Fourier transform A mathematical operation that analyzes an arbitrary waveform into its constituent sinusoids (of different frequencies and amplitudes). This relationship is stated as

where s(t) is the waveform to be decomposed into a sum of sinusoids, S(f) is the Fourier transform of s(t), and i = √–1. An analogous formula gives s(t) in terms of S(f), but with a normalizing factor, 1/2π. Sometimes, for symmetry, the normalizing factor is split between the two relations.

The Fourier transform pair, s(t) and S(f), has to be modified before it is amenable to computation on a digital computer. This modified pair, called the discrete Fourier transform (DFT) must approximate as closely as possible the continuous Fourier transform. The continuous time function is approximated by N samples at time intervals T: g(kT), k = 0,1,… n–1

The continuous Fourier transform is also approximated by N samples at frequency intervals 1/NT: G(n/NT), n = 0,1,… N–1

Since the N values of time and frequency are related by the continuous Fourier transform, then a discrete relationship can be derived:

Fourier analysis The method whereby any periodic function can be broken down into a covergent trigonometric series of the form f(χ) = a₀/2 + Σ_∞ⁿ⁼¹(a_ncos nχ + b_nsin nχ) where a_n and b_n are constant coefficients. Fourier analysis is the process of determining the frequency domain function from a time function (e.g. a seismic-trace waveform). See also FOURIER TRANSFORM.

Fourier transform The mathematical formulae by which a time function (e.g. a seismic trace) is converted into a frequency domain function and vice versa. See also FOURIER ANALYSIS; and FOURIER SYNTHESIS.

Fourier analysis The analysis of an arbitrary waveform into its constituent sinusoids (of different frequencies and amplitudes). See Fourier transform. See also orthonormal basis.

Fourier series The infinite trigonometric series

By suitable choice of the coefficients a_i and b_i, the series can be made equal to any function of x defined on the interval (–π, π). If f is such a function, the Fourier coefficients are given by the formulas