Signal interaction and the devil function
John R. Hershey, Peder A. Olsen, Steven J. Rennie · 2010
Abstract It is common in signal processing to model signals in the logpower spectrum domain. In this domain, when multiple signalsare present, they combine in a nonlinear way. If the phases ofthe signals are independent, then we can analyze the interactionin terms of a probability density we call the “devil function,”after its treacherous form. This paper derives an analytical ex-pression for the devil function, and discusses its properties withrespect to model-based signal enhancement. Exact inference inthis problem requires integrals involving the devil function thatare intractable. Previous methods have used approximations toderive closed-form solutions. However it is unknown how theseapproximations differ from the true interaction function in termsof performance. We propose Monte-Carlo methods for approx-imating the required integrals. Tests are conducted on a speechseparation and recognition problem to compare these methodswith past approximations. 1. Introduction Signals are often analyzed and classified using models of theirlog power spectra. In the context of noise, or any other inter-fering signal, such model-based classifiers must compensate forthe noise in some way. Model-based noise compensation re-quires a signal interaction model, which describes the effect ofadding two signals on the resulting acoustic features. Tradition-ally the influence of phase has either been ignored through theuse of approximate interaction models, or has been diminishedby averaging, especially when working in the log spectrum do-main.We describe and illustrate the signal interaction model, whichwe call the “devil function.” Exact inference using this func-tion is difficult because the required integrals are intractable.Even efficient approximate inference can be elusive. Just howimportant it is to accurately model this signal interaction is anempirical question. To address this question we perform exper-iments using accelerated Monte Carlo simulations that can ap-proximate inference arbitrarily well if enough samples are used,and compare the results to simpler approximations on a speechseparation and recognition task.