## Fourier Series and FFT

by AppliedMathematics.info

The decomposition of a periodic function as a fourier series is of great importance in many areas of physics such as vibration, acoustics and electromagnetics. It is also a fundamental technique of digital signal processing.

__Fourier Series__

A periodic function *f*(*t*) of period *T* can be expanded in the form

*f*(*t*) = | *a*_{0} 2
| + | ¥ å *n* = 1
| *a*_{n} cos( | 2 *n* p *T*
| )+ | ¥ å *n* = 1
| *b*_{n} cos( | 2 *n* p *T*
| ) | |

where the *a*_{n} and *b*_{n} are real numbers defined by

*a*_{0} = | 2 *T*
| | ó õ | *T*
0
| *f*(*t*) *d**t* , | |

*a*_{n} = | 2 *T*
| | ó õ | *T*
0
| *f*(*t*) cos( | 2*n* p*t* *T*
| ) *d**t* , | |

*b*_{n} = | 2 *T*
| | ó õ | *T*
0
| *f*(*t*) sin( | 2*n* p*t* *T*
| ) *d**t* . | |

Note that the domain of integration may be taken to be [-^{T}/_{2}, ^{T}/_{2}]. Alternatively, we may use the complex form of the series,

*f*(*t*) = *R**e* | é ê ë | | ¥ å *n* = 0
| (*a*_{n} - *i* *b*_{n}) exp( *i* | 2 *n* p *T*
| *t* ) | ù ú û | | |

with

*a*_{n} - *i* *b*_{n} = | 2 *T*
| | ó õ | *T*
0
| *f*(*t*) exp(- *i* | 2*n* p *T*
| *t* ) *d**t* . | |

__Fourier Transform of Discretely Sampled Data__

Usually the periodic function *f*(*t*) will be realised only at discrete sample points. Suppose that *f*_{1}, *f*_{2}, ... *f*_{N} is a sequence of discrete values of *f*(*t*); *f*_{k} = *f*( [(*k*-1 )/*N*] *T*). By substitution of these values into the Fourier transform equation above and using the rectangle integration rule,

*a*_{n} - *i* *b*_{n} » *c*_{n} = | 1 *N*
| | *N* å *k* = 0
| *f*_{k} *e*^{-2 i n pk/N} | |

The Fast Fourier Transform is an efficient algorithm for obtaining the *c*_{n} from the *f*_{k}.

# www.science-books.net