Abstract
This paper addresses the use of multirate filter banks in the context of error-correction coding. An in-depth study of these filter banks is presented, motivated by earlier results and applications based on the filter bank representation of Reed-Solomon (RS) codes, such as Soft-In Soft-Out RS-decoding or RS-OFDM. The specific structure of the filter banks (critical subsampling) is an important aspect in these applications. The goal of the paper is twofold. First, the filter bank representation of RS codes is now explained based on polynomial descriptions. This approach allows us to gain new insight in the correspondence between RS codes and filter banks. More specifically, it allows us to show that the inherent periodically time-varying character of a critically subsampled filter bank matches remarkably well with the cyclic properties of RS codes. Secondly, an extension of these techniques toward the more general class of BCH codes is presented. It is demonstrated that a BCH code can be decomposed into a sum of critically subsampled filter banks.
Highlights
Multirate filter banks have long been known to be powerful tools for image and audio processing [1], for example, in video/audio compression [2, Chapter 14]
Multirate filter banks essentially work in a block oriented fashion, that is, the data are divided in blocks of N
This paper presents an in-depth investigation of filter bank representations for RS and BCH codes, motivated by a number of applications presented earlier
Summary
Multirate filter banks have long been known to be powerful tools for image and audio processing [1], for example, in video/audio compression [2, Chapter 14]. There is, an important distinction between this work and the literature mentioned so far; the filter banks discussed in this paper operate in a finite field (Galois field) and represent Reed-Solomon or BCH codes. We developed a critically subsampled filter bank representation of RS codes, which is the starting point in building a novel SISO RS decoder [11, 12]. In this paper, a novel way to describe the correspondence between filter banks and RS codes is developed using a polynomial description. An nth root of unity in a finite field is denoted as αn. a | b denotes “a divides b.”
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