Fourier analysis

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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:

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Fourier analysis The method whereby any periodic function can be broken down into a covergent trigonometric series of the form f(χ) = a0/2 + Σn=1(ancos nχ + bnsin nχ) where an and bn 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.

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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.

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

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Fourier series The infinite trigonometric series

By suitable choice of the coefficients ai and bi, 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