Two-band wavelets and filterbanks over finite fields with connections to error control coding

Recently, we have developed a new framework to study error-control coding using finite-field wavelets and filterbanks (FBs). This framework reveals a rich set of signal processing techniques that can be exploited to investigate new error correcting codes and to simplify encoding and decoding techniques for some existing ones. The paper introduces the theory of wavelet decompositions of signals in vector spaces defined over Galois fields. To avoid the limitations of the number theoretic Fourier transform, our wavelet transform relies on a basis decomposition in the time rather than the frequency domain. First, by employing a symmetric, nondegenerate canonical bilinear form, we obtain a necessary and sufficient condition that the basis functions defined over finite fields must satisfy in order to construct an orthogonal wavelet transform. Then, we present a design methodology to generate the mother wavelet and scaling function over finite fields by relating the wavelet transform to two-channel paraunitary (PU) FBs. Finally, we describe the application of this transform to the construction of error correcting codes. In particular, we give examples of double circulant codes that are generated by wavelets.

I. Introduction

Wavelets and filterbanks (FBs) that operate on real or complex signals are already well established as powerful signal processing tools. They are applied in audio and video compression [1], [2], time-frequency analysis [3], radar [4], and a host of other areas. While wavelets defined over real or complex fields have been studied extensively for years [5] [6]– [10], the finite-field wavelet transform has received little attention, possibly because of the limited applications of this transform in the signal and image processing areas.