Fourier Series Representation of Periodic Functions

Abstract

With few exceptions, periodic functions can be expressed as a Fourier series of sine and cosine functions. If x(t)

• 1. is a single-valued, periodic function of the variable t with a period T,

• 2. satisfies the Dirichlet conditions (i.e., it is continuous except for a finite number of discontinuities and has only a finite number of maxima and minima), and

• 3. is a bounded function, i.e., $\int_0^T {\left| {x(t)} \right|} dt \leqslant c < \infty ,$ it can then be represented by the following Fourier series:

  $$ x(t) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {({a_n}\cos (2\pi n{f_0}t)} + {b_n}\sin (2\pi n{f_0}t)),{f_0} = \frac{1}{T}.$$

  (1.1)