Fourier Series (FS) and Relation to DFT

Chapter Contents:

Search Mathematics of the DFT

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In continuous time, a periodic signal , with period 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

th harmonic frequency (rad/sec). The generally complex value $X(\omega_k)$ is called the

th Fourier series coefficient. The normalization by

is optional, but often included to make the Fourier series coefficients independent of the fundamental frequency

, and thereby depend only on the shape of one period of the time waveform.

written by 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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