# Fourier Series Function and Formula

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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 (wot + q1) + a2 cos (2 wot + q2)   + ... + ... + aN cos (N wot + qN)

where
the fundamental frequency wo 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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