The Stationary Phase Approximation, Time-Frequency Decomposition and Auditory Processing

The principle of stationary phase (PSP) is re-examined in the context of linear time-frequency (TF) decomposition using Gaussian, gammatone and gamma chirp 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 do not 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 and ii) a test for phase rate dominance in that plane. Together this is a TF stationary phase approximation (TFSPA) 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.