# Dictionary:Fourier analysis

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(foor’ ēā,) The analytical representation of a waveform as a weighted sum of sinusoidal functions. Determining the amplitude and phase of cosine (or sine) waves of different frequencies into which a waveform can be decomposed. Fourier analysis can be thought of as a subset of the Fourier transform (q.v.). See Figure F-18. Opposite of Fourier synthesis. Named for Jean Baptiste Joseph Fourier (1768–1830), French mathematician.

FIG. F-18. (a) Fourier analysis involves finding the amplitude of frequency components for a waveform. The frequency-domain representation or spectrum G(f) of a discrete time function gt (waveform, seismic record trace, etc.) can be decomposed into a series of sinusoids by any of the following equivalent equations:
{\displaystyle {\begin{aligned}g_{t}&=a_{0}/2+\sum [a_{n}\cos(2\pi f_{n}t)+b_{n}\cos(2\pi ft)]\\&=c_{0}/2+\sum c_{n}\cos(2\pi f_{n}t-\gamma _{n})=\sum \alpha _{n}exp[j2\pi f_{n}t]\end{aligned}}}
Where
{\displaystyle {\begin{aligned}a_{n}&=(2/T)\sum g_{i}\cos(2\pi f_{i}t),\\b_{n}&=(2/T)\sum g_{i}\sin(2\pi f_{i}t),\\c_{n}&=(2/T)\sum g_{i}\cos(2\pi f_{i}t-\gamma _{i}),\\\gamma _{n}&=0,\;\gamma _{n}=\tan ^{-1}(b_{n}/a_{n}),\\&n>0,\;\alpha =(2/T)\sum g_{i}exp[j2\pi f_{i}t]\end{aligned}}}
If ${\displaystyle g(t)}$ is a continuous waveform, the sum signs become integrals. (b) Fourier synthesis involves superimposing the components to reconstitute the waveform. For an antisymmetric sawtooth waveform, the first four components are:
${\displaystyle \sin x;-(1/2)\sin 2x;(1/3)\sin 3x;-(1/4)\sin 4x}$
. For a Fourier transform the limits are ${\displaystyle 0}$ and ${\displaystyle \pm \infty ,}$ and ${\displaystyle G(f)}$ and ${\displaystyle g(t)}$ constitute a Fourier-transform pair; see Figure F-19.