Abstract
The principle of stationary phase (PSP) is re-examined in the context of linear time-frequency (TF) decomposition using Gaussian, gammatone and gammachirp filters at uniform, logarithmic and cochlear spacings in frequency. This necessitates consideration of the use the PSP on non-asymptotic integrals and leads to the introduction of a test for phase rate dominance. Regions of the TF plane that pass the test and don't contain stationary phase points contribute little or nothing to the final output. Analysis values that lie in these regions can thus be set to zero, i.e. sparsity. In regions of the TF plane that fail the test or are in the vicinity of stationary phase points, synthesis is performed in the usual way. A new interpretation of the location parameters associated with the synthesis filters leads to: (i) a new method for locating stationary phase points in the TF plane; (ii) a test for phase rate dominance in that plane. Together this is a TF stationary phase approximation (TFSFA) for both analysis and synthesis. The stationary phase regions of several elementary signals are identified theoretically and examples of reconstruction given. An analysis of the TF phase rate characteristics for the case of two simultaneous tones predicts and quantifies a form of simultaneous masking similar to that which characterizes the auditory system.
Highlights
T HE principle of stationary phase (PSP) [1] is a result from asymptotics that can provide closed-form approximations, in the limit as, to often intractable oscillatory integrals of the form (1) where
This is a minor modification to the gammachirp filter of [16] in order to decouple the dependency between the location of the peak gain in the frequency response and the chirp rate parameter
The starting point for this paper was an examination of the application of the PSP to non-asymptotic integrals in general and TF synthesis in particular
Summary
T HE principle (or method) of stationary phase (PSP) [1] is a result from asymptotics that can provide closed-form approximations, in the limit as. The motivation is the recent resurgence of interest in analogue filter banks both as part of a synthetic cochlea and as a means to provide power efficient implementations of analysis filter banks [5] The desire with both is to extract salient features from the TF decomposition using the limited functionality associated with analogue circuitry. The primary contribution of this paper is a TF stationary phase approximation (TFSPA) for both linear signal analysis and synthesis (Section IV). It reduces both the extraction of salient features from TF analysis and the selection of components for synthesis to simple tests that can be performed instantaneously at the output of an analysis filter bank. Aspects of this work were reported briefly in [19]
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