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Notation and Terminology

If $ X$ is the DFT of $ x$ , we say that $ x$ and $ X$ form a transform pair and write

$\displaystyle \zbox {x\;\longleftrightarrow\;X}$   $\displaystyle \mbox{(\lq\lq $x$\ corresponds to $X$'')}$$\displaystyle . \protect$

Another notation we'll use is

\begin{eqnarray*}
\hbox{\sc DFT}(x)&\isdef & X,\quad\mbox{and} \\
\hbox{\sc DFT}_k(x)&\isdef & X(k).
\end{eqnarray*}

If we need to indicate the length of the DFT explicitly, we will write $ \hbox{\sc DFT}_N(x) = X$ and $ \hbox{\sc DFT}_{N,k}(x) = X(k)$ . As we've already seen, time-domain signals are consistently denoted using lowercase symbols such as ``$ x(n)$ ,'' while frequency-domain signals (spectra), are denoted in uppercase (`` $ X(\omega_k)$ '').


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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 © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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