Sequences and Spectra

Abstract

Traditionally, digital signal processing deals with vectors whose components are elements of the complex number system or the real number system. The discrete Fourier transform is commonly defined in the complex field and it forms a fundamental tool in the subject of digital signal processing. Likewise, many of the tasks of error-control codes can be described in terms of the discrete Fourier transform in a Galois field. The rich set of properties of the Fourier transform — which the engineer has internalized to form part of his intuition — are valid in the complex field and in the Galois fields. These properties underlie much of the structure of digital signal processing and of error-control codes. Indeed, these properties hold in an arbitrary field and can be developed without mention of any particular field.