This chapter introduces the Discrete Fourier Transform (DFT) and points out the mathematical elements that will be explicated in this book. To find motivation for a detailed study of the DFT, the reader might first peruse Chapter 8 to get a feeling for some of the many practical applications of the DFT. (See also the preface on page .)

Before we get started on the DFT, let's look for a moment at the
*Fourier transform* (FT) and explain why we are not talking about
it instead. The Fourier transform of a continuous-time signal
may be defined as

Thus, right off the bat, we need

where the various quantities in this formula are defined on the next page. Calculus is not needed to define the DFT (or its inverse, as we will see), and with finite summation limits, we cannot encounter difficulties with infinities (provided is finite, which is always true in practice). Moreover, in the field of digital signal processing, signals and spectra are processed only in

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