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This is a list of [[linear transformation]]s of [[function (mathematics)|function]]s related to [[Fourier analysis]]. Such transformations [[Map (mathematics)|map]] a function to a set of [[coefficient]]s of [[basis function]]s, where the basis functions are [[trigonometric function|sinusoidal]] and are therefore strongly localized in the [[frequency spectrum]].  (These transforms are generally designed to be invertible.)  In the case of the Fourier transform, each basis function corresponds to a single [[frequency]] component.
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== Continuous transforms ==
Applied to functions of continuous arguments, Fourier-related transforms include:
* [[Two-sided Laplace transform]]
* [[Mellin transform]], another closely related integral transform
* [[Laplace transform]]
* [[Fourier transform]], with special cases''':'''
** [[Fourier series]]
*** When the input function/waveform is periodic, the Fourier transform output is a [[Dirac comb]] function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called '''Fourier series coefficients'''.  The term '''Fourier series''' actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients.
*** When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued.  But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed. The same discrete set is obtained by treating the duration of the segment as one period of a periodic function and computing the Fourier series coefficients.
** [[Sine and cosine transforms]]''':'''  When the input function has odd or even symmetry around the origin, the Fourier transform reduces to a sine or cosine transform.
* [[Hartley transform]]
* [[Short-time Fourier transform]] (or short-term Fourier transform) (STFT)
* [[Chirplet transform]]
* [[Fractional Fourier transform]] (FRFT)
* [[Hankel transform]]: related to the Fourier Transform of radial functions.
 
== Discrete transforms ==
For usage on [[computer]]s, number theory and algebra, discrete arguments (e.g. functions of a series of discrete samples) are often more appropriate, and are handled by the transforms (analogous to the continuous cases above):
 
* [[Discrete-time Fourier transform]] (DTFT)''':''' Equivalent to the Fourier transform of a "continuous" function that is constructed from the discrete input function by using the sample values to modulate a [[Dirac comb]]. When the sample values are derived by sampling a function on the real line, ƒ(''x''), the DTFT is equivalent to a [[periodic summation]] of the Fourier transform of '''ƒ'''. The DTFT output is always [[Period (physics)|periodic]] (cyclic).  An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded (or ''finite''), the length of one cycle.
** [[discrete Fourier transform]] (DFT)''':'''
*** When the input sequence is periodic, the DTFT output is also a [[Dirac comb]] function, modulated by the coefficients of a Fourier series<ref>The Fourier series represents <math>\scriptstyle \sum_{n=-\infty}^{\infty}f(nT)\cdot \delta(t-nT),\,</math> where T is the interval between samples.</ref> which can be computed as a DFT of one cycle of the input sequence. The number of discrete values in one cycle of the DFT is the same as in one cycle of the input sequence.
*** When the non-zero portion of the input sequence has finite duration, the DTFT is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed.  The same discrete set is obtained by treating the duration of the segment as one cycle of a periodic function and computing the DFT'''.
** Discrete [[sine and cosine transforms]]''':'''  When the input sequence has odd or even symmetry around the origin, the DTFT reduces to a [[discrete sine transform]] (DST) or [[discrete cosine transform]] (DCT).
***[[Regressive discrete Fourier series]], in which the period is determined by the data rather than fixed in advance.
**[[Discrete chebyshev transforms]] (on the 'roots' grid and the 'extrema' grid of the chebyshev polynomials of the first kind). This transform is of much importance in the field of spectral methods for solving differential equations because it can be used to swiftly and efficient go from grid point values to chebyshev series coefficients.
* Generalized DFT (GDFT), a generalization of the DFT and constant modulus transforms where phase functions might be of linear with integer and real valued slopes, or even non-linear phase bringing flexibilities for optimal designs of various metrics, e.g. auto- and cross-correlations.
* [[Z-transform]], a generalization of the DTFT.
* [[Modified discrete cosine transform]] (MDCT)
* [[Discrete Hartley transform]] (DHT)
* Also the discretized STFT (see above).
* [[Hadamard transform]] ([[Walsh function]]).
 
The use of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a [[fast Fourier transform]] (FFT).  The [[Nyquist&ndash;Shannon sampling theorem]] is critical for understanding the output of such discrete transforms.
 
==Notes==
{{reflist}}
 
==See also==
* [[Integral transform]]
* [[Wavelet transform]]
* [[Fourier transform spectroscopy]]
* [[Harmonic analysis]]
* [[List of transforms]]
* [[List of operators]]
* [[Bispectrum]]
 
== References==
* A. D. Polyanin and A. V. Manzhirov, ''Handbook of Integral Equations'', CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
* A.N. Akansu and H. Agirman-Tosun, [http://web.njit.edu/~akansu/PAPERS/AkansuIEEE-TSP2010.pdf "''Generalized Discrete Fourier Transform With Nonlinear Phase''"], IEEE ''Transactions on Signal Processing'', vol. 58, no. 9, pp. 4547-4556, Sept. 2010.
[[Category:Fourier analysis| ]]
[[Category:Image processing]]
[[Category:Integral transforms|Fourier transforms]]
[[Category:Mathematics-related lists|Fourier transforms]]
 
[[de:Liste der Fourier-Transformationen]]

Latest revision as of 19:33, 20 December 2014

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The sleek driving expertise of mud tires can be skilled especially on rugged terrain. Another good tip to remember is to always ensure that you are equipped with helpful instruments for those situations that may depart you stranded by the roadside. If that ought to happen you would a minimum of have the instruments wanted for somebody that can assist you. Proudly owning your individual set of instruments will let you lower your expenses as a result of it is possible for you to to make repairs needed in your automobile instead of having to go to a repair store. If you discover stores with the tires on sale, it might develop into a great place for you to find the spare instruments you want.

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The far bigger recall of Firestone tires, to which 88 deaths have been linked, includes the smaller of two sizes of tires put in on Ford Explorers. Bridgestone/Firestone is recalling 15-inch tires in that case, but not 16-inch tires. Smaller tires can carry less weight than larger tires of the same stress, so they're subject to more pressure when mounted on large, heavy sport utility automobiles. Ford's advice that the Explorer tires be inflated to 26 pounds per square inch has been criticized by some lawmakers and personal harm attorneys as inadequate. But the beneficial stress for the sixteen-inch tires on the Navigator is 30 pounds for the entrance tires and 35 kilos for the rear tires. Higher stress tires can carry extra weight.

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