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See Also

Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Fourier Theorems for the DFT
       The DFT and its Inverse Restated
          Notation and Terminology

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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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Next: Modulo Indexing, Periodic Extension

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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