Fourier Series and the Fast Fourier Transform

Trigonometric series are useful to represent periodic functions. A function deÞned for−∞ < x < ∞ has a period of 2π if f(x+2π) = f(x) for all x. In most practical situations, such a function can be expressed as a complex Fourier series

f(x) = ∞∑

=−∞ ce

ıx where ı = √−1.