Flexible Rational-Exponent Auditory Filters

Authors: Samiya Alkhairy1

1MIT, KACST

Background: Auditory filters and filterbanks are useful for several applications such as cochlear implants, vocoders, cochlea-inspired frequency multiplexers, and potentially hearing aids. One such set of auditory filters (Generalized Auditory Filters – GAFs, and All-pole Gammatone Filters – APGFs) are represented by a rational transfer function with a pair of complex conjugate poles raised to an integer power. The integer exponent imposes limitations on the types of behavior realizable by these filters.

Method: In order to extend the filters to achieve behavior on a continuum, rather than having discrete constraints on behavior, we generalized auditory filters to rational exponent filters. We study their causality and stability in the Laplace domain and derive equivalent representations for the filter for flexible implementations using transform methods and methods from complex analysis.

Results: We derive conditions for the causality and stability of rational-exponent GAFs/APGFs, and demonstrate the added flexibility in filter behavior afforded by extensions to the rational exponents. The implementable representations we derive include impulse responses and integral representations. For impulse response representations, we also derive approximations in terms of generalizations of the widely used gammatone filters.

Conclusion: Our extension of auditory filters to have rational exponents allows for imposing arbitrary continuous specification of filter characteristics such as the ratio of various quality factors to one another – which specify the shape of the frequency response magnitude, and the ratio of quality factor to group delay which relates frequency selectivity to delay. We derive representations towards software implementations of the rational-exponent filters, including an integral representation which is appropriate for real-time implementations without requirements on preprocessing.