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Fourier Transforms for Continuous/Discrete Time/Frequency

The Fourier transform can be defined for signals which are

This results in four cases. Quite naturally, the frequency domain has the same four cases, When time is discrete, the frequency axis is finite, and vice versa.

This book has been concerned almost exclusively with the discrete-time, discrete-frequency case (the DFT), and in that case, both the time and frequency axes are finite in length. In the following sections, we briefly summarize the other three cases. Table B.1 summarizes all four Fourier-transform cases corresponding to discrete or continuous time and/or frequency.


Table B.1: Four cases of sampled/continuous time and frequency.
\begin{table}\begin{center} \begin{displaymath} \begin{array}{\vert c\vert c\vert c\vert} \hline \multicolumn{3}{\vert c\vert}{\hbox{Time Duration}} \\ \hline \hbox{Finite} & \hbox{Infinite} & \\ \hline \hbox{Discrete FT (DFT)} & \hbox{Discrete Time FT (DTFT)} & \hbox{discr.} \\ X(k)=\displaystyle\sum_{n=0}^{N-1} x(n)e^{-j\omega_k n} & \displaystyle X(\omega)=\displaystyle\sum_{n=-\infty}^{+\infty} x(n)e^{-j\omega n} & \hbox{time} \\ k=0,1, \dots, N-1 & \omega \in [ - \pi, +\pi ) & \hbox{$n$} \\ \hline \hbox{Fourier Series (FS)} & \hbox{Fourier Transform (FT)} & \hbox{cont.} \\ X(k)={\scriptstyle\frac{1}{P}} \displaystyle\int_0^Px(t)e^{-j\omega_kt}dt & X(\omega)= \displaystyle\int_{-\infty}^{+\infty}x(t)e^{-j\omega t} dt & \hbox{time} \\ k = - \infty, \ldots, +\infty & \omega \in ( - \infty, +\infty) & \hbox{$t$} \\ \hline \hbox{discrete freq. } k & \hbox{continuous freq. } \omega & \\ \hline \end{array}\end{displaymath} \end{center} \end{table}



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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
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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