Blahut R.E. Algebraic Methods for Signal Processing and Communications Coding

Springer, 1992. — 150 p. — ISBN10: 146127687X, ISBN13: 978-1461276876.Algorithms for computation are a central part of both digital signal processing and decoders for error-control codes and the central algorithms of the two subjects share many similarities. Each subject makes extensive use of the discrete Fourier transform, of convolutions, and of algorithms for the inversion of Toeplitz systems of equations.
Digital signal processing is now an established subject in its own right; it no longer needs to be viewed as a digitized version of analog signal processing. Algebraic structures are becoming more important to its development. Many of the techniques of digital signal processing are valid in any algebraic field, although in most cases at least part of the problem will naturally lie either in the real field or the complex field because that is where the data originate. In other cases the choice of field for computations may be up to the algorithm designer, who usually chooses the real field or the complex field because of familiarity with it or because it is suitable for the particular application. Still, it is appropriate to catalog the many algebraic fields in a way that is accessible to students of digital signal processing, in hopes of stimulating new applications to engineering tasks.
An error-control code can be viewed as a technique of digital signal processing intended to prevent random errors or burst errors caused by noise in a communication channel. Error-control codes can be developed in an arbitrary field; this is so despite the fact that finite fields of characteristic 2, GF(2^m), seem to be the best choice in most applications. Even though the underlying data may originate as real numbers and the codewords must be represented as another sequence of real or complex numbers for modulation into a communication waveform, it is better to represent the data temporarily by elements of a finite field for coding. However, there has not been much study of error-control codes in algebraic fields other than finite fields. Further insights should emerge as time goes by.
The primary purpose of this monograph is to explore the ties between digital signal processing and error-control codes, with the thought of eventually making them the two components of a unified theory, or of making a large part of the theory of error-control codes a subset of digital signal processing. By studying the properties of the Fourier transform in an arbitrary field, a perspective emerges in which the two subjects are unified. Because there are many fields and many Fourier transforms in most of those fields, the unified view will also uncover a rich set of mathematical tools, many of which have yet to find an engineering application.
The secondary purposes of the monograph are of four kinds: First, a parallel development of the two subjects makes the subject of error-control coding more accessible to the many engineers familiar with digital signal processing. Second, the many fast algorithms of digital signal processing can be borrowed for use in the field of error-control codes. Third, the techniques of error-control can be borrowed for use in the field of digital signal processing to protect against impulsive noise. And fourth, perhaps the design of VLSI chips that perform a variety of signal processing and errorcontrol tasks would become simplified if these tasks could be merged and performed in the same algebraic field.
The monograph attempts to combine many elements from digital signal processing and error-control codes into a single study. I hope that this attempt at unification justifies repeating some of these topics here. In part the book is based on a short course I have given in Zurich under the sponsorship of Advanced Technology Seminars in which aspects of digital signal processing were integrated with topics on error-control codes.Mathematical Fundamentals
Sequences and Spectra
Cyclic Codes and Related Codes
Fast Algorithms for Convolution
Solving Toeplitz Systems 8
Fast Algorithms for the Fourier Transform
Decoding of Cyclic Codes