A theory of filter banks which represent signals as holomorphic functions on the time-frequency domain

Holomorphic functions have fine mathematical properties such as the Cauchy-Rieman relation, infinite differentiability, conformality, and Laurent expandability. For treating these functions, we can apply various useful theorems such as Cauchy´s integral theorem, residue theorem, etc. In this paper, in order to introduce such a powerful mathematical foundation into the time-frequency analysis, we consider a filter bank and a complex logarithmic mapping which transform an input signal into a holomorphic function in the time-frequency domain. As an application of this framework, we show that zeros of the input signal are transformed into poles in the time-frequency domain, hence they can be used as salient and acculate features to describe the input signal. We show several experimental results of this principle.