# Fourier Series Function and Formula

### - basics of the Fourier mathematical series and the Fourier series function used in a variety of math or maths calculations and in particular for analysing electrical and other waveforms.

### In this section

The Fourier series and its resulting Fourier analysis is used in mathematics and also within electronics engineering to analyse waveforms and process them in a process known as digital signal processing (DSP).

A Fourier series uses mathematical processes and decomposes a periodic function into a sum of simple oscillating functions, i.e. sines and cosines. By manipulating these series using mathematical processes it is possible to analyse and process waveforms using computer techniques. Often this is done in real time to enable complex processing to replace analogue circuitry. This has the advantage that waveforms can be processed more exactly to give very high levels of performance.

## Basic Fourier series formula and function

For a continuous-time, T-periodic signal x(t), the N-harmonic Fourier series approximation can be written as the following function or formula:

x(t) = |
a0 + a1 cos (w_{o}t + q1) + a2 cos (2 w_{o}t + q2) + ... |

+ ... + aN cos (N w_{o}t + qN) |

where

the fundamental frequency w_{o} is 2pi /T rad/sec;

the amplitude coefficients a1, ..., aN are non-negative

the radian phase angles satisfy 0 £ q1 , ..., qN < 2pi

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