Fourier analysis

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

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions. The attempt to understand functions (or other objects) by breaking them into basic pieces that are easier to understand is one of the central themes in Fourier analysis. Fourier analysis is named after Joseph Fourier who showed that representing a function by a trigonometric series greatly simplified the study of heat propagation.

Today the subject of Fourier analysis encompasses a vast spectrum of mathematics with parts that, at first glance, may appear quite different. In the sciences and engineering the process of decomposing a function into simpler pieces is often called an analysis. The corresponding operation of rebuilding the function from these pieces is known as synthesis. In this context the term Fourier synthesis describes the act of rebuilding and the term Fourier analysis describes the process of breaking the function into a sum of simpler pieces. In mathematics, the term Fourier analysis often refers to the study of both operations.

In Fourier analysis, the term Fourier transform often refers to the process that decomposes a given function into the basic pieces. This process results in another function that describes how much of each basic piece are in the original function. However, the transform is often 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.

Applications

Fourier analysis has many scientific applications — in physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, optics, diffraction, geometry, and other areas.

This wide applicability stems from many useful properties of the transforms:

Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses a variant of the Fourier transformation (discrete cosine transform) 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.

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:

Variants of Fourier analysis

Fourier analysis has different forms, some of which have different names. Below are given several of the most common variants. 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 argument, such as time (t). In this case the Fourier transform describes a function ƒ(t) in terms of basic complex exponentials of various frequencies. In terms of ordinary frequency ν, the Fourier transform is given by the complex number:

$F(\nu) = \int_{-\infty}^{\infty} f(t) \cdot e^{- 2\pi \cdot i \cdot \nu \cdot t} dt.$

Evaluating this quantity for all values of ν produces the frequency-domain function.

• the inverse transform, F(ν) → ƒ(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

Fourier series

Fourier analysis for functions defined on a circle, or equivalently for periodic functions, mainly focuses on the study of Fourier series. Suppose that ƒ(x) is periodic function with period 2π, in this case one can attempt to decompose ƒ(x) as a sum of complex exponentials functions. The coefficients F(n) of the complex exponential in the sum are referred to as the Fourier coefficients for ƒ and are analogous to the "Fourier transform" of a function on the line (Katznelson 1976). The term Fourier series expansion or simply Fourier series refers to the infinite series that appears in the inverse transform. The Fourier coefficients of ƒ(x) are given by:

$F(n) = \frac{1}{2\pi} \int_{0}^{2\pi} f(x) e^{-i n x} \, dx,$

for all integers n. And the Fourier series of ƒ(x) is given by:

$f(x)=\sum_{n=-\infty}^\infty F(n) e^{i n x}.$

Equality may not always hold in the equation above and the study of the convergence of Fourier series is a central part of Fourier analysis of the circle.

Analysis of periodic functions or functions with limited duration

When ƒ(x) has finite duration (or compact support), a discrete subset of the values of its continuous Fourier transform is sufficient to reconstruct/represent the function ƒ(x) on its support. One such discrete set is obtained by treating the duration of the segment as if it is the period of a periodic function and computing the Fourier coefficients. Putting convergence issues aside, the Fourier series expansion will be a periodic function not the finite-duration function ƒ(x); but one period of the expansion will give the values of ƒ(x) on its support.

• Fourier series expansions for general periods,
• transform properties,
• historical development,
• special cases and generalizations.

Discrete-time Fourier transform (DTFT)

For functions of an integer index, the discrete-time Fourier transform (DTFT) provides a useful frequency-domain transform.

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 and some time-interval T. If information is lost, then only an approximation to the original transform, S(f), can be obtained by looking at one period of the periodic function:

$S_T(f) = \sum_{k=-\infty}^{\infty} S\left(f - \frac{k}{T}\right) \equiv \sum_{n=-\infty}^{\infty} \underbrace{T\cdot s(nT)}_{s[n]} \cdot e^{-i 2\pi f n T},$

which is the DTFT. The identity above is a result of the Poisson summation formula. The DTFT is also equivalent to the Fourier transform of a "continuous" function that is constructed by using the s[n] sequence to modulate a Dirac comb.

Applications of the DTFT are not limited to sampled functions. It can be applied to any discrete sequence. 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

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,  ST(ƒ) 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),  ST(ƒ) 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 ST(ƒ) 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, ST(ƒ), 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.

• the inverse transform
• transform properties
• applications
• tabulated transforms of specific functions

Fourier Transforms Summary

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 Function's
Domain property Function property Domain property Function property Energy Average Power
(Continuous) Fourier transform (FT) Continuous Aperiodic Continuous Aperiodic Finite Infinitesimal
Discrete-time Fourier transform (DTFT) Discrete Aperiodic Continuous Periodic (ƒs) Finite Infinitesimal
Fourier Series (FS) Continuous Periodic (τ) Discrete Aperiodic Infinite Finite
Discrete Fourier Series[1] (DFS) Discrete Periodic (N)[2] Discrete Periodic (N) Infinite Finite

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.

Time–frequency transforms

In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information.

As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform or fractional Fourier transform, or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform.

History

A primitive form of harmonic series dates back to ancient Babylonian mathematics, where they were used to compute ephemeris (tables of astronomical positions).[3][4][5][6]

In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit,[7] which has been described as the first formula for the DFT,[8] and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string.[9] Technically, Clairaut's work was a cosine-only series (a form of discrete cosine transform), while Lagrange's work was a sine-only series (a form of discrete sine transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits.[10] Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.[9]

An early modern development toward Fourier analysis was the 1770 paper Réflexions sur la résolution algébrique des équations by Lagrange, which in the method of Lagrange resolvents used a complex Fourier decomposition to study the solution of a cubic:[11] Lagrange transformed the roots x1,x2,x3 into the resolvents:

\begin{align} r_1 &= x_1 + x_2 + x_3\\ r_2 &= x_1 + \zeta x_2 + \zeta^2 x_3\\ r_3 &= x_1 + \zeta^2 x_2 + \zeta x_3 \end{align}

where ζ is a cubic root of unity, which is the DFT of order 3.

A number of authors, notably Jean le Rond d'Alembert, , and Carl Friedrich Gauss used trigonometric series to study the heat equation,[citation needed] but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier, whose crucial insight was to model all functions by trigonometric series, introducing the Fourier series.

Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: Daniel Bernoulli and Leonhard Euler had introduced trigonometric representations of functions,[8] and Lagrange had given the Fourier series solution to the wave equation,[8] so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.[8]

The subsequent development of the field is known as harmonic analysis, and is also an early instance of representation theory.

The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gauss when interpolating measurements of the orbit of the asteroids Juno and Pallas, although that particular FFT algorithm is more often attributed to its modern rediscoverers Cooley and Tukey.[10][12]

Interpretation in terms of time and frequency

In signal processing, the Fourier transform often takes a time series or a function of continuous time, and maps it into a frequency spectrum. That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.

When the function ƒ 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 the phase of F.

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. This justifies their use in branches such diverse as image processing, heat conduction and automatic control.

Notes

1. ^ The Discrete Fourier Series (DFS) are practically the same as the Discrete Fourier transform (DFT).
2. ^ Or N is simply the length of a finite sequence.  In either case, the inverse DFT formula produces a periodic function, s[n].
3. ^ Prestini, Elena (2004), The evolution of applied harmonic analysis: models of the real world, Birkhäuser, ISBN 978 0 81764125 2 , p. 62
4. ^ Rota, Gian-Carlo; Palombi, Fabrizio (1997), Indiscrete thoughts, Birkhäuser, ISBN 978 0 81763866 5 , p. 11
5. ^ Neugebauer, Otto (1969), The Exact Sciences in Antiquity, Courier Dover Publications,, ISBN 978 0 48622332 2
6. ^ Brack-Bernsen, Lis; Brack, Matthias, Analyzing shell structure from Babylonian and modern times
7. ^  , p. 30
8. ^ a b c d Briggs, William L.; Henson, Van Emden (1995), The DFT : an owner's manual for the discrete Fourier transform, SIAM, ISBN 978 0 89871342 8 , p. 4
9. ^ a b Briggs, William L.; Henson, Van Emden (1995), The DFT: an owner's manual for the discrete Fourier transform, SIAM, ISBN 978 0 89871342 8 , p. 2
10. ^ a b Heideman, M. T., D. H. Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier transform," IEEE ASSP Magazine, 1, (4), 14–21 (1984)
11. ^ Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978 0 81763248 9 , p. 501
12. ^  , p. 31

References

• Conte, S. D.; de Boor, Carl (1980), Elementary Numerical Analysis (Third ed.), New York: McGraw Hill, Inc., ISBN 0-07-012447-7
• Evans, Lawrence (1998), Partial Differential Equations, American Mathematical Society
• Edward W. Kamen, Bonnie S. Heck, "Fundamentals of Signals and Systems Using the Web and Matlab", ISBN 0-13-017293-6
• Knuth, Donald E. (1997), The Art of Computer Programming Volume 2: Seminumerical Algorithms (3rd ed.), Section 4.3.3.C: Discrete Fourier transforms, pg.305: Addison-Wesley Professional, ISBN 0201896842
• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
• Rudin, Walter (1990), Fourier Analysis on Groups, Wiley-Interscience, ISBN 047152364X
• Smith, Steven W. (1999), The Scientist and Engineer's Guide to Digital Signal Processing (Second ed.), San Diego, Calif.: California Technical Publishing, ISBN 0-9660176-3-3
• E. M. Stein, G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton University Press, 1971. ISBN 0-691-08078-X