Fourier Transforms, Uncertainty, and Convolutions

This chapter covers the continuous-time Fourier transform and the related topics of the uncertainty principle and convolutions. The space of square integrable functions, images2, is explained and used to state the Fourier transform and other theorems in this chapter.

The purpose of the Fourier transform is to describe a specific signal in terms of its frequency components. The Fourier transform can be defined for both continuous-time signals and for discrete-time signals. The continuous-time case is the subject of this chapter, whereas Chapter 9 covers the discrete-time case.

Both of these two chapters on Fourier transforms are essential material for digital signal processing. However, not all of the material is strictly required for the project in Part III. You might want to skip to Part III now, and use the rest of Part II as reference material.

There are four principal ideas and results in this chapter. The first and most fundamental of these is the notion of a set of orthogonal functions, found in Section 8.3, which is basic to all numeric transforms, not only the Fourier transform. The second is the Fourier transform theorem itself, which is properly called the Plancherel theorem (Section 8.5). The third central point is the Parseval theorem, which, as a special case, shows that the Fourier transform coefficients represent ...

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