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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. As you might expect, the frequency domain has the same cases: When time is discrete, the frequency axis is finite, and vice versa.

Reference [264] develops the DFT in detail--the discrete-time, discrete-frequency case. In the DFT, both the time and frequency axes are finite in length.

Table 2.1 (next page) summarizes the four Fourier-transform cases corresponding to discrete or continuous time and/or frequency.

Table 2.1: Four cases of sampled/continuous finite/infinite time and frequency. (Often the FS coefficients are divided by the period $ P$ .)
\begin{array}{\vert c\vert c\vert c\vert}
\multicolumn{3}{\vert c\vert}{\hbox{Time Duration}} \\
\hbox{Finite} & \hbox{Infinite} & \\
\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{$n$}
\hbox{Fourier Series (FS)} & \hbox{Fourier Transform (FT)}
& \hbox{cont.}
& X(\omega)= \displaystyle\int_{-\infty}^{+\infty}x(t)e^{-j\omega t} dt
& \hbox{time}
k = - \infty, \ldots, +\infty
& \omega \in ( - \infty, +\infty)
& \hbox{$t$}
\hbox{discrete freq. } k & \hbox{continuous freq. } \omega & \\

In all four cases, the Fourier transform can be interpreted as the inner product of the signal $ x$ with a complex sinusoid at radian frequency $ \omega$ [264]:

$\displaystyle X(\omega) = \left<x,s_\omega\right>$ (3.1)

where $ s_\omega$ is appropriately adapted, e.g.,

s_\omega(t) &=& e^{j\omega t}\qquad\qquad\qquad\quad\;\mbox{(Fourier Transform)}\\
s_{\omega_k}(t_n) &=& e^{j\omega_k t_n} \eqsp e^{j2\pi nk/N} \quad\mbox{(DFT)} \\
s_\omega(t_n) &=& e^{j\omega t_n} \eqsp \, e^{j\omega n} \quad\qquad\;\!\mbox{(DTFT)}.

In spectral modeling of audio, we usually deal with indefinitely long signals. Fourier analysis of an indefinitely long discrete-time signal is carried out using the Discrete Time Fourier Transform (DTFT).3.1Below, the DTFT is defined, and selected Fourier theorems are stated and proved for the DTFT case. Additionally, for completeness, the Fourier Transform (FT) is defined, and selected FT theorems are stated and proved as well. Theorems for the DFT case are detailed in [264].3.2

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University