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The DFT and its Inverse Restated

Let $ x(n), n=0,1,2,\ldots,N-1$ , denote an $ N$ -sample complex sequence, i.e., $ x\in{\bf C}^N$ . Then the spectrum of $ x$ is defined by the Discrete Fourier Transform (DFT):

$\displaystyle \zbox {X(k) \isdef \sum_{n=0}^{N-1}x(n) e^{-j 2\pi nk/N},\quad k=0,1,2,\ldots,N-1}
$

The inverse DFT (IDFT) is defined by

$\displaystyle \zbox {x(n) = \frac{1}{N}\sum_{k=0}^{N-1}X(k) e^{j 2\pi nk/N},\quad n=0,1,2,\ldots,N-1.}
$

In this chapter, we will omit mention of an explicit sampling interval $ T=1/f_s$ , as this is most typical in the digital signal processing literature. It is often said that the sampling frequency is $ f_s=1$ . In this case, a radian frequency $ \omega_k \isdef 2\pi k/N$ is in units of ``radians per sample.'' Elsewhere in this book, $ \omega_k$ usually means ``radians per second.'' (Of course, there's no difference when the sampling rate is really $ 1$ .) Another term we use in connection with the $ f_s=1$ convention is normalized frequency: All normalized radian frequencies lie in the range $ [-\pi,\pi)$ , and all normalized frequencies in Hz lie in the range $ [-0.5,0.5)$ .7.1 Note that physical units of seconds and Hertz can be reintroduced by the substitution

$\displaystyle e^{j 2\pi nk/N} = e^{j 2\pi k (f_s/N) nT} \isdef e^{j \omega_k t_n}.
$



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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