Stochastic process modeling of sound energy decay in rooms

Abstract

The discrete parameter first order Markov chain model of the reverberation process developed at Göttingen has, as a limitation, an inability to deal with unequal free path lengths. A continuous parameter Markov chain model has been developed which eliminates this difficulty, but which yields results which are much worse predictors of the reverberation characteristics of the scale model room tested at Göttingen. The success of the discrete parameter model has been found to be a beneficial, but erroneous, result of the equal free path assumption. The overall failure of the Markov Model was traced to its “memorylessness” property, which in effect forces all reflections to diffuse. In the two rooms investigated, specular reflections predominate, and so the assumption implied by the use of the first order Markov model was manifested in its poor predicting power. A more general stochastic process model is proposed which permits the use of a mixture of low-order and high-order terms to describe, respectively, scattering and specular reflection. This model may be considered a multivariate digital filter, since, as formulated here, it operates in discrete time, but without encountering the problems noted for the Göttingen discrete parameter Markov model. In addition to having the potential for far superior predicting power, it should also present fewer computational problems than multiple order Markov chains.