Fourier analysis

Fourier analysis is a method of defining periodic waveform s in terms of trigonometric function s. The method gets its name from a French mathematician and physicist named Jean Baptiste Joseph, Baron de Fourier, who lived during the 18th and 19th centuries. Fourier analysis is used in electronics, acoustics, and communications.

Many waveforms consist of energy at a fundamental frequency and also at harmonic frequencies (multiples of the fundamental). The relative proportions of energy in the fundamental and the harmonics determines the shape of the wave. The wave function (usually amplitude , frequency, or phase versus time ) can be expressed as of a sum of sine and cosine function s called a Fourier series , uniquely defined by constants known as Fourier coefficient s. If these coefficients are represented by a , a ₁ , a ₂ , a ₃ , ..., a _n , ... and b ₁ , b ₂ , b ₃ , ..., b _n , ..., then the Fourier series F ( x ), where x is an independent variable (usually time), has the following form:

F ( x ) = a /2 + a ₁ cos x + b ₁ sin x + a ₂ cos 2 x + b ₂ sin 2 x + ...
+ a _n cos nx + b _n sin nx + ...

In Fourier analysis, the objective is to calculate coefficients a , a ₁ , a ₂ , a ₃ , ..., a _n and b ₁ , b ₂ , b ₃ , ..., b _n up to the largest possible value of n . The greater the value of n (that is, the more terms in the series whose coefficients can be determined), the more accurate is the Fourier-series representation of the waveform.

Compare Fourier synthesis .