Fourier analysis

## Introduction

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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

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## The trigonometric form

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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:

a n cos ⁡ ( ω n t ) + b n sin ⁡ ( ω n t ) {\displaystyle a_{n}\cos(\omega _{n}t)+b_{n}\sin(\omega _{n}t)}

## The exponential form

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Consider f(x) as real valued function

f ( x ) = ∑ n = − ∞ ∞ A n e i n x {\displaystyle f(x)=\sum _{n=-\infty }^{\infty }A_{n}e^{inx}} [1]

1. ↑ http://mathworld.wolfram.com/FourierSeries.html