A Unified View of Fourier Transforms

Chapter 9 introduced the concept of the discrete Fourier transform and examined four cases generalizing the Fourier transform introduced in Chapter 8. This chapter is an introduction to Abelian harmonic analysis, the study of functions defined on a commutative group. By the end of this chapter, you will understand the underlying connections among all the manifestations of the Fourier transform presented individually in Chapters 8 and 9. All those Fourier transform theorems are special cases of Theorem 11.5 on page 199 in this chapter. All of the theory presented in this chapter is directed toward this one result.

Should you read this chapter, or should you skip ahead? That depends on your goals and your previous knowledge. This chapter introduces the concepts of a group and of analysis on commutative groups, a special case of harmonic analysis. These topics are certainly not mainstream engineering—at least they haven't been in the past. However, as digital signal processing, information theory, and coding theory intrude more and more into daily practice, the subjects of group theory and harmonic analysis become more important.

The concepts presented in this chapter are not difficult to master and will provide you with a firm foundation. The alternative to studying these beautiful (to the eye of a mathematician) subjects is to develop several ad hoc ways of explaining Fourier transforms, particularly the fast Fourier transform algorithm. ...

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