Fourier analysis
Fourier transforms |
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In mathematics, Fourier analysis (/ˈfʊrieɪ, -iər/)[1] is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations.
The decomposition process itself is called a
To use Fourier analysis, data must be equally spaced. Different approaches have been developed for analyzing unequally spaced data, notably the least-squares spectral analysis (LSSA) methods that use a least squares fit of sinusoids to data samples, similar to Fourier analysis.[2][3] Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in long gapped records; LSSA mitigates such problems.[4]
Applications
Fourier analysis has many scientific applications – in
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).[5]
- The transforms are usually invertible.
- The linear time-invariant systemcan 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 signal filtering, polynomial multiplication, and multiplying large numbers.[7]
- The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms.[8]
In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum. The FT method is used to decode the measured signals and record the wavelength data. And by using a computer, these Fourier calculations are rapidly carried out, so that in a matter of seconds, a computer-operated FT-IR instrument can produce an infrared absorption pattern comparable to that of a prism instrument.[9]
Fourier transformation is also useful as a compact representation of a signal. For example,
In
When a function is a function of time and represents a physical
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 such diverse branches as
When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate narrowband components of a compound waveform, concentrating them for easier detection 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.[10]
Some examples include:
- bandpass filters;
- Digital radio reception without a superheterodyne circuit, as in a modern cell phone or radio scanner;
- radio frequency interferencein a digital camera;
- Cross correlationof 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, including infrared and nuclear magnetic resonance spectroscopies;
- Generation of sound spectrograms used to analyze sounds;
- Passive sonar used to classify targets based on machinery noise.
Variants of Fourier analysis
(Continuous) 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 (initial) function is time (t), and the domain of the output (final) function is ordinary frequency, the transform of function s(t) at frequency f is given by the complex number:
Evaluating this quantity for all values of f produces the frequency-domain function. Then s(t) can be represented as a recombination of
which is the inverse transform formula. The complex number, S(f), conveys both amplitude and phase of frequency f.
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
The Fourier transform of a periodic function, sP(t), with period P, becomes a
- (where ∫P is the integral over any interval of length P).
The inverse transform, known as Fourier series, is a representation of sP(t) in terms of a summation of a potentially infinite number of harmonically related sinusoids or
Any sP(t) can be expressed as a periodic summation of another function, s(t):
and the coefficients are proportional to samples of S(f) at discrete intervals of 1/P:
Note that any s(t) whose transform has the same discrete sample values can be used in the periodic summation. A sufficient condition for recovering s(t) (and therefore S(f)) from just these samples (i.e. from the Fourier series) 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)
The DTFT is the mathematical dual of the time-domain Fourier series. Thus, a convergent periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:
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.[B]
The Fourier series coefficients (and inverse transform), are defined by:
Parameter T corresponds to the sampling interval, and this Fourier series can now be recognized as a form of the Poisson summation formula. Thus we have the important result that when a discrete data sequence, s[n], is proportional to samples of an underlying continuous function, s(t), one can observe a periodic summation of the continuous Fourier transform, S(f). Note that any s(t) with the same discrete sample values produces the same DTFT But under certain idealized conditions one can theoretically recover S(f) and s(t) exactly. A sufficient condition for perfect recovery is that the non-zero portion of S(f) be confined to a known frequency interval of width 1/T. When that interval is [−1/2T, 1/2T], the applicable reconstruction formula is the Whittaker–Shannon interpolation formula. This is a cornerstone in the foundation of digital signal processing.
Another reason to be interested in S1/T(f) 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)
Similar to a Fourier series, the DTFT of a periodic sequence, , with period , becomes a Dirac comb function, modulated by a sequence of complex coefficients (see
- (where Σn is the sum over any sequence of length N).
The S[k] sequence is what is customarily known as the DFT of one cycle of sN. It is also N-periodic, so it is never necessary to compute more than N coefficients. The inverse transform, also known as a discrete Fourier series, is given by:
- where Σk is the sum over any sequence of length N.
When sN[n] is expressed as a periodic summation of another function:
- and [C]
the coefficients are proportional to samples of S1/T(f) at disrete intervals of 1/P = 1/NT:
Conversely, when one wants to compute an arbitrary number (N) of discrete samples of one cycle of a continuous DTFT, S1/T(f), it can be done by computing the relatively simple DFT of sN[n], as defined above. 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 S1/T(f). Decreasing N, causes overlap (adding) in the time-domain (analogous to
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 (not discussed above) is to handle that divergence via
It is common in practice for the duration of s(•) to be limited to the period, P or N. But these formulas do not require that condition.
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Symmetry properties
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:[11]
From this, various relationships are apparent, for example:
- The transform of a real-valued function (sRE + sRO) is the even symmetric function SRE + i SIO. Conversely, an even-symmetric transform implies a real-valued time-domain.
- The transform of an imaginary-valued function (i sIE + i sIO) is the odd symmetric function SRO + i SIE, and the converse is true.
- The transform of an even-symmetric function (sRE + i sIO) is the real-valued function SRE + SRO, and the converse is true.
- The transform of an odd-symmetric function (sRO + i sIE) is the imaginary-valued function i SIE + i SIO, and the converse is true.
History
An early form of harmonic series dates back to ancient
The Classical Greek concepts of
In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit,[16] which has been described as the first formula for the DFT,[17] and in 1759 by
An early modern development toward Fourier analysis was the 1770 paper
where ζ is a cubic root of unity, which is the DFT of order 3.
A number of authors, notably
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.[17]
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.[18][16]
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
Fourier transforms on arbitrary locally compact abelian topological groups
The Fourier variants can also be generalized to Fourier transforms on arbitrary
More specific, Fourier analysis can be done on cosets,[21] even discrete cosets.
See also
- Conjugate Fourier series
- Generalized Fourier series
- Fourier–Bessel series
- Fourier-related transforms
- Laplace transform (LT)
- Two-sided Laplace transform
- Mellin transform
- Non-uniform discrete Fourier transform (NDFT)
- Quantum Fourier transform (QFT)
- Number-theoretic transform
- Basis vectors
- Bispectrum
- Characteristic function (probability theory)
- Orthogonal functions
- Schwartz space
- Spectral density
- Spectral density estimation
- Spectral music
- Walsh function
- Wavelet
Notes
- ^
- ^ We may also note that:
- ^ Note that this definition intentionally differs from the DTFT section by a factor of T. This facilitates the " transforms" table. Alternatively, can be defined as in which case
- ^
References
- ^ "Fourier". Dictionary.com Unabridged (Online). n.d.
- ISBN 0-7923-6084-2.
- ISBN 0-521-85370-2.
- ISBN 978-0-521-88068-8.
- ^
Rudin, Walter (1990). Fourier Analysis on Groups. Wiley-Interscience. ISBN 978-0-471-52364-2.
- ^
Evans, L. (1998). Partial Differential Equations. American Mathematical Society. ISBN 978-3-540-76124-2.
- ^
Knuth, Donald E. (1997). ISBN 978-0-201-89684-8.
- ^
Conte, S. D.; de Boor, Carl (1980). Elementary Numerical Analysis (Third ed.). New York: McGraw Hill, Inc. ISBN 978-0-07-066228-5.
- ^ Saferstein, Richard (2013). Criminalistics: An Introduction to Forensic Science.
- ^
Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ. ISBN 9780139141010.)
{{cite book}}
: CS1 maint: location missing publisher (link - ^
Proakis, John G.; Manolakis, Dimitri G. (1996), Digital Signal Processing: Principles, Algorithms and Applications (3 ed.), New Jersey: Prentice-Hall International, p. 291, ISBN 9780133942897, sAcfAQAAIAAJ
- ^
Prestini, Elena (2004). The Evolution of Applied Harmonic Analysis: Models of the Real World. Birkhäuser. p. 62. ISBN 978-0-8176-4125-2.
- ^
ISBN 978-0-8176-3866-5.
- ^
PMID 14884919.
- ^
S2CID 15704235.
- ^ a b
ISBN 978-0-521-45718-7.
- ^ a b c d
Briggs, William L.; Henson, Van Emden (1995). The DFT: An Owner's Manual for the Discrete Fourier Transform. SIAM. pp. 2–4. ISBN 978-0-89871-342-8.
- ^ a b
Heideman, M.T.; Johnson, D. H.; Burrus, C. S. (1984). "Gauss and the history of the fast Fourier transform". IEEE ASSP Magazine. 1 (4): 14–21. S2CID 10032502.
- ^
Knapp, Anthony W. (2006). Basic Algebra. Springer. p. 501. ISBN 978-0-8176-3248-9.
- ^
Narasimhan, T.N. (February 1999). "Fourier's heat conduction equation: History, influence, and connections". Reviews of Geophysics. 37 (1): 151–172. S2CID 38786145.
- ^ Forrest, Brian. (1998). Fourier Analysis on Coset Spaces. Rocky Mountain Journal of Mathematics. 28. 10.1216/rmjm/1181071828.
Further reading
- Howell, Kenneth B. (2001). Principles of Fourier Analysis. CRC Press. ISBN 978-0-8493-8275-8.
- Kamen, E.W.; Heck, B.S. (2 March 2000). Fundamentals of Signals and Systems Using the Web and Matlab (2 ed.). Prentiss-Hall. ISBN 978-0-13-017293-8.
- Müller, Meinard (2015). The Fourier Transform in a Nutshell (PDF). Springer. In Fundamentals of Music Processing, Section 2.1, pp. 40–56. (PDF) from the original on 8 April 2016.
- Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 978-0-8493-2876-3.
- Smith, Steven W. (1999). The Scientist and Engineer's Guide to Digital Signal Processing (Second ed.). San Diego: California Technical Publishing. ISBN 978-0-9660176-3-2.
- Stein, E. M.; Weiss, G. (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press. ISBN 978-0-691-08078-9.
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
- Moriarty, Philip; Bowley, Roger (2009). "Σ Summation (and Fourier Analysis)". Sixty Symbols. Brady Haran for the University of Nottingham.
- Introduction to Fourier analysis of time series at Medium