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Fourier Series (FS) and Relation to DFT

In continuous time, a periodic signal $ x(t)$ , with period $ P$ seconds,B.2 may be expanded into a Fourier series with coefficients given by

$\displaystyle X(\omega_k) \isdef \frac{1}{P}\int_0^P x(t) e^{-j\omega_k t} dt, \quad k=0,\pm1,\pm2,\dots \protect$ (B.5)

where $ \omega_k \isdef 2\pi k/P$ is the $ k$ th harmonic frequency (rad/sec). The generally complex value $ X(\omega_k)$ is called the $ k$ th Fourier series coefficient. The normalization by $ 1/P$ is optional, but often included to make the Fourier series coefficients independent of the fundamental frequency $ 1/P$ , and thereby depend only on the shape of one period of the time waveform.



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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
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