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Fourier transform
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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Cite this article
JOHN DAINTITH. "Fourier transform." A Dictionary of Computing. 2004. Encyclopedia.com. 1 Jun. 2012 <http://www.encyclopedia.com>. JOHN DAINTITH. "Fourier transform." A Dictionary of Computing. 2004. Encyclopedia.com. (June 1, 2012). http://www.encyclopedia.com/doc/1O11-Fouriertransform.html JOHN DAINTITH. "Fourier transform." A Dictionary of Computing. 2004. Retrieved June 01, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O11-Fouriertransform.html |
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Fourier analysis
Fourier analysis A technique used to find the frequencies present in a complex signal; also known as frequency analysis. A series of measurements made at different times is broken down mathematically into the sum of simple oscillations of various frequencies. The lowest frequency present is called the fundamental frequency, and the higher frequencies are then harmonics (whole multiples) of the fundamental frequency. The brightness fluctuations of some variable stars, for example, can be broken down into two or three simple sinusoidal variations by the use of Fourier analysis. The technique was invented by the French mathematician ( Jean-Baptiste) Joseph Fourier (1768–1830).
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Cite this article
"Fourier analysis." A Dictionary of Astronomy. 1997. Encyclopedia.com. 1 Jun. 2012 <http://www.encyclopedia.com>. "Fourier analysis." A Dictionary of Astronomy. 1997. Encyclopedia.com. (June 1, 2012). http://www.encyclopedia.com/doc/1O80-Fourieranalysis.html "Fourier analysis." A Dictionary of Astronomy. 1997. Retrieved June 01, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O80-Fourieranalysis.html |
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Fourier transform
Fourier transform The mathematical breakdown of a non-repeating phenomenon into its component frequencies. It may be regarded as Fourier analysis applied to a function with an infinite period. The technique is widely used in image processing to reduce the effects of periodic and random errors. It is also needed to convert the outputs from aperture synthesis telescopes and Fourier transform spectrometers into images and spectra respectively.
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Cite this article
"Fourier transform." A Dictionary of Astronomy. 1997. Encyclopedia.com. 1 Jun. 2012 <http://www.encyclopedia.com>. "Fourier transform." A Dictionary of Astronomy. 1997. Encyclopedia.com. (June 1, 2012). http://www.encyclopedia.com/doc/1O80-Fouriertransform.html "Fourier transform." A Dictionary of Astronomy. 1997. Retrieved June 01, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O80-Fouriertransform.html |
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Fourier analysis
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(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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Cite this article
AILSA ALLABY and MICHAEL ALLABY. "Fourier analysis." A Dictionary of Earth Sciences. 1999. Encyclopedia.com. 1 Jun. 2012 <http://www.encyclopedia.com>. AILSA ALLABY and MICHAEL ALLABY. "Fourier analysis." A Dictionary of Earth Sciences. 1999. Encyclopedia.com. (June 1, 2012). http://www.encyclopedia.com/doc/1O13-Fourieranalysis.html AILSA ALLABY and MICHAEL ALLABY. "Fourier analysis." A Dictionary of Earth Sciences. 1999. Retrieved June 01, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O13-Fourieranalysis.html |
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Fourier series
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 |
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Cite this article
JOHN DAINTITH. "Fourier series." A Dictionary of Computing. 2004. Encyclopedia.com. 1 Jun. 2012 <http://www.encyclopedia.com>. JOHN DAINTITH. "Fourier series." A Dictionary of Computing. 2004. Encyclopedia.com. (June 1, 2012). http://www.encyclopedia.com/doc/1O11-Fourierseries.html JOHN DAINTITH. "Fourier series." A Dictionary of Computing. 2004. Retrieved June 01, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O11-Fourierseries.html |
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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.
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Cite this article
AILSA ALLABY and MICHAEL ALLABY. "Fourier transform." A Dictionary of Earth Sciences. 1999. Encyclopedia.com. 1 Jun. 2012 <http://www.encyclopedia.com>. AILSA ALLABY and MICHAEL ALLABY. "Fourier transform." A Dictionary of Earth Sciences. 1999. Encyclopedia.com. (June 1, 2012). http://www.encyclopedia.com/doc/1O13-Fouriertransform.html AILSA ALLABY and MICHAEL ALLABY. "Fourier transform." A Dictionary of Earth Sciences. 1999. Retrieved June 01, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O13-Fouriertransform.html |
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Fourier analysis
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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Cite this article
JOHN DAINTITH. "Fourier analysis." A Dictionary of Computing. 2004. Encyclopedia.com. 1 Jun. 2012 <http://www.encyclopedia.com>. JOHN DAINTITH. "Fourier analysis." A Dictionary of Computing. 2004. Encyclopedia.com. (June 1, 2012). http://www.encyclopedia.com/doc/1O11-Fourieranalysis.html JOHN DAINTITH. "Fourier analysis." A Dictionary of Computing. 2004. Retrieved June 01, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O11-Fourieranalysis.html |
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