Movatterモバイル変換

[0]ホーム

Jump to content

Fourier analysis

Edit links

From Wikipedia, the free encyclopedia

Branch of mathematics

Bass guitar time signal of open string A note (55 Hz).

Fourier transform of bass guitar time signal of open string A note (55 Hz). Fourier analysis reveals the oscillatory components of signals andfunctions.

Fourier transforms
Fourier transform Fourier series Discrete-time Fourier transform Discrete Fourier transform Discrete Fourier transform over a ring Fourier transform on finite groups Fourier analysis Related transforms

Inmathematics,Fourier analysis (/ˈfʊrieɪ,-iər/)^[1] is the study of the way generalfunctions may be represented or approximated by sums of simplertrigonometric functions. Fourier analysis grew from the study ofFourier series, and is named afterJoseph Fourier, who showed that representing a function as asum of trigonometric functions greatly simplifies the study ofheat transfer.

The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function intooscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known asFourier synthesis. For example, determining what componentfrequencies 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 termFourier analysis often refers to the study of both operations.

The decomposition process itself is called aFourier transformation. Its output, theFourier transform, is often given a more specific name, which depends on thedomain 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 asharmonic analysis. Eachtransform used for analysis (seelist of Fourier-related transforms) has a correspondinginverse transform that can be used for synthesis.

To use Fourier analysis, data must be equally spaced. Different approaches have been developed for analyzing unequally spaced data, notably theleast-squares spectral analysis (LSSA) methods that use aleast squares fit ofsinusoids 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

[edit]

Fourier analysis has many scientific applications – inphysics,partial differential equations,number theory,combinatorics,signal processing,digital image processing,probability theory,statistics,forensics,option pricing,cryptography,numerical analysis,acoustics,oceanography,sonar,optics,diffraction,geometry,protein structure analysis, and other areas.

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

The transforms arelinear operators and, with proper normalization, areunitary as well (a property known asParseval's theorem or, more generally, as thePlancherel theorem, and most generally viaPontryagin duality).^[5]
The transforms are usually invertible.
Theexponential functions areeigenfunctions ofdifferentiation, which means that this representation transforms lineardifferential equations withconstant coefficients into ordinary algebraic ones.^[6] Therefore, the behavior of alinear time-invariant system can be analyzed at each frequency independently.
By theconvolution theorem, Fourier transforms turn the complicatedconvolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as signal filtering,polynomial multiplication, andmultiplying large numbers.^[7]
Thediscrete version of the Fourier transform (see below) can be evaluated quickly on computers usingfast 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,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 lowerarithmetic precision, and weak components are eliminated, 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.

Insignal processing, the Fourier transform often takes atime series or a function ofcontinuous time, and maps it into afrequency spectrum. That is, it takes a function from the time domain into thefrequency domain; it is adecomposition of a function intosinusoids of different frequencies; in the case of aFourier series ordiscrete Fourier transform, the sinusoids areharmonics of the fundamental frequency of the function being analyzed.

When a function $s(t)$ is a function of time and represents a physicalsignal, the transform has a standard interpretation as the frequency spectrum of the signal. Themagnitude of the resulting complex-valued function $S(f)$ at frequency $f {\displaystyle f}$ represents theamplitude of a frequency component whoseinitial phase is given by the angle of $S(f)$ (polar coordinates).

Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyzespatial frequencies, and indeed for nearly any function domain. This justifies their use in such diverse branches asimage processing,heat conduction, andautomatic control.

When processing signals, such asaudio,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:

Equalization of audio recordings with a series ofbandpass filters;
Digital radio reception without asuperheterodyne circuit, as in a modern cell phone orradio scanner;
Image processing to remove periodic oranisotropic artifacts such asjaggies frominterlaced video, strip artifacts fromstrip aerial photography, or wave patterns fromradio 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, includinginfrared andnuclear magnetic resonance spectroscopies;
Generation of soundspectrograms used to analyze sounds;
Passivesonar used to classify targets based on machinery noise.

Variants of Fourier analysis

[edit]

A Fourier transform and 3 variations caused by periodic sampling (at interval $T {\displaystyle T}$ ) and/or periodic summation (at interval $P {\displaystyle P}$ ) of the underlying time-domain function. The relative computational ease of the DFT sequence and the insight it gives into $S(f)$ make it a popular analysis tool.

A Fourier transform and 3 variations caused by periodic sampling (at interval $T {\displaystyle T}$ ) and/or periodic summation (at interval $P {\displaystyle P}$ ) of the underlying time-domain function. The relative computational ease of the DFT sequence and the insight it gives into $S(f)$ make it a popular analysis tool.

(Continuous) Fourier transform

[edit]

Main article:Fourier transform

Most often, the unqualified termFourier transform refers to the transform of functions of a continuousreal argument, and it produces a continuous function of frequency, known as afrequency distribution. One function is transformed into another, and the operation is reversible. When the domain of the input (initial) function is time ( $t {\displaystyle t}$ ), and the domain of the output (final) function isordinary frequency, the transform of function $s(t)$ at frequency $f {\displaystyle f}$ is given by thecomplex number:

S(f)=\int _{-\infty }^{\infty }s(t)\cdot e^{-i2\pi ft}\,dt.

Evaluating this quantity for all values of $f {\displaystyle f}$ produces thefrequency-domain function. Then $s(t)$ can be represented as a recombination ofcomplex exponentials of all possible frequencies:

s(t)=\int _{-\infty }^{\infty }S(f)\cdot e^{i2\pi ft}\,df,

which is the inverse transform formula. The complex number, $S(f),$ conveys both amplitude and phase of frequency $f . {\displaystyle f.}$

SeeFourier 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

[edit]

Main article:Fourier series

The Fourier transform of a periodic function, $s_{_{P}}(t),$ with period $P, {\displaystyle P,}$ becomes aDirac comb function, modulated by a sequence of complexcoefficients:

S[k]={\frac {1}{P}}\int _{P}s_{_{P}}(t)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt,\quad k\in \mathbb {Z} ,

(where

\int _{P}

is the integral over any interval of length

P {\displaystyle P}

The inverse transform, known asFourier series, is a representation of $s_{_{P}}(t)$ in terms of a summation of a potentially infinite number of harmonically related sinusoids orcomplex exponential functions, each with an amplitude and phase specified by one of the coefficients:

s_{_{P}}(t)\ \ =\ \ {\mathcal {F}}^{-1}\left\{\sum _{k=-\infty }^{+\infty }S[k]\,\delta \left(f-{\frac {k}{P}}\right)\right\}\ \ =\ \ \sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{P}}t}.

Any $s_{_{P}}(t)$ can be expressed as aperiodic summation of another function, $s(t)$ :

s_{_{P}}(t)\,\triangleq \,\sum _{m=-\infty }^{\infty }s(t-mP),

and the coefficients are proportional to samples of $S(f)$ at discrete intervals of ${\frac {1}{P}}$ :

S[k]={\frac {1}{P}}\cdot S\left({\frac {k}{P}}\right).

^[A]

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 $P, {\displaystyle P,}$ which is the frequency domain dual of theNyquist–Shannon sampling theorem.

SeeFourier series for more information, including the historical development.

Discrete-time Fourier transform (DTFT)

[edit]

Main article:Discrete-time Fourier transform

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

S_{\tfrac {1}{T}}(f)\ \triangleq \ \underbrace {\sum _{k=-\infty }^{\infty }S\left(f-{\frac {k}{T}}\right)\equiv \overbrace {\sum _{n=-\infty }^{\infty }s[n]\cdot e^{-i2\pi fnT}} ^{\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 theDTFT of the $s[n]$ sequence is also theFourier transform of the modulatedDirac comb function.^[B]

The Fourier series coefficients (and inverse transform), are defined by:

s[n]\ \triangleq \ T\int _{\frac {1}{T}}S_{\tfrac {1}{T}}(f)\cdot e^{i2\pi fnT}\,df=T\underbrace {\int _{-\infty }^{\infty }S(f)\cdot e^{i2\pi fnT}\,df} _{\triangleq \,s(nT)}.

Parameter $T {\displaystyle T}$ corresponds to the sampling interval, and this Fourier series can now be recognized as a form of thePoisson 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 ${\tfrac {1}{T}}.$ When that interval is $\left[-{\tfrac {1}{2T}},{\tfrac {1}{2T}}\right],$ the applicable reconstruction formula is theWhittaker–Shannon interpolation formula. This is a cornerstone in the foundation ofdigital signal processing.

Another reason to be interested in $S_{\tfrac {1}{T}}(f)$ is that it often provides insight into the amount ofaliasing caused by the sampling process.

Applications of the DTFT are not limited to sampled functions. SeeDiscrete-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)

[edit]

Main article:Discrete Fourier transform

Similar to a Fourier series, the DTFT of a periodic sequence, $s_{_{N}}[n],$ with period $N {\displaystyle N}$ , becomes a Dirac comb function, modulated by a sequence of complex coefficients (seeDTFT § Periodic data):

S[k]=\sum _{n}s_{_{N}}[n]\cdot e^{-i2\pi {\frac {k}{N}}n},\quad k\in \mathbb {Z} ,

(where

\sum _{n}

is the sum over any sequence of length

N . {\displaystyle N.}

)

The $S[k]$ sequence is customarily known as theDFT of one cycle of $s_{_{N}}.$ It is also $N {\displaystyle N}$ -periodic, so it is never necessary to compute more than $N {\displaystyle N}$ coefficients. The inverse transform, also known as adiscrete Fourier series, is given by:

s_{_{N}}[n]={\frac {1}{N}}\sum _{k}S[k]\cdot e^{i2\pi {\frac {n}{N}}k},

where

\sum _{k}

is the sum over any sequence of length

N . {\displaystyle N.}

When $s_{_{N}}[n]$ is expressed as aperiodic summation of another function:

s_{_{N}}[n]\,\triangleq \,\sum _{m=-\infty }^{\infty }s[n-mN],

and

s[n]\,\triangleq \,T\cdot s(nT),

the coefficients are samples of $S_{\tfrac {1}{T}}(f)$ at discrete intervals of ${\tfrac {1}{P}}={\tfrac {1}{NT}}$ :

S[k]=S_{\tfrac {1}{T}}\left({\frac {k}{P}}\right).

Conversely, when one wants to compute an arbitrary number $(N)$ of discrete samples of one cycle of a continuous DTFT, $S_{\tfrac {1}{T}}(f),$ it can be done by computing the relatively simple DFT of $s_{_{N}}[n],$ as defined above. In most cases, $N {\displaystyle N}$ is chosen equal to the length of the non-zero portion of $s[n].$ Increasing $N, {\displaystyle N,}$ known aszero-padding orinterpolation, results in more closely spaced samples of one cycle of $S_{\tfrac {1}{T}}(f).$ Decreasing $N, {\displaystyle N,}$ causes overlap (adding) in the time-domain (analogous toaliasing), which corresponds to decimation in the frequency domain. (seeDiscrete-time Fourier transform § L=N×I) In most cases of practical interest, the $s[n]$ sequence represents a longer sequence that was truncated by the application of a finite-lengthwindow function orFIR filter array.

The DFT can be computed using afast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers.

SeeDiscrete Fourier transform for much more information, including:

transform properties
applications
tabulated transforms of specific functions

Summary

[edit]

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 viaDirac delta andDirac 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.

It is common in practice for the duration ofs(•) to be limited to the period,P orN. But these formulas do not require that condition.

$s(t)$ transforms (continuous-time)
	Continuous frequency	Discrete frequencies
Transform	$S(f)\,\triangleq \,\int _{-\infty }^{\infty }s(t)\cdot e^{-i2\pi ft}\,dt$	$\underbrace {{\frac {1}{P}}\cdot S\left({\frac {k}{P}}\right)} _{S[k]}\,\triangleq \,{\frac {1}{P}}\int _{-\infty }^{\infty }s(t)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\equiv {\frac {1}{P}}\int _{P}s_{_{P}}(t)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt$
Inverse	$s(t)=\int _{-\infty }^{\infty }S(f)\cdot e^{i2\pi ft}\,df$	$\underbrace {s_{_{P}}(t)=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{P}}t}} _{\text{Poisson summation formula (Fourier series)}}\,$

$s(nT)$ transforms (discrete-time)
	Continuous frequency	Discrete frequencies
Transform	$\underbrace {S_{\tfrac {1}{T}}(f)\,\triangleq \,\sum _{n=-\infty }^{\infty }s[n]\cdot e^{-i2\pi fnT}} _{\text{Poisson summation formula (DTFT)}}$	${\begin{aligned}\underbrace {S_{\tfrac {1}{T}}\left({\frac {k}{NT}}\right)} _{S[k]}\,&\triangleq \,\sum _{n=-\infty }^{\infty }s[n]\cdot e^{-i2\pi {\frac {kn}{N}}}\\&\equiv \underbrace {\sum _{N}s_{_{N}}[n]\cdot e^{-i2\pi {\frac {kn}{N}}}} _{\text{DFT}}\,\end{aligned}}$
Inverse	$s[n]=\underbrace {T\int _{\frac {1}{T}}S_{\tfrac {1}{T}}(f)\cdot e^{i2\pi fnT}\,df} _{\text{Fourier series coefficient}}$ $\sum _{n=-\infty }^{\infty }s[n]\cdot \delta (t-nT)=\underbrace {\int _{-\infty }^{\infty }S_{\tfrac {1}{T}}(f)\cdot e^{i2\pi ft}\,df} _{\text{inverse Fourier transform}}\,$	$s_{_{N}}[n]=\underbrace {{\frac {1}{N}}\sum _{N}S[k]\cdot e^{i2\pi {\frac {kn}{N}}}} _{\text{inverse DFT}}$

Symmetry properties

[edit]

When the real and imaginary parts of a complex function are decomposed into theireven 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]

{\begin{array}{rccccccccc}{\text{Time domain}}&s&=&s_{_{\text{RE}}}&+&s_{_{\text{RO}}}&+&is_{_{\text{IE}}}&+&\underbrace {i\ s_{_{\text{IO}}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\text{Frequency domain}}&S&=&S_{\text{RE}}&+&\overbrace {\,i\ S_{\text{IO}}\,} &+&iS_{\text{IE}}&+&S_{\text{RO}}\end{array}}

From this, various relationships are apparent, for example:

The transform of a real-valued function $(s_{_{RE}}+s_{_{RO}})$ is theconjugate symmetric function $S_{RE}+i\ S_{IO}.$ Conversely, aconjugate symmetric transform implies a real-valued time-domain.
The transform of an imaginary-valued function $(i\ s_{_{IE}}+i\ s_{_{IO}})$ is theconjugate antisymmetric function $S_{RO}+i\ S_{IE},$ and the converse is true.
The transform of aconjugate symmetric function $(s_{_{RE}}+i\ s_{_{IO}})$ is the real-valued function $S_{RE}+S_{RO},$ and the converse is true.
The transform of aconjugate antisymmetric function $(s_{_{RO}}+i\ s_{_{IE}})$ is the imaginary-valued function $i\ S_{IE}+i\ S_{IO},$ and the converse is true.

History

[edit]

An early form of harmonic series dates back to ancientBabylonian mathematics, where they were used to computeephemerides (tables of astronomical positions).^[12]^[13]^[14]^[15]

The Classical Greek concepts ofdeferent and epicycle in thePtolemaic system of astronomy were related to Fourier series (seeDeferent and epicycle § Mathematical formalism).

In modern times, variants of the discrete Fourier transform were used byAlexis Clairaut in 1754 to compute an orbit,^[16]which has been described as the first formula for the DFT,^[17]and in 1759 byJoseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string.^[17] Technically, Clairaut's work was a cosine-only series (a form ofdiscrete cosine transform), while Lagrange's work was a sine-only series (a form ofdiscrete sine transform); a true cosine+sine DFT was used byGauss in 1805 fortrigonometric interpolation ofasteroid orbits.^[18]Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.^[17]

An early modern development toward Fourier analysis was the 1770 paperRéflexions sur la résolution algébrique des équations by Lagrange, which in the method ofLagrange resolvents used a complex Fourier decomposition to study the solution of a cubic:^[19]Lagrange transformed the roots $x_{1},$ $x_{2},$ $x_{3}$ into the resolvents:

{\begin{aligned}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{aligned}}

whereζ is a cubicroot of unity, which is the DFT of order 3.

A number of authors, notablyJean le Rond d'Alembert, andCarl Friedrich Gauss usedtrigonometric series to study theheat equation,^[20] but the breakthrough development was the 1807 paperMémoire sur la propagation de la chaleur dans les corps solides byJoseph Fourier, whose crucial insight was to modelall functions by trigonometric series, introducing the Fourier series. Independently of Fourier, astronomerFriedrich Wilhelm Bessel also introduced Fourier series to solveKepler's equation. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822.^[21]

Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory:Daniel Bernoulli andLeonhard 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 asharmonic analysis, and is also an early instance ofrepresentation theory.

The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 byCarl Friedrich Gauss when interpolating measurements of the orbit of the asteroidsJuno andPallas, although that particular FFT algorithm is more often attributed to its modern rediscoverersCooley and Tukey.^[18]^[16]

Time–frequency transforms

[edit]

Further information:Time–frequency analysis

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

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

Fourier transforms on arbitrary locally compact abelian topological groups

[edit]

The Fourier variants can also be generalized to Fourier transforms on arbitrarylocally compact Abelian topological groups, which are studied inharmonic analysis; there, the Fourier transform takes functions on a group to functions on the dual group. This treatment also allows a general formulation of theconvolution theorem, which relates Fourier transforms andconvolutions. See also thePontryagin duality for the generalized underpinnings of the Fourier transform.

More specific, Fourier analysis can be done on cosets,^[22] even discrete cosets.

Notes

[edit]

^ $\int _{P}\left(\sum _{m=-\infty }^{\infty }s(t-mP)\right)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt=\underbrace {\int _{-\infty }^{\infty }s(t)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt} _{\triangleq \,S\left({\frac {k}{P}}\right)}$
^We may also note that:
${\begin{aligned}\sum _{n=-\infty }^{+\infty }T\cdot s(nT)\delta (t-nT)&=\sum _{n=-\infty }^{+\infty }T\cdot s(t)\delta (t-nT)\\&=s(t)\cdot T\sum _{n=-\infty }^{+\infty }\delta (t-nT).\end{aligned}}$
Consequently, a common practice is to model "sampling" as a multiplication by theDirac comb function, which of course is only "possible" in a purely mathematical sense.

References

[edit]

^"Fourier".Dictionary.com Unabridged (Online). n.d.
^Cafer Ibanoglu (2000).Variable Stars As Essential Astrophysical Tools. Springer.ISBN 0-7923-6084-2.
^D. Scott Birney; David Oesper; Guillermo Gonzalez (2006).Observational Astronomy. Cambridge University Press.ISBN 0-521-85370-2.
^Press (2007).Numerical Recipes (3rd ed.). Cambridge University Press.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).The Art of Computer Programming Volume 2: Seminumerical Algorithms (3rd ed.). Addison-Wesley Professional. Section 4.3.3.C: Discrete Fourier transforms, pg.305.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. Prentice-Hall.ISBN 9780139141010.OCLC 602011570.
^Proakis, John G.; Manolakis, Dimitri G. (1996),Digital Signal Processing: Principles, Algorithms and Applications (3 ed.), New Jersey: Prentice-Hall International, p. 291,ISBN 978-0-13-394289-7, sAcfAQAAIAAJ
^Prestini, Elena (2004).The Evolution of Applied Harmonic Analysis: Models of the Real World. Birkhäuser. p. 62.ISBN 978-0-8176-4125-2.
^Rota, Gian-Carlo; Palombi, Fabrizio (1997).Indiscrete Thoughts. Birkhäuser. p. 11.ISBN 978-0-8176-3866-5.
^Neugebauer, Otto (1969) [1957].The Exact Sciences in Antiquity. Acta Historica Scientiarum Naturalium et Medicinalium. Vol. 9 (2nd ed.).Dover Publications. pp. 1–191.ISBN 978-0-486-22332-2.PMID 14884919.
^Brack-Bernsen, Lis; Brack, Matthias (2004). "Analyzing shell structure from Babylonian and modern times".International Journal of Modern Physics E.13 (1): 247.arXiv:physics/0310126.Bibcode:2004IJMPE..13..247B.doi:10.1142/S0218301304002028.S2CID 15704235.
^^a ^bTerras, Audrey (1999).Fourier Analysis on Finite Groups and Applications.Cambridge University Press. pp. 30-32.ISBN 978-0-521-45718-7.
^^a ^b ^c ^dBriggs, 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 ^bHeideman, 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.doi:10.1109/MASSP.1984.1162257.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.Bibcode:1999RvGeo..37..151N.CiteSeerX 10.1.1.455.4798.doi:10.1029/1998RG900006.ISSN 1944-9208.OCLC 5156426043.S2CID 38786145.
^Dutka, Jacques (1995). "On the early history of Bessel functions".Archive for History of Exact Sciences.49 (2):105–134.doi:10.1007/BF00376544.
^Forrest, Brian (1998). "Fourier Analysis on Coset Spaces".Rocky Mountain Journal of Mathematics.28 (1):170–190.doi:10.1216/rmjm/1181071828.JSTOR 44238164.

External links

[edit]

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 theUniversity of Nottingham.
Introduction to Fourier analysis of time series at Medium

v t e Major topics inmathematical analysis
Calculus:Integration Differentiation Differential equations ordinary partial stochastic Fundamental theorem of calculus Calculus of variations Vector calculus Tensor calculus Matrix calculus Lists of integrals Table of derivatives
Real analysis Complex analysis Hypercomplex analysis (quaternionic analysis) Functional analysis Fourier analysis Least-squares spectral analysis Harmonic analysis P-adic analysis (P-adic numbers) Measure theory Representation theory Functions Continuous function Special functions Limit Series Infinity
Mathematics portal

Lp spaces

Basic concepts

L¹ spaces

L² spaces

L^{\infty }

spaces

Maps

Inequalities

Results

ForLebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related