Fourier Series

4. FOURIER SERIES

Any periodic function or waveform can be representing as a (possibly infinite) sum of sine and cosine functions. This type of representation is called Fourier series. It is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms.

$f(x) = \frac{a_0}{2} + \sum_{k=1}^{\infty} a_k \cos \frac{2 \pi k x}{T}  + \sum_{k=1}^{\infty} b_k \sin \frac{2 \pi k x}{T} ,$

$a_k =  \frac{2}{T} \int_0^T f(x) \cos \frac{2 \pi k x}{T}\:dx, \quad b_k = \frac{2}{T} \int_0^T f(x) \sin \frac{2 \pi k x}{T}\:dx$

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