(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 F18. Opposite of Fourier synthesis. Named for Jean Baptiste Joseph Fourier (1768–1830), French mathematician.
FIG. F18. (
a)
Fourier analysis involves finding the amplitude of frequency components for a waveform. The frequencydomain representation or spectrum
G(
f) of a discrete time function
g_{t} (waveform, seismic record trace, etc.) can be decomposed into a series of sinusoids by any of the following equivalent equations:
${\begin{aligned}g_{t}&=a_{0}/2+\sum [a_{n}\cos(2\pi f_{n}t)+b_{n}\cos(2\pi ft)]\\&=c_{0}/2+\sum c_{n}\cos(2\pi f_{n}t\gamma _{n})=\sum \alpha _{n}exp[j2\pi f_{n}t]\end{aligned}}$ Where
${\begin{aligned}a_{n}&=(2/T)\sum g_{i}\cos(2\pi f_{i}t),\\b_{n}&=(2/T)\sum g_{i}\sin(2\pi f_{i}t),\\c_{n}&=(2/T)\sum g_{i}\cos(2\pi f_{i}t\gamma _{i}),\\\gamma _{n}&=0,\;\gamma _{n}=\tan ^{1}(b_{n}/a_{n}),\\&n>0,\;\alpha =(2/T)\sum g_{i}exp[j2\pi f_{i}t]\end{aligned}}$ If
$g(t)$ 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:
$\sin x;(1/2)\sin 2x;(1/3)\sin 3x;(1/4)\sin 4x$. For a Fourier transform the limits are
$0$ and
$\pm \infty ,$ and
$G(f)$ and
$g(t)$ constitute a Fouriertransform pair; see Figure
F19.
External links
find literature about Fourier analysis







