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DFT Definition

The Discrete Fourier Transform (DFT) of a signal $ x$ may be defined by

$\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 ` $ \isdeftext $ ' means ``is defined as'' or ``equals by definition'', and

\begin{eqnarray*}
\sum_{n=0}^{N-1} f(n) &\isdef & f(0) + f(1) + \dots + f(N-1)\\
x(t_n) &\isdef & \mbox{input signal \emph{amplitude} (real or complex) at time $t_n$\ (sec)}
\\
t_n &\isdef & nT = \mbox{$n$th sampling instant (sec), $n$\ an integer $\ge 0$}\\
T &\isdef & \mbox{sampling interval (sec)} \\
X(\omega_k ) &\isdef & \mbox{\emph{spectrum}\index{spectrum\vert textbf} of $x$\ (complex valued), at frequency $\omega_k $}\\
\omega_k &\isdef & k\Omega = \mbox{$k$th frequency sample (radians per second)} \\
\Omega &\isdef & \frac{2\pi}{NT}
= \mbox{radian-frequency sampling interval (rad/sec)} \\
f_s &\isdef & 1/T = \mbox{\emph{sampling rate}\index{sampling rate\vert textbf} (samples/sec, or Hertz (Hz))}\index{Hertz\vert textbf}\index{Hz\vert textbf}\\
N &=& \mbox{number of time samples = no.\ frequency samples (integer).}
\end{eqnarray*}

The sampling interval $ T$ is also called the sampling period. For a tutorial on sampling continuous-time signals to obtain non-aliased discrete-time signals, see Appendix D.

When all $ N$ signal samples $ x(t_n)$ are real, we say $ x\in{\bf R}^N$ . If they may be complex, we write $ x\in{\bf C}^N$ . Finally, $ n\in{\bf Z}$ means $ n$ is any integer.


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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