Fourier analysis

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

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions. Fourier analysis is named after Joseph Fourier, who showed that representing a function by a trigonometric series greatly simplifies the study of heat propagation.

Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into simpler pieces is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. In mathematics, the term Fourier analysis often refers to the study of both operations.

The decomposition process itself is called a Fourier transform. The transform is often given a more specific name which depends upon the domain and other properties of the function being transformed. 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:

• 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)(Rudin 1990).
• The transforms are usually invertible.
• The exponential functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones (Evans 1998). Therefore, the behavior of a linear time-invariant system can be analyzed at each frequency 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 (Knuth 1997).
• The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms. (Conte & de Boor 1980)

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 image square is reassembled from the preserved approximate Fourier-transformed components, which are 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:

• 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);
• 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 crystal 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.
• Many other forms of spectroscopy also rely upon Fourier Transforms to determine the three-dimensional structure and/or identity of the sample being analyzed, including Infrared and Nuclear Magnetic Resonance spectroscopies.
• Generation of sound spectrograms used to analyze sounds.

Variants of Fourier analysis

Illustration of using Dirac comb functions and the convolution theorem to model the effects of sampling and/or periodic summation. At lower left is a DTFT, the result of sampling s(t). The graphs on the right side depict the (finite) coefficients that modulate the infinite amplitudes of a comb function whose teeth are spaced at intervals of 1/P. The coefficients in the upper and lower figures are respectively computed by the Fourier series integral and the DFT summation. The relative computational ease of the DFT and its similarities to the upper-left figure make it a popular analysis tool.

(Continuous) Fourier transform

Main article: Fourier Transform

Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, and it produces a continuous function of frequency, known as a frequency distribution. One function is transformed into another, and the operation is reversible. When the domain of the input function is time (t), and the domain of the output function is ordinary frequency, the transform of function s(t) at frequency ƒ is given by 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 ƒ produces the frequency-domain function. Then s(t) can be represented as a recombination of complex exponentials of all possible frequencies:

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

which is the inverse transform formula. The complex number, S(ƒ), conveys both amplitude and phase of frequency ƒ.

See Fourier transform for much more information, including:

• 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

Main article: Fourier series

The Fourier transform of a periodic function, s_P(t), with period P, becomes a Dirac comb function, modulated by a sequence of complex coefficients:

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

and where $\scriptstyle \int_P$  is the integral over any interval of length P.

The inverse transform, known as Fourier series, is a representation of s_P(t) in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients:

$s_P(t)=\sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{P} t} \quad\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \sum_{k=-\infty}^{+\infty} S[k]\ \delta \left(f-\frac{k}{P}\right).$

When s_P(t), is expressed as a periodic summation of another function, s(t):   $s_P(t)\ \stackrel{\text{def}}{=}\ \sum_{k=-\infty}^{\infty} s(t-kP),$

the coefficients are proportional to samples of S(ƒ) at discrete intervals of 1/P:   $S[k] =\frac{1}{P}\cdot S\left(\frac{k}{P}\right).\,$

A sufficient condition for recovering s(t) (and therefore S(ƒ)) from just these samples is that the non-zero portion of s(t) be confined to a known interval of duration P, which is the frequency domain dual of the Nyquist–Shannon sampling theorem.

See Fourier series for more information, including the historical development.

Discrete-time Fourier transform (DTFT)

Main article: Discrete-time Fourier transform

The DTFT is the mathematical dual of the time-domain Fourier series. Thus, any periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:

$S_{1/T}(f)\ \stackrel{\text{def}}{=}\ \underbrace{\sum_{k=-\infty}^{\infty} S\left(f - \frac{k}{T}\right) \equiv \overbrace{\sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i 2\pi f n T}}^{\text{Fourier series (DTFT)}}}_{\text{Poisson summation formula}} = \mathcal{F} \left \{ \sum_{n=-\infty}^{+\infty} s[n]\ \delta(t-nT)\right \},\,$

which is known as the DTFT. Thus the DTFT of the s[n] sequence is also the Fourier transform of the modulated Dirac comb function.

The Fourier series coefficients, defined by:

$s[n] = T \int_{1/T} S_{1/T}(f)\cdot e^{i 2\pi f nT} df,\,$

is the inverse transform. And it can be readily shown that the coefficients are just samples of s(t) at discrete intervals of T:  s[n] = T·s(nT).

Thus we have the important result that when a discrete data sequence, s[n], represents samples of an underlying continuous function, s(t), one can deduce something about its Fourier transform, S(ƒ). That is a cornerstone in the foundation of digital signal processing. Furthermore, under certain idealized conditions one can theoretically recover S(ƒ) and s(t) exactly. A sufficient condition for perfect recovery is that the non-zero portion of S(ƒ) be confined to a known frequency interval of width 1/T. When that interval is [-0.5/T, 0.5/T], the applicable reconstruction formula is the Whittaker–Shannon interpolation formula.

Another reason to be interested in S_1/T(ƒ) is that it often provides insight into the amount of aliasing caused by the sampling process.

Applications of the DTFT are not limited to sampled functions. See Discrete-time Fourier transform for more information on this and other topics, including:

• normalized frequency units
• windowing (finite-length sequences)
• transform properties
• tabulated transforms of specific functions

Discrete Fourier transform (DFT)

Main article: Discrete Fourier transform

The DTFT of a periodic sequence, s_N[n], with period N, becomes another Dirac comb function, modulated by the coefficients of a Fourier series.  And the integral formula for the coefficients simplifies to a summation:

$S_N[k] =\frac{1}{NT} \underbrace{\sum_N s_N[n]\cdot e^{-i 2\pi \frac{k}{N} n}}_{S_k},\,$     where $\scriptstyle \sum_N$  is the sum over any n-sequence of length N.

The S_k sequence is what's customarily known as the DFT of s_N. It is also N-periodic, so it is never necessary to compute more than N coefficients. In terms of S_k, the inverse transform is given by:

$s_N[n] = \frac{1}{N} \sum_{N} S_k\cdot e^{i 2\pi \frac{n}{N}k},\,$     where $\scriptstyle \sum_N$  is the sum over any k-sequence of length N.

When s_N[n] is expressed as a periodic summation of another function, s[n] = T·s(nT):   $s_N[n]\ \stackrel{\text{def}}{=}\ \sum_{k=-\infty}^{\infty} s[n-kN],\,$

the coefficients are equivalent to samples of S_1/T(ƒ) at discrete intervals of 1/P = 1/NT:   $S_k = S_{1/T}\left(\frac{k}{P}\right).\,$

In most cases, N is chosen equal to the length of non-zero portion of s[n]. Increasing N, known as zero-padding or interpolation, results in more closely spaced samples of one cycle of  S_1/T(ƒ). Decreasing N, causes overlap (adding) in the time-domain (analogous to aliasing), which corresponds to decimation in the frequency domain. (see Sampling the DTFT) In most cases of practical interest, the s[n] sequence represents a longer sequence that was truncated by the application of a finite-length window function or FIR filter array.

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:

• transform properties
• applications
• tabulated transforms of specific functions

Summary

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components (Fourier series), and the transforms diverge at those frequencies. One common practice is to handle that divergence via Dirac delta and Dirac comb functions. But the same spectral information can be discerned from just one cycle of the periodic function, since all the other cycles are identical. Similarly, finite-duration functions can be represented as a Fourier series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact. The formulas in the right hand columns below apply to both cases, where in one case  $s\,$ is the finite duration function to be analyzed, and in the other case its periodic summation,  $s_P,\,$  is the function under analysis. We note in passing that none of the formulas actually require the duration of $s\,$ to be limited to the period, P or N. But that is the most common situation.

$s(t)\,$  transforms (continuous-time)

	Continuous frequency	Discrete frequencies
Transform	$S(f)\ \stackrel{\text{def}}{=}\ \int_{-\infty}^{\infty} s(t)\ e^{-i 2\pi f t}\,dt$	$\underbrace{\frac{1}{P}\cdot S\left(\frac{k}{P}\right)}_{S[k]}\ \stackrel{\text{def}}{=}\ \frac{1}{P} \int_{-\infty}^{\infty} s(t)\ e^{-i 2\pi \frac{k}{P} t}\,dt \equiv \frac{1}{P} \int_P s_P(t)\ e^{-i 2\pi \frac{k}{P} t}\,dt$
Inverse	$s(t) = \int_{-\infty}^{\infty} S(f)\ e^{ i 2 \pi f t}\,df$	$\underbrace{s_P(t) = \sum_{k=-\infty}^{\infty} S[k] \cdot e^{i 2\pi \frac{k}{P} t}}_{\text{Poisson summation formula (Fourier series)}}$

$s[n]\,$  transforms (discrete-time)

	Continuous frequency	Discrete frequencies
Transform	$\underbrace{S_{1/T}(f) = \sum_{n=-\infty}^{\infty} \overbrace{T\cdot s(nT)}^{s[n]}\cdot e^{-i 2\pi f nT}}_{\text{Poisson summation formula (DTFT)}}$	$\underbrace{\overbrace{S_{1/T}\left(\frac{k}{NT}\right)}^{S_k} = \sum_{n=-\infty}^{\infty} s[n]\cdot e^{-i 2\pi \frac{kn}{N}}}_{\text{Poisson summation formula}} \equiv \underbrace{\sum_{N} s_N[n]\cdot e^{-i 2\pi \frac{kn}{N}}}_{\text{DFT}}$
Inverse	$s[n] = T \int_{1/T} S_{1/T}(f)\cdot e^{i 2\pi f nT} \,df$ $\sum_{n=-\infty}^{\infty} s[n]\cdot \delta(t-nT) = \underbrace{\int_{-\infty}^{\infty} S_{1/T}(f)\cdot e^{i 2\pi f t}\,df}_{\text{inverse Fourier transform}}$	$s_N[n] = \underbrace{\frac{1}{N} \sum_{N} S_k\cdot e^{i 2\pi \frac{kn}{N}}}_{\text{inverse DFT}}$ $s_P(nT) = \frac{1}{T}\cdot s_N[n] = \sum_{N} \underbrace{\frac{1}{P}\cdot S_{1/T}\left(\frac{k}{P}\right)}_{S_{N}[k]} \cdot e^{i 2\pi \frac{kn}{N}}$

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 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

For more details on this topic, see Time–frequency analysis.

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, the Gabor 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

See also: Fourier series#Historical development

A primitive form of harmonic series dates back to ancient Babylonian mathematics, where they were used to compute ephemerides (tables of astronomical positions).

In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit, which has been described as the first formula for the DFT, and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string. 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. Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.

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: 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, and Lagrange had given the Fourier series solution to the wave equation, so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.

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.

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.

See also

• Fourier-related transforms
• Laplace transform (LT)
• Two-sided Laplace transform
• Mellin transform
• Fast Fourier transform (FFT)
• Non-uniform discrete Fourier transform (NDFT)
• Fractional Fourier transform (FRFT)
• Quantum Fourier transform (QFT)
• Number-theoretic transform
• Least-squares spectral analysis
• Basis vectors
• Bispectrum
• Characteristic function (probability theory)
• Orthogonal functions
• Pontryagin duality
• Schwartz space
• Spectral density
• Spectral density estimation
• Wavelet

References

• Conte, S. D.; de Boor, Carl (1980), Elementary Numerical Analysis (Third ed.), New York: McGraw Hill, Inc., ISBN 0070662282 
• Evans, L. (1998), Partial Differential Equations, American Mathematical Society, ISBN 3540761241 
• Howell, Kenneth B. (2001). Principles of Fourier Analysis, CRC Press. ISBN 9780849382758
• Kamen, E.W., and B.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 
• Polyanin, A.D., and A.V. Manzhirov (1998). Handbook of Integral Equations, CRC Press, Boca Raton. 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, http://www.dspguide.com/pdfbook.htm 
• Stein, E.M., and G. Weiss (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press. ISBN 0-691-08078-X

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