Dictionary:Fig F-18

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 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 Fourier-transform pair; see Figure F-19.