<title>Optimal warping function design for discrete time-warped Fourier transforms</title>

Abstract

We have recently introduced the class of generalized scale transforms and its subclass of warped Fourier transforms. Members in each class are defined by continuous time warping functions. While the two transforms admit a mathematically elegant analysis of warp-shift invariant systems it is still unclear how to design warping functions that deliver optimal representations for a given class of signals or systems. In many cases we can obtain an optimal choice for the warping function via a closed form analysis of the system that generates the signal of interest. In cases in which a closed form analysis is not possible we have to rely on a warp function estimation method. The approach we are taking in this paper is founded in information theory. We consider the observed signal as a random process. A power estimate of the warped Fourier transform parameterized by an underlying warping function is obtained from a finite number of realizations. We treat the power estimate as a probability density in warp-frequency and minimize its differential entropy over the space of admissible warping functions. We use an iterative numerical method for the minimization process. A proper formulation of a discrete time warped Fourier transform is employed as a foundation for the numerical analysis. Applications of the proposed algorithm can be found in detection, system identification, and data-compression.