Fourier analysis

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Fourier transforms
Continuous Fourier transform
Fourier series
Discrete Fourier transform
Discrete-time Fourier transform
Related transforms

Fourier analysis, named after Joseph Fourier's introduction of the Fourier series, is the decomposition of a function in terms of sinusoidal^[1] functions (called basis functions) 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 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 or 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 "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 used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis.

1 Applications
2 Variants of Fourier analysis
3 Interpretation in terms of time and frequency
4 Applications in signal processing
5 How it works (a basic explanation)
6 See also
7 Notes
8 References
9 External links

[edit] Applications

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Fourier analysis has many scientific applications — in physics, number theory, combinatorics, signal processing, 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 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 exponential 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.

[edit] 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 $f\,$ , for instance, generally represents frequency in hertz (SI units), or a normalized frequency in cycles per sample. Also popular is the variable $\omega\,$ , 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.

[edit] (Continuous) Fourier transform

Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, such as time ( $t\,$ ). The amplitude and phase of a sinusoidal component of function s(t) depends on the component's frequency. In terms of ordinary frequency ( $f\,$ ), it is the complex number:

$S(f) = \int_{-\infty}^{\infty} s(t) \cdot e^{-i 2\pi f t} dt$

Evaluating this quantity for all values of $f\,$ produces the frequency-domain function.

Also see 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

[edit] Discrete-time Fourier transform (DTFT)

For use on computers, a useful "discrete-time" function can be obtained by 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 $s[n] \ \stackrel{\mathrm{def}}{=}\ T\cdot s(nT)$ to modulate a Dirac comb. In that case, the integral formula above simplifies to a summation:

$S_T(f) = \sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i 2\pi f n T} = \sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i 2\pi \frac{f}{f_s} n},$

which is a periodic function, with period $f_s = 1/T.\,$ An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded (or finite), with span $f_s.\,$

The DTFT can be applied to any discrete sequence. But in the particular case where s[n] are samples of s(t), $S_T(f)\,$ is closely related to $S(f).\,$ 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

[edit] Analysis of periodic functions or functions with limited duration

[edit] Fourier series

When $s(t)\,$ is periodic, with period $\tau\,$ , $S(f)\,$ 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:

$S[k] = \frac{1}{\tau} \int_{0}^{\tau} s(t) \cdot e^{-i 2\pi \frac{k}{\tau} t} dt,$ for all integer values of k.

This sequence is called the Fourier series coefficients for $s(t)\,$ . The inverse transform, which reconstructs $s(t)\,$ 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 $S(f).\,$

When $s(t)\,$ is not periodic, but its non-zero portion has finite duration, $S(f)\,$ is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of $s(t)\,$ that was analyzed. The same discrete set is obtained by treating the duration of the segment as if it is the period, $\tau\,$ , 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 $s(t).\,$

See Fourier series for more information, including:

the inverse transform (Fourier series expansion)
transform properties
historical development
special case of real-valued s(t)

[edit] 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 $s[n]\,$ is periodic, with period N, $S_T(f)\,$ is another Dirac comb function, modulated by the coefficients of a Fourier series. And the integral formula for the coefficients simplifies to:

$S[k] = \sum_{n=0}^{N-1} s[n] \cdot e^{-i 2 \pi \frac{k}{N} n}$ for all integer values of k.

Since the DTFT is periodic, so is $S[k]\,$ . 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 $s[n]\,$ is not periodic, but its non-zero portion has finite duration (N), $S_T(f)\,$ is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of $s[n]\,$ 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 $S[k]\,$ does not produce the finite-length sequence, $s[n],\,$ when evaluated for all values of n. (It takes the inverse of $S_T(f)\,$ 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, $S_T(f)\,$ , with the discrete subset, $S[k]\,$ . N can be larger than the actual non-zero portion of $s[n]\,$ . 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).

Name	Time domain		Frequency domain
Name	Domain property	Function property	Domain property	Function property
(Continuous) Fourier transform	Continuous	Aperiodic	Continuous	Aperiodic
Discrete-time Fourier transform	Discrete	Aperiodic	Continuous	Periodic ( $f s$ )
Fourier series	Continuous	Periodic ( $τ$ )	Discrete	Aperiodic
Discrete Fourier transform	Discrete	Periodic (N)^[2]	Discrete	Periodic (N)

[edit] 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 groups, 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 convolutions. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform.

[edit] Time-frequency transforms

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 an (mathematical) uncertainty principle.

[edit] 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 at frequency ω represents the amplitude of a frequency component whose 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.

[edit] Applications in signal processing

When processing signals, such as audio, radio waves, 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 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 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.

[edit] 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 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 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.

[edit] See also

[edit] Notes

^ An explanation: http://www.4p8.com/eric.brasseur/fouren.html
^ Or N is simply the length of a finite sequence. In either case, the inverse DFT formula produces a periodic function, $s[n].\,$

[edit] 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
Smith, Steven W. (1999). The Scientist and Engineer's Guide to Digital Signal Processing, Second Edition, San Diego, Calif.: California Technical Publishing. ISBN 0-9660176-3-3.

[edit] External links

Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
An Intuitive Explanation of Fourier Theory by Steven Lehar.
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

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