Fourier analysis

Introduction

Fourier analysis is a method of analysing functions. These functions may be electrical signals (say, from an electronic circuit being tested), pure mathematical functions, or any kind of data being analysed on a computer. Regardless, if the function is single-valued, Fourier analysis can be used to produce an imperfect approximation.

Basics

The trigonometric form

Fourier analysis works by breaking down the function being considered into a Fourier Series. The Fourier Series, in simplest terms, is a summation of sine and cosine functions. Each of these trigonometric functions looks something like this:

<math>a_n \cos(\omega_n t) + b_n \sin(\omega_n t)</math>

The exponential form

Consider f(x) as real valued function

<math>f(x)=\sum_{n=-\infty}^{\infty} A_n e^{inx}</math> <ref>http://mathworld.wolfram.com/FourierSeries.html</ref>