Fourier analysis
11659
224341491
2008-07-08T12:42:17Z
Strangealibi
1876342
/* Time-frequency transforms */ the sentence did not make sense. ~~~~
{{Fourier transforms}}
'''Fourier analysis''', named after [[Joseph Fourier]]'s introduction of the [[Fourier series]], is the decomposition of a function in terms of [[trigonometric function|sinusoidal]]<ref> An explanation: http://www.4p8.com/eric.brasseur/fouren.html</ref> functions (called [[basis function]]s) of different frequencies that can be recombined to obtain the original function. The recombination process is called '''Fourier synthesis''' (in which case, ''Fourier analysis'' refers specifically to the decomposition process).
The result of the decomposition is the amount (i.e. amplitude) and the [[Phase (waves)|phase]] to be imparted to each basis function (each frequency) in the reconstruction. It is therefore also a function (of frequency), whose value can be represented as a [[complex number]], in either [[Polar coordinates|polar]] or [[Cartesian coordinate system|rectangular]] coordinates. And it is referred to as the ''[[frequency domain]] representation'' of the original function. A useful analogy is the waveform produced by a musical chord and the set of musical notes (the frequency components) that it comprises.
The term [[Fourier transform]] can refer to either the frequency domain representation of a function or to the process/formula that "[[Transform (mathematics)|transforms]]" one function into the other. However, the transform is usually given a more specific name depending upon the domain and other properties of the function being transformed, as elaborated below. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as [[harmonic analysis]].
Each [[Transform (mathematics)|transform]] used for analysis (see [[list of Fourier-related transforms]]) has a corresponding [[Inverse (mathematics)|inverse]] transform that can be used for synthesis.
==Applications==
{{Expand-section|date=January 2008}}
Fourier analysis has many scientific applications — in [[physics]], [[number theory]], [[combinatorics]], [[signal processing]], [[imaging]], [[probability theory]], [[statistics]], [[option pricing]], [[cryptography]], [[numerical analysis]], [[acoustics]], [[oceanography]], [[optics]] and [[diffraction]], [[geometry]], and other areas.
This wide applicability stems from many useful properties of the transforms''':'''
* The transforms are [[linear operator]]s and, with proper normalization, are [[unitary operator|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 exponential basis functions are [[eigenfunctions]] of [[derivative|differentiation]], which means that this representation transforms linear [[differential equation]]s 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 [[multiplication algorithm|multiplying large numbers]].
* The [[Discrete Fourier transform|discrete]] version of the Fourier transform (see below) can be evaluated quickly on computers using [[fast Fourier transform]] (FFT) algorithms.
==Variants of Fourier analysis==
Fourier analysis has different forms, some of which have different names. Variations within the same name are caused by differences in scale factors ("normalization") and/or the units that are used. The variable <math>f\,</math>, for instance, generally represents frequency in [[hertz]] ([[SI]] units), or a [[normalized frequency]] in ''cycles per sample''. Also popular is the variable <math>\omega\,</math>, which represents [[angular frequency]] units, or a normalized frequency in ''radians per sample''.
Variations with different names usually reflect different properties of the function or data being analyzed. The resultant transforms can be seen as special cases or generalizations of each other.
=== (Continuous) Fourier transform ===
Most often, the unqualified term '''Fourier transform''' refers to the transform of functions of a continuous [[real number|real]] argument, such as time (<math>t\,</math>). The amplitude and phase of a sinusoidal component of function ''s''(''t'') depends on the component's frequency. In terms of ordinary frequency (<math>f\,</math>), it is the complex number''':'''
:<math>S(f) = \int_{-\infty}^{\infty} s(t) \cdot e^{-i 2\pi f t} dt</math>
Evaluating this quantity for all values of <math>f\,</math> produces the frequency-domain function.
Also see [[#How it works (a basic explanation)|How it works]], below. And see [[Continuous Fourier transform]] for even more information, including''':'''
* the inverse transform, ''S''(''f'') → ''s''(''t'')
* conventions for amplitude normalization and frequency scaling/units
* transform properties
* tabulated transforms of specific functions
* an extension/generalization for functions of multiple dimensions, such as '''images'''
=== Discrete-time Fourier transform (DTFT) ===
For use on computers, a useful "discrete-time" function can be obtained by [[Sampling (signal processing)|sampling]] a "continuous-time" function, ''s''(''t''), which produces a sequence, ''s''(''nT''), for integer values of ''n''. The '''DTFT''' is equivalent to the Fourier transform of a "continuous" function that is constructed by using the sequence <math>s[n] \ \stackrel{\mathrm{def}}{=}\ T\cdot s(nT)</math> to modulate a [[Dirac comb]]. In that case, the integral formula above simplifies to a summation''':'''
:<math>S_T(f) = \sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i 2\pi f n T} = \sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i 2\pi \frac{f}{f_s} n},</math>
which is a '''periodic''' function, with period <math>f_s = 1/T.\,</math> An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded (or ''finite''), with span <math>f_s.\,</math>
The DTFT can be applied to any discrete sequence. But in the particular case where ''s''[''n''] are samples of ''s''(''t''), <math>S_T(f)\,</math> is closely related to <math>S(f).\,</math> See [[Discrete-time Fourier transform]] for more information on this and other topics, including''':'''
* the inverse transform
* normalized frequency units
* windowing (finite-length sequences)
* transform properties
* tabulated transforms of specific functions
=== Analysis of periodic functions or functions with limited duration ===
==== Fourier series ====
When <math>s(t)\,</math> is periodic, with period <math>\tau\,</math>, <math>S(f)\,</math> is a [[Dirac comb]] function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. The sequence is given by''':'''
:<math>S[k] = \frac{1}{\tau} \int_{0}^{\tau} s(t) \cdot e^{-i 2\pi \frac{k}{\tau} t} \, dt,</math>
for all integer values of ''k''.
This sequence is called the ''[[Fourier_series#Fourier_series_on_a_general_interval_.5Ba.2Cb.5D |Fourier series]] coefficients'' for <math>s(t)\,</math>. The inverse transform, which reconstructs <math>s(t)\,</math> from the coefficients, is called a ''Fourier series expansion'' or just ''Fourier series''. It is a simplification/special-case of the more general inverse Fourier transform of <math>S(f).\,</math>
When <math>s(t)\,</math> is not periodic, but its non-zero portion has '''finite''' duration, <math>S(f)\,</math> is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of <math>s(t)\,</math> that was analyzed. The same discrete set is obtained by treating the duration of the segment as if it is the period, <math>\tau\,</math>, of a periodic function and computing the Fourier series coefficients, as above. The Fourier series expansion is always a periodic function, not the finite-duration function; but one period of the expansion can match <math>s(t).\,</math>
See [[Fourier series]] for more information, including''':'''
* the inverse transform (''Fourier series expansion'')
* transform properties
* historical development
* special case of real-valued ''s''(''t'')
==== Discrete Fourier transform (DFT) ====
Since the DTFT is also a continuous Fourier transform (of a comb function), the Fourier series also applies to it. Thus, when <math>s[n]\,</math> is periodic, with period '''N''', <math>S_T(f)\,</math> is another [[Dirac comb]] function, modulated by the coefficients of a '''Fourier series'''. And the integral formula for the coefficients simplifies to''':'''
:<math>S[k] = \sum_{n=0}^{N-1} s[n] \cdot e^{-i 2 \pi \frac{k}{N} n}</math> for all integer values of '''k'''.
Since the DTFT is periodic, so is <math>S[k]\,</math>. And it has the same period ('''N''') as the input function. This transform is also called '''DFT''', particularly when only one period of the output sequence is computed from one period of the input sequence.
When <math>s[n]\,</math> is not periodic, but its non-zero portion has finite duration ('''N'''), <math>S_T(f)\,</math> is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of <math>s[n]\,</math> that was analyzed. The same discrete set is obtained by treating '''N''' as if it is the period of a periodic function and computing the '''Fourier series coefficients / DFT'''.
* The inverse transform of <math>S[k]\,</math> does not produce the finite-length sequence, <math>s[n],\,</math> when evaluated for all values of '''n'''. (It takes the inverse of <math>S_T(f)\,</math> to do that.) The inverse DFT can only reproduce the entire time-domain if the input happens to be periodic (forever). Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain, discrete-time functions. An alternative viewpoint is that the periodicity is the time-domain consequence of approximating the continuous-domain function, <math>S_T(f)\,</math>, with the discrete subset, <math>S[k]\,</math>. '''N''' can be larger than the actual non-zero portion of <math>s[n]\,</math>. The larger it is, the better the approximation (also known as '''zero-padding''').
The DFT can be computed using a [[fast Fourier transform]] (FFT) algorithm, which makes it a practical and important transformation on computers.
See [[Discrete Fourier transform]] for much more information, including''':'''
* the inverse transform
* transform properties
* applications
* tabulated transforms of specific functions
The following table recaps the four basic forms discussed above, highlighting the '''duality''' of the properties of ''discreteness'' and ''periodicity''. I.e., if the signal representation in one domain has either (or both) of those properties, then its transform representation to the other domain has the other property (or both).
{| class="wikitable"
|-
! rowspan="2" | Name
! colspan="2" | Time domain
! colspan="2" | Frequency domain
|-
| align="center" | Domain property
| align="center" | Function property
| align="center" | Domain property
| align="center" | Function property
|-
|(Continuous) [[Fourier transform]]
| align="center" | Continuous
| align="center" | Aperiodic
| align="center" | Continuous
| align="center" | Aperiodic
|-
|[[Discrete-time Fourier transform]]
| align="center" | Discrete
| align="center" | Aperiodic
| align="center" | Continuous
| align="center" | Periodic (<math>f_s</math>)
|-
|[[Fourier series]]
| align="center" | Continuous
| align="center" | Periodic (<math>\tau</math>)
| align="center" | Discrete
| align="center" | Aperiodic
|-
|[[Discrete Fourier transform]]
| align="center" | Discrete
| align="center" | Periodic (N)<ref>
Or N is simply the length of a finite sequence. In either case, the <u>inverse</u> DFT formula produces a periodic function, <math>s[n].\,</math>
</ref>
| align="center" | Discrete
| align="center" | Periodic (N)
|}
===Fourier transforms on arbitrary locally compact abelian topological groups===
The Fourier variants can also be generalized to Fourier transforms on arbitrary [[locally compact]] [[abelian]] [[topological group]]s, which are studied in [[harmonic analysis]]; there, the Fourier transform takes functions on a group to functions on the [[dual group]]. This treatment also allows a general formulation of the [[convolution theorem]], which relates Fourier transforms and [[convolution]]s. See also the [[Pontryagin duality]] for the generalized underpinnings of the Fourier transform.
===Time-frequency transforms===
[[Time-frequency transform]]s such as the [[short-time Fourier transform]], [[wavelet transform]]s, [[chirplet transform]]s, 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 the (mathematical) [[uncertainty principle]].
==Interpretation in terms of time and frequency==
In terms of [[Signal (information theory)|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 [[orthogonal system|decomposition]] of a function into [[harmonic]]s of different frequencies.
When the function ''f'' is a function of time and represents a physical [[Signal (information theory)|signal]], the transform has a standard interpretation as the frequency spectrum of the signal. The [[magnitude (mathematics)|magnitude]] of the resulting complex-valued function ''F'' at frequency ω represents the [[amplitude]] of a frequency component whose [[phase (waves)|initial phase]] is given by: ''arctan (imaginary part/real part)''.
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==
When processing signals, such as [[Sound|audio]], [[radio wave]]s, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection and/or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.
Some examples include:
* Telephone dialing; the [[touch tone|touch-tone]] signals for each telephone key, when pressed, are each a sum of two separate tones (frequencies). Fourier analysis can be used to separate (or ''analyze'') the telephone signal, to reveal the two component tones and therefore which button was pressed.
* 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 filter]]s;
* 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 [[precision (arithmetic)|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.
== How it works (a basic explanation) ==
To measure the amplitude and phase of a particular frequency component, the transform process multiplies the original function (the one being analyzed) by a sinusoid with the same frequency (called a ''basis function''). If the original function contains a ''component'' with the same shape (i.e. same frequency), its shape (but not its amplitude) is effectively squared.
*''Squaring'' implies that at every point on the product waveform, the contribution of the matching component to that product is a positive contribution, even though the component might be negative.
*''Squaring'' describes the case where the [[Phase (waves)|phases]] happen to match. What happens more generally is that a constant phase difference produces ''vectors'' at every point that are all aimed in the same direction, which is determined by the difference between the two phases. To make that happen actually requires two sinusoidal basis functions, cosine and sine, which are combined into a basis function that is complex-valued (see [[Complex exponential]]). The ''vector'' analogy refers to the [[Polar coordinates|polar]] coordinate representation.
The complex numbers produced by the product of the original function and the basis function are subsequently summed into a single result.
*Note that if the functions are continuous, rather than sets of discrete points, this step requires [[integral calculus]] or [[numerical integration]]. But the basic concept is just addition.
The contributions from the component that matches the basis function all have the same sign (or vector direction). The other components contribute values that alternate in sign (or vectors that rotate in direction) and tend to cancel out of the summation. The final value is therefore dominated by the component that matches the basis function. The stronger it is, the larger is the measurement. Repeating this measurement for all the basis functions produces the frequency-domain representation.
==See also==
* [[Fourier series]]
* [[Bispectrum]]
* [[Characteristic function (probability theory)]]
* [[Fractional Fourier transform]]
* [[Laplace transform]]
* [[Least-squares spectral analysis ]]
* [[Mellin transform]]
* [[Number-theoretic transform]]
* [[Orthogonal functions]]
* [[Pontryagin duality]]
* [[Schwartz space]]
* [[Spectral density]]
* [[Spectral density estimation]]
* [[Two-sided Laplace transform]]
* [[Wavelet]]
==Notes==
{{reflist}}
==References==
* Edward W. Kamen, Bonnie S. Heck, "Fundamentals of Signals and Systems Using the Web and Matlab", ISBN 0-13-017293-6
* E. M. Stein, G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton University Press, 1971. ISBN 0-691-08078-X
* A. D. Polyanin and A. V. Manzhirov, ''Handbook of Integral Equations'', CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
* {{cite book
| last = Smith | first = Steven W.
| url = http://www.dspguide.com/pdfbook.htm
| title = The Scientist and Engineer's Guide to Digital Signal Processing
| edition = Second Edition
| location = San Diego, Calif.
| publisher = California Technical Publishing
| year=1999
| id=ISBN 0-9660176-3-3
}}
==External links==
*[http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
*[http://cns-alumni.bu.edu/~slehar/fourier/fourier.html An Intuitive Explanation of Fourier Theory] by Steven Lehar.
*[http://www.archive.org/details/Lectures_on_Image_Processing Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 6 is on the 1- and 2-D Fourier Transform. Lectures 7-15 make use of it.], by Alan Peters
[[Category:Fourier analysis]]
[[Category:Integral transforms]]
[[Category:Digital signal processing]]
[[Category:Estimation theory]]
[[Category:Signal processing]]
[[Category:Transforms]]
[[Category:Computational science]]
[[Category:Mathematical analysis]]
[[Category:Applied mathematics]]
[[Category:Mathematics of computing]]
[[Category:Time series analysis]]
[[Category:Statistical theory]]
[[ar:تحويل فوريي]]
[[be:Пераўтварэнне Фур'е]]
[[cs:Fourierova transformace]]
[[de:Fourier-Transformation]]
[[es:Transformada de Fourier]]
[[eu:Fourierren transformaketa]]
[[fa:تبدیل فوریه]]
[[fr:Transformée de Fourier]]
[[gl:Transformada de Fourier]]
[[id:Transformasi Fourier]]
[[is:Fourier–vörpun]]
[[it:Analisi di Fourier]]
[[nl:Fouriertransformatie]]
[[ja:フーリエ変換]]
[[pl:Transformacja Fouriera]]
[[pt:Transformada de Fourier]]
[[ru:Преобразование Фурье]]
[[sr:Фуријеова трансформација]]
[[fi:Fourier'n muunnos]]
[[sv:Fourier-transform]]
[[th:การแปลงฟูริเยร์]]
[[vi:Biến đổi Fourier]]
[[tr:Fourier dönüşümü]]
[[zh:傅立叶分析]]