<title>Applications of multirate filter banks in error correction codes: partial response channels</title>

Abstract

ABSTRACT Both filterbank theory and error correction codes have been studied extensively during the recent decades. In thispaper, we first build a bridge between these two theories. We restate some results in error correction coding theory by using filterbank terminologies over finite fields. We then apply filterbank theory to error correction codes for partialresponse channels (PRC), such as magnetic recording systems. We present necessary and sufficient conditions foruniquely decodable convolutional codes for PRC. We also extend the results to multihead and multitrack recordingsystems. Examples of uniquely decodable codes are given. With these conditions one is able to check whether a convolutional code in a PRC is uniquely decodable. Keywords: Partial response channels, multirate filterbanks over finite fields. 1. Introduction Multirate filterbank theory has been studied extensively in the last decade and found applications in many areasincluding signal and image processing. The main advantage of multirate filterbanks is that they can be used to splita signal into several different frequency bands, each of which may be processed differently according to differentneeds. This advantage holds for real and complex valued signals. Multirate filterbanks taking values in finite fields(called fihterbanks over finiie fields) have also been studied, see for example [9-11]. However, due to the difficultyof the interpretation of the frequency bands less applications have been found, although their application in imageprocessing are visible. This is because of the elimination of the round off errors for finite field arithmetics. Inthis paper, we focus on multirate filterbanks over finite fields with applications in channel coding. We describeconvolutional codes in terms of multirate filterbank terminologies and show that convolutional codes are equivalentto nonmaximally decimated FIR multirate filterbanks. We also show that a noncatastrophic convolutional code isequivalent to an FIR nonmaximally decimated multirate filterbank with an FIR inverse. This was also observed by