# Fourier series

A Fourier (pronounced foor-YAY) series is a specific type of infinite mathematical series involving trigonometric functions. The series gets its name from a French mathematician and physicist named Jean Baptiste Joseph, Baron de Fourier, who lived during the 18th and 19th centuries. Fourier series are used in applied mathematics, and especially in physics and electronics, to express periodic functions such as those that comprise communications signal waveform s.

Let { *a* _{} , *a* _{1} , *a* _{2} , *a* _{3} , ..., *a _{n}* , ...} and {

*b*

_{1},

*b*

_{2},

*b*

_{3}, ...,

*b*, ...} be infinite sets of constant s. These constants are called the Fourier coefficient s. Let

_{n}*x*be a variable. The general Fourier series is given by:

*F* ( *x* ) = *a* _{} /2 + *a* _{1} cos *x* + *b* _{1} sin *x* + *a* _{2} cos 2 *x* + *b* _{2} sin 2 *x* + ...

+ *a _{n}* cos

*nx*+

*b*sin

_{n}*nx*+ ...

Some waveforms are simple, such as the pure sine wave , but these are theoretical ideals. In the real world, most waveforms contain energy at harmonic frequencies (whole-number multiples of the lowest, or fundamental, frequency). The proportion of energy at harmonic frequencies, compared with the energy at the fundamental, depends on the waveform. Fourier series mathematically define such waveforms as functions of displacement (usually amplitude , frequency , or phase ) versus time .

As the number of calculated terms in a Fourier series increases, the series more and more closely approximates the exact function that defines a complex signal waveform. Computers can calculate Fourier series out to hundreds, thousands, or millions of terms.