Fourier transform
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Template:Fourier transformsThe Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i.e. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). There are many closely related variations of this transform, summarized below, depending upon the type of function being transformed. See also: List of Fourier-related transforms.
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Applications
Fourier transforms have many scientific applications — in physics, number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, oceanography, optics, geometry, and other areas. (In signal processing and related fields, the Fourier transform is typically thought of as decomposing a signal into its component frequencies and their amplitudes.) This wide applicability stems from several useful properties of the transforms:
- The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality).
- The transforms are invertible, and in fact the inverse transform has almost the same form as the forward transform.
- The sinusoidal basis functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. (For example, in a linear time-invariant physical system, frequency is a conserved quantity, so the behavior at each frequency can be solved independently.)
- By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.
- The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms.
Variants of the Fourier transform
Continuous Fourier transform
Most often, the unqualified term "Fourier transform" refers to the continuous Fourier transform, representing any square-integrable function <math>f \left( t \right)</math> as a sum of complex exponentials with angular frequencies <math>\omega</math> and complex amplitudes <math>F\left( \omega \right)</math>:
- <math>
f \left( t \right) = \left( \mathcal{F}^{-1}F \right) \left( t \right)
= \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^\infty F\left( \omega\right) e^{i\omega t}\,d\omega.
</math>
This is actually the inverse continuous Fourier transform, whereas the Fourier transform expresses <math>F\left( \omega \right)</math> in terms of <math>f \left( t \right)</math>; the original function and its transform are sometimes called a transform pair. See continuous Fourier transform for more information, including a table of transforms, discussion of the transform properties, and the various conventions. A generalization of this transform is the fractional Fourier transform, by which the transform can be raised to any real "power".
When <math>f \left( t \right)</math> is an even or odd function, the sine or cosine terms disappear and one is left with the cosine transform or sine transform, respectively. Another important case is where <math>f \left( t \right)</math> is purely real, where it follows that <math>F\left( - \omega \right) = F\left( \omega \right)^*</math> (where the <math>*</math> denotes complex conjugation.) Similar special cases appear for all other variants of the Fourier transform as well.
Multi-dimensional version
The formulation for the Fourier transform given above applies in one dimension. The Fourier transform, however, can be expanded to arbitrary dimension <math>n</math>. The more generalised version of this transform in dimension <math>n</math>, notated by <math>\mathcal{F}_n</math> is:
- <math>f \left( \mathbf{x} \right) = \left( \mathcal{F}^{-1}_n F \right) \left( \mathbf{x} \right) = \frac{1}{(2\pi)^{n/2}} \int F \left( \boldsymbol{\omega} \right) e^{i \left\langle \boldsymbol{\omega} ,\mathbf{x} \right\rangle} \, d \boldsymbol{\omega},</math>
where <math>\mathbf{x}</math> and <math>\boldsymbol{\omega}</math> are <math>n</math>-dimensional vectors, <math>\left\langle \boldsymbol{\omega} ,\mathbf{x} \right\rangle</math> is the inner product of these two vectors, and the integration is performed over all <math>n</math> dimensions.
Fourier series
The continuous transform is itself actually a generalization of an earlier concept, a Fourier series, which was specific to periodic (or finite-domain) functions <math>f \left( x \right)</math> (with period <math>2\pi</math>), and represents these functions as a series of sinusoids:
- <math>f(x) = \sum_{n=-\infty}^{\infty} F_n \,e^{inx} ,</math>
where <math>F_n</math> is the (complex) amplitude. Or, for real-valued functions, the Fourier series is often written:
- <math>f(x) = \frac{1}{2}a_0 + \sum_{n=1}^\infty\left[a_n\cos(nx)+b_n\sin(nx)\right],</math>
where <math>a_n</math> and <math>b_n</math> are the (real) Fourier series amplitudes.
Discrete Fourier transform
For use on computers, both for scientific computation and digital signal processing, one must have functions xk that are defined over discrete instead of continuous domains, again finite or periodic. In this case, one uses the discrete Fourier transform (DFT), which represents xk as the sum of sinusoids:
- <math>x_k = \frac{1}{n} \sum_{j=0}^{n-1} f_j e^{2\pi ijk/n} \quad \quad k = 0,\dots,n-1</math>
where fj are the Fourier amplitudes. Although applying this formula directly would require O(n2) operations (see Big O notation), it can be computed in only O(n log n) operations using a fast Fourier transform (FFT) algorithm, which makes FFT a practical and important operation on computers.
Other variants
The DFT is a special case of (and is sometimes used as an approximation for) the discrete-time Fourier transform (DTFT), in which the xk are defined over discrete but infinite domains, and thus the spectrum is continuous and periodic. The DTFT is essentially the inverse of the Fourier series.
These Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topological groups, which are studied in harmonic analysis; there, one transforms from a group to its dual group. This treatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform.
Time-frequency transforms such as the short-time Fourier transform, wavelet transforms, chirplet transforms, and the fractional Fourier transform try to obtain frequency information from a signal as a function of time (or whatever the independent variable is), although the ability to simultaneously resolve frequency and time is limited by a (mathematical) uncertainty principle.
Family of Fourier transforms
The following table summarizes the family of Fourier transforms. We see that discreteness in one domain implies periodicity in the conjugate domain and that continuity in one domain implies aperiodicity in the conjugate domain. Moreover: reality in one domain implies symmetry in the conjugate domain.
| Transform | Time | Frequency |
|---|---|---|
| Continuous Fourier transform | Continuous, Aperiodic | Continuous, Aperiodic |
| Fourier series | Continuous, Periodic | Discrete, Aperiodic |
| Discrete-time Fourier transform | Discrete, Aperiodic | Continuous, Periodic |
| Discrete Fourier transform | Discrete, Periodic | Discrete, Periodic |
Interpretation in terms of time and frequency
In terms of signal processing, the transform takes a time series representation of a signal function and maps it into a frequency spectrum, where ω is angular frequency. That is, it takes a function in the time domain into the frequency domain; it is a decomposition of a function into harmonics of different frequencies.
When the function f is a function of time and represents a physical signal, the transform has a standard interpretation as the frequency spectrum of the signal. The magnitude of the resulting complex-valued function F represents the amplitudes of the respective frequencies (ω), while the phase shifts are given by arctan (imaginary parts/real parts).
However, it is important to realize that Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze spatial frequencies, and indeed for nearly any function domain.
Applications in signal processing
In signal processing, Fourier transformation can isolate individual components of a complex signal, concentrating them for easier detection and/or removal. A large family of signal processing techniques consist of Fourier-transforming a signal (such as a clip of audio or an image), manipulating the Fourier-transformed data in a simple way, and reversing the transformation. Some examples include:
- Removal of unwanted frequencies from an audio recording (used to eliminate hum from leakage of AC power into the signal, to eliminate the stereo subcarrier from FM radio recordings, or to create karaoke tracks with the vocals removed);
- Noise gating of audio recordings to remove quiet background noise by eliminating Fourier components that do not exceed a preset amplitude;
- Equalization of audio recordings with a series of bandpass filters;
- Digital radio reception with no superheterodyne circuit, as in a modern cell phone or radio scanner;
- Image processing to remove periodic or anisotropic artifacts such as jaggies from interlaced video, stripe artifacts from strip aerial photography, or wave patterns from radio frequency interference in a digital camera;
- Cross correlation of similar images for co-alignment;
- X-ray crystallography to reconstruct a protein's structure from its diffraction pattern;
- Fourier transform ion cyclotron resonance mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field.
Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses Fourier transformation of small square pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision, and weak components are eliminated entirely, so that the remaining components can be stored very compactly. In image reconstruction, each Fourier-transformed image square is reassembled from the preserved approximate components, and then inverse-transformed to produce an approximation of the original image.
References
- A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
- Smith, Steven W. The Scientist and Engineer's Guide to Digital Signal Processing, 2nd edition. San Diego: California Technical Publishing, 1999. ISBN 0-9660176-3-3. (also available online: [1])
See also
- Bispectrum
- Characteristic function (probability theory)
- Chirplet
- Number-theoretic transform
- Laplace transform
- Mellin transform
- Orthogonal functions
- Pontryagin duality
- Schwartz space
- Two-sided Laplace transform
- Wavelet
External links
- Online Computation of the transform or inverse transform, wims.unice.fr
- Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
- An Intuitive Explanation of Fourier Theory by Steven Lehar.ar:تحويل فوريي
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