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Fourier Series Special Case

In the very special case of truly periodic signals $ x(t) =
x(t+NT)$ , for all $ t\in(-\infty,\infty)$ , the DFT may be regarded as computing the Fourier series coefficients of $ x(t)$ from one period of its sampled representation $ x(nT)$ , $ n=0,1,\dots,N-1$ . The period of $ x$ must be exactly $ NT$ seconds for this to work. For the details, see §B.3.


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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