frequency. Modal theory implies that groups of typically more than eight propagation paths must be coherently related in order to give rise to reverberation. As a result, fewer degrees of freedom may be available than expected from the number of propagation paths involved, thus leading to a slower convergence to Gaussian propagation models. The stochastic model introduced is further applied in order to understand how far IRs can locally fluctuate away from their root-mean-square amplitude profiles. All theoretical predictions are supported by experimental results.