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Introduction to the DFT

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 $ x(t)$ may be defined as

$\displaystyle X(\omega) = \int_{-\infty}^\infty x(t)e^{-j\omega t} dt,
\qquad \omega\in(-\infty,\infty).
$

Thus, right off the bat, we need calculus. The DFT, on the other hand, replaces the infinite integral with a finite sum:

$\displaystyle X(\omega_k ) \isdef \sum_{n=0}^{N-1}x(t_n)e^{-j\omega_k t_n}, \qquad k=0,1,2,\ldots,N-1,
$

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 $ x(t_n)$ is finite, which is always true in practice). Moreover, in the field of digital signal processing, signals and spectra are processed only in sampled form, so that the DFT is what we really need anyway (implemented using an FFT when possible). In summary, the DFT is simpler mathematically, and more relevant computationally than the Fourier transform. At the same time, the basic concepts are the same. Therefore, we begin with the DFT, and address FT-specific results in the appendices.



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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA