Frames, Irregular Sampling, and a Wavelet Auditory Model

Abstract

14A Wavelet Auditory Model (WAM) is constructed in tenns of wavelet frames and an irregular sampling algorithm for Fourier frames. Its theoretical effectiveness is demonstrated in the context of speech coding, and its original fonnulation is found in [8][9]. The presentation of WAM in this chapter emphasizes its underlying mathematical ideas, and, in particular, develops the notions from the theory of frames and irregular sampling that arise naturally in constructing WAM. IntroductionWe shall develop some of the basic theory of frames, as well as conse-, quences ofthat theory from the area of irregular sampling. Moreover, we shall see how these ideas playa natural role in modelling parts of the mammalian auditory process with the purpose of using this process to devise signal reconstruction algorithms in the field of speech coding.The theory of frames is due to Duffin and Schaeffer [22], and it was developed to address problems in non-harmonic Fourier series. Prior to [22], these problems were concerned with finding criteria on real sequences {tn} so that the closed linear span, span{e t }, of exponentials e t (f) = e'2rtjtnf would be equal to the space L2 [-0, 0] offinit~ energy signals defin~d on [-0, 0]. The origins of

On stability of sampling-reconstruction models

“…The sampling and reconstruction problem includes devising efficient methods for representing a signal (function) in terms of a discrete (finite or countable) set of its samples (values) and reconstructing the original signal from the samples (see e.g., [1,3,8,9,17,22] and the reference therein). In this paper we consider a very general sampling model where the signal is assumed to belong to a finitely generated shift invariant space and the sampling is performed on an irregular separated set and is averaged by finite Borel measures.…”