Evaluation of decay times in coupled spaces: Bayesian parameter estimation

Abstract

Determination of sound decay times in coupled spaces often demands considerable effort. Based on Schroeder's backward integration of room impulse responses, it is often difficult to distinguish different portions of multirate sound energy decay functions. A model-based parameter estimation method, using Bayesian probabilistic inference, proves to be a powerful tool for evaluating decay times. A decay model due to one of the authors [N. Xiang, J. Acoust. Soc. Am. 98, 2112-2121 (1995)] is extended to multirate decay functions. Following a summary of Bayesian model-based parameter estimation, the present paper discusses estimates in terms of both synthesized and measured decay functions. No careful estimation of initial values is required, in contrast to gradient-based approaches. The resulting robust algorithmic estimation of more than one decay time, from experimentally measured decay functions, is clearly superior to the existing nonlinear regression approach.

Sound Decay Analysis in Acoustically Coupled Spaces Using a Re-Parameterized Decay Model

“…Previous works [1,2] apply Bayesian inferential methods based on Bretthorst's approach [5] to architectural-acoustic applications. The prior information expressed in (6), however, was not incorporated into the Bayesian approach in these works.…”

“…This curvature, illustrated in Fig. 2 (a) and discussed in [1] and [7], can impede the determination of the decay parameters. The decay model in (4) is in the form of a general linear model.…”

“…Following [7], the background noise is supposed to be additive, with mean-square value n 2 . By applying Schroeder's backward integration to the RIR written as a noiseless RIR, h f (t i ), plus independent background noise (see [1,7] for more detail), the normalized decay function d(t i ) can be expressed as…”

“…This is traditionally accomplished, with difficulty, by visual inspection, since no algorithmic approach has been available. Our recent works [1,2] have applied Bayesian inferential methods, based on Bretthorst's approach [5], to more efficiently determine the decay times in coupled rooms using Schroeder's energy decay functions [6] as data along with the multi-rate decay models established in [1] and [7]. In these works, relatively little prior information about parameters of the decay model is incorporated in the inference calculations in spite of the fact that much information regarding both the possible range of the parameters and the relationship between the parameters is known.…”

A note on the definition of signal-to-noise ratio of room impulse responses

“…Model fitting methods [5][6][7] are not considered in this paper since the determined SNR in these approaches is often not used for decay time estimation, they are computationally more expensive than the simpler methods, and finally, they rely on a priori information of the decay and their applicability depend on the appropriateness of the selected model.…”