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In mathematics, Fourier analysis
Fourier analysis
(English: /ˈfʊəriˌeɪ/)[1] is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis
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. Today, the subject of Fourier analysis
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
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
Fourier analysis
often refers to the study of both operations. The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on 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.

Contents

1 Applications

1.1 Applications in signal processing

2 Variants of Fourier analysis

2.1 (Continuous) Fourier transform 2.2 Fourier series 2.3 Discrete-time Fourier transform
Fourier transform
(DTFT) 2.4 Discrete Fourier transform
Fourier transform
(DFT) 2.5 Summary 2.6 Fourier transforms on arbitrary locally compact abelian topological groups 2.7 Time–frequency transforms

3 History 4 Interpretation in terms of time and frequency 5 See also 6 Notes 7 References 8 Further reading 9 External links

Applications[edit] Fourier analysis
Fourier analysis
has many scientific applications – in physics, 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 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
Fourier transform
(see below) can be evaluated quickly on computers using Fast Fourier Transform
Fast Fourier Transform
(FFT) algorithms. (Conte & de Boor 1980)

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.[2] Fourier transformation
Fourier transformation
is also useful as a compact representation of a signal. For example, JPEG
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[edit] When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis
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.[3] Some examples include:

Equalization of audio recordings with a series of bandpass filters; Digital radio reception without a 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, strip artifacts from strip aerial photography, or wave patterns from radio frequency interference in a digital camera; Cross correlation
Cross correlation
of similar images for co-alignment; X-ray crystallography
X-ray crystallography
to reconstruct a crystal structure from its diffraction pattern; Fourier transform
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[edit]

A Fourier transform
Fourier transform
and 3 variations caused by periodic sampling (at interval T) and/or periodic summation (at interval 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 term Fourier transform
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:

S ( f ) =

− ∞

s ( t ) ⋅

e

− 2 i π f t

d t .

displaystyle S(f)=int _ -infty ^ infty s(t)cdot e^ -2ipi ft ,dt.

Evaluating this quantity for all values of f produces the frequency-domain function. Then s(t) can be represented as a recombination of complex exponentials of all possible frequencies:

s ( t ) =

− ∞

S ( f ) ⋅

e

2 i π f t

d f ,

displaystyle s(t)=int _ -infty ^ infty S(f)cdot e^ 2ipi ft ,df,

which is the inverse transform formula. The complex number, S( f ), conveys both amplitude and phase of frequency f. See Fourier transform
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[edit] Main article: Fourier series The Fourier transform
Fourier transform
of a periodic function, sP(t), with period P, becomes a Dirac comb
Dirac comb
function, modulated by a sequence of complex coefficients:

S [ k ] =

1 P

P

s

P

( t ) ⋅

e

− 2 i π

k P

t

d t

displaystyle S[k]= frac 1 P int _ P s_ P (t)cdot e^ -2ipi frac k P t ,dt

for all integer values of k, and 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 complex exponential functions, each with an amplitude and phase specified by one of the coefficients:

s

P

( t ) =

k = − ∞

S [ k ] ⋅

e

2 i π

k P

t

F

k = − ∞

+ ∞

S [ k ]

δ

(

f −

k P

)

.

displaystyle s_ P (t)=sum _ k=-infty ^ infty S[k]cdot e^ 2ipi frac k P t quad stackrel displaystyle mathcal F Longleftrightarrow quad sum _ k=-infty ^ +infty S[k],delta left(f- frac k P right).

When sP(t), is expressed as a periodic summation of another function, s(t):

s

P

( t )

=

def

m = − ∞

s ( t − m P ) ,

displaystyle s_ P (t), stackrel text def = ,sum _ m=-infty ^ infty s(t-mP),

the coefficients are proportional to samples of S( f ) at discrete intervals of 1/P:

S [ k ] =

1 P

⋅ S

(

k P

)

.

displaystyle S[k]= frac 1 P cdot Sleft( frac k P right).

[note 1]

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
Fourier series
for more information, including the historical development. Discrete-time Fourier transform
Fourier transform
(DTFT)[edit] Main article: Discrete-time Fourier transform 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:

S

1 T

( f )  

=

def

 

k = − ∞

S

(

f −

k T

)

n = − ∞

s [ n ] ⋅

e

− 2 i π f n T

Fourier series
Fourier series
(DTFT)

Poisson summation formula

=

F

n = − ∞

s [ n ]   δ ( t − n T )

,

displaystyle S_ frac 1 T (f) stackrel text def = underbrace sum _ k=-infty ^ infty Sleft(f- frac k T right)equiv overbrace sum _ n=-infty ^ infty s[n]cdot e^ -2ipi fnT ^ text Fourier series
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
Fourier transform
of the modulated Dirac comb
Dirac comb
function.[note 2] The Fourier series
Fourier series
coefficients (and inverse transform), are defined by:

s [ n ]  

=

d e f

  T

1 T

S

1 T

( f ) ⋅

e

2 i π f n T

d f = T

− ∞

S ( f ) ⋅

e

2 i π f n T

d f

=

d e f

s ( n T )

.

displaystyle s[n] stackrel mathrm def = Tint _ frac 1 T S_ frac 1 T (f)cdot e^ 2ipi fnT ,df=Tunderbrace int _ -infty ^ infty S(f)cdot e^ 2ipi fnT ,df _ stackrel mathrm def = ,s(nT) .

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 ). That is a cornerstone in the foundation of digital signal processing. Furthermore, 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. 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
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
Fourier transform
(DFT)[edit] Main article: Discrete Fourier transform Similar to a Fourier series, the DTFT of a periodic sequence, sN[n], with period N, becomes a Dirac comb
Dirac comb
function, modulated by a sequence of complex coefficients (see DTFT/Periodic data):

S [ k ] =

n

s

N

[ n ] ⋅

e

− 2 i π

k N

n

,

displaystyle S[k]=sum _ n s_ N [n]cdot e^ -2ipi frac k N n ,

  where ∑n is the sum over any sequence of length N.

The S[k] sequence is what is customarily known as the DFT of sN.  It is also N-periodic, so it is never necessary to compute more than N coefficients. The inverse transform is given by:

s

N

[ n ] =

1 N

k

S [ k ] ⋅

e

2 i π

n N

k

,

displaystyle s_ N [n]= frac 1 N sum _ k S[k]cdot e^ 2ipi frac n N k ,

  where ∑k is the sum over any sequence of length N.

When sN[n] is expressed as a periodic summation of another function:

s

N

[ n ]

=

def

m = − ∞

s [ n − m N ] ,

displaystyle s_ N [n], stackrel text def = ,sum _ m=-infty ^ infty s[n-mN],

  and  

s [ n ]

=

def

s ( n T ) ,

displaystyle s[n], stackrel text def = ,s(nT),

[note 3]

the coefficients are proportional to samples of S1/T( f ) at discrete intervals of 1/P = 1/NT:

S [ k ] =

1 T

S

1 T

(

k P

)

.

displaystyle S[k]= frac 1 T cdot S_ frac 1 T left( frac k P right).

[note 4]

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 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
FIR filter
array. The DFT can be computed using a fast Fourier transform
Fourier transform
(FFT) algorithm, which makes it a practical and important transformation on computers. See Discrete Fourier transform
Fourier transform
for much more information, including:

transform properties applications tabulated transforms of specific functions

Summary[edit] For periodic functions, both the Fourier transform
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 Dirac delta
Dirac delta
and Dirac comb
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. We also note that 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.

s(t) transforms (continuous-time)

Continuous frequency Discrete frequencies

Transform

S ( f )

=

def

− ∞

s ( t ) ⋅

e

− 2 i π f t

d t

displaystyle S(f), stackrel text def = ,int _ -infty ^ infty s(t)cdot e^ -2ipi ft ,dt

1 P

⋅ S

(

k P

)

S [ k ]

=

def

1 P

− ∞

s ( t ) ⋅

e

− 2 i π

k P

t

d t ≡

1 P

P

s

P

( t ) ⋅

e

− 2 i π

k P

t

d t

displaystyle overbrace frac 1 P cdot Sleft( frac k P right) ^ S[k] , stackrel text def = , frac 1 P int _ -infty ^ infty s(t)cdot e^ -2ipi frac k P t ,dtequiv frac 1 P int _ P s_ P (t)cdot e^ -2ipi frac k P t ,dt

Inverse

s ( t ) =

− ∞

S ( f ) ⋅

e

2 i π f t

d f

displaystyle s(t)=int _ -infty ^ infty S(f)cdot e^ 2ipi ft ,df

s

P

( t ) =

k = − ∞

S [ k ] ⋅

e

2 i π

k P

t

Poisson summation formula (Fourier series)

displaystyle underbrace s_ P (t)=sum _ k=-infty ^ infty S[k]cdot e^ 2ipi frac k P t _ text Poisson summation formula (Fourier series) ,

s(nT) transforms (discrete-time)

Continuous frequency Discrete frequencies

Transform

1 T

S

1 T

( f )

=

def

n = − ∞

s ( n T ) ⋅

e

− 2 i π f n T

Poisson summation formula (DTFT)

displaystyle underbrace frac 1 T S_ frac 1 T (f), stackrel text def = ,sum _ n=-infty ^ infty s(nT)cdot e^ -2ipi fnT _ text Poisson summation formula (DTFT)

1 T

S

1 T

(

k

N T

)

S [ k ]

=

def

n = − ∞

s ( n T ) ⋅

e

− 2 i π

k n

N

n

s

P

( n T ) ⋅

e

− 2 i π

k n

N

DFT

displaystyle begin aligned overbrace frac 1 T S_ frac 1 T left( frac k NT right) ^ S[k] ,& stackrel text def = ,sum _ n=-infty ^ infty s(nT)cdot e^ -2ipi frac kn N \&equiv underbrace sum _ n s_ P (nT)cdot e^ -2ipi frac kn N _ text DFT ,end aligned

Inverse

s ( n T ) = T

1 T

1 T

S

1 T

( f ) ⋅

e

2 i π f n T

d f

displaystyle s(nT)=Tint _ frac 1 T frac 1 T S_ frac 1 T (f)cdot e^ 2ipi fnT ,df

n = − ∞

s ( n T ) ⋅ δ ( t − n T ) =

− ∞

1 T

 

S

1 T

( f ) ⋅

e

2 i π f t

d f

inverse Fourier transform

displaystyle sum _ n=-infty ^ infty s(nT)cdot delta (t-nT)=underbrace int _ -infty ^ infty frac 1 T S_ frac 1 T (f)cdot e^ 2ipi ft ,df _ text inverse Fourier transform
Fourier transform
,

s

P

( n T )

=

1 N

k

S [ k ] ⋅

e

2 i π

k n

N

inverse DFT

=

1 P

k

S

1 T

(

k P

)

e

2 i π

k n

N

displaystyle begin aligned s_ P (nT)&=overbrace frac 1 N sum _ k S[k]cdot e^ 2ipi frac kn N ^ text inverse DFT \&= tfrac 1 P sum _ k S_ frac 1 T left( frac k P right)cdot e^ 2ipi frac kn N end aligned

Fourier transforms on arbitrary locally compact abelian topological groups[edit] 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
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[edit] Further information: 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
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
Gabor transform
or fractional Fourier transform
Fourier transform
(FRFT), or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform
Fourier transform
being the continuous wavelet transform. History[edit] See also: Fourier series
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).[4][5][6][7] The classical Greek concepts of deferent and epicycle in the Ptolemaic system of astronomy were related to Fourier series
Fourier series
(see Deferent and epicycle: Mathematical formalism). In modern times, variants of the discrete Fourier transform
Fourier transform
were used by Alexis Clairaut
Alexis Clairaut
in 1754 to compute an orbit,[8] which has been described as the first formula for the DFT,[9] and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string.[10] 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.[11] Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.[10] An early modern development toward Fourier analysis
Fourier analysis
was the 1770 paper Réflexions sur la résolution algébrique des équations
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:[12] Lagrange transformed the roots x1, x2, x3 into the resolvents:

r

1

=

x

1

+

x

2

+

x

3

r

2

=

x

1

+ ζ

x

2

+

ζ

2

x

3

r

3

=

x

1

+

ζ

2

x

2

+ ζ

x

3

displaystyle 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

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,[13] but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides
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
Daniel Bernoulli
and Leonhard Euler had introduced trigonometric representations of functions,[9] and Lagrange had given the Fourier series
Fourier series
solution to the wave equation,[9] so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.[9] 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
Fourier transform
(FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gauss
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.[11][14] Interpretation in terms of time and frequency[edit] In signal processing, the Fourier transform
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
Fourier series
or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed. 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 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 such diverse branches as image processing, heat conduction, and automatic control. See also[edit]

Generalized Fourier series Fourier-Bessel series Fourier-related transforms Laplace transform
Laplace transform
(LT) Two-sided Laplace transform Mellin transform Non-uniform discrete Fourier transform
Fourier transform
(NDFT) Quantum Fourier transform
Fourier transform
(QFT) Number-theoretic transform Least-squares spectral analysis Basis vectors Bispectrum Characteristic function (probability theory) Orthogonal functions Schwartz space Spectral density Spectral density
Spectral density
estimation Spectral music Wavelet

Notes[edit]

^

P

(

m = − ∞

s ( t − m P )

)

e

− 2 i π

k P

t

d t =

− ∞

s ( t ) ⋅

e

− 2 i π

k P

t

d t

=

d e f

S

(

k P

)

displaystyle int _ P left(sum _ m=-infty ^ infty s(t-mP)right)cdot e^ -2ipi frac k P t ,dt=underbrace int _ -infty ^ infty s(t)cdot e^ -2ipi frac k P t ,dt _ stackrel mathrm def = ,Sleft( frac k P right)

^ We may also note that:

n = − ∞

+ ∞

T ⋅ s ( n T ) δ ( t − n T )

=

n = − ∞

+ ∞

T ⋅ s ( t ) δ ( t − n T )

= s ( t ) ⋅ T

n = − ∞

+ ∞

δ ( t − n T ) .

displaystyle begin aligned sum _ n=-infty ^ +infty Tcdot s(nT)delta (t-nT)&=sum _ n=-infty ^ +infty Tcdot s(t)delta (t-nT)\&=s(t)cdot Tsum _ n=-infty ^ +infty delta (t-nT).end aligned

Consequently, a common practice is to model "sampling" as a multiplication by the Dirac comb
Dirac comb
function, which of course is only "possible" in a purely mathematical sense. ^ Note that this definition differs from the DTFT section by a factor of T. ^

n = 0

N − 1

(

m = − ∞

s ( [ n − m N ] T )

)

e

− 2 i π

k N

n

=

n = − ∞

s ( n T ) ⋅

e

− 2 i π

k N

n

=

d e f

1 T

S

1 T

(

k

N T

)

displaystyle sum _ n=0 ^ N-1 left(sum _ m=-infty ^ infty s([n-mN]T)right)cdot e^ -2ipi frac k N n =underbrace sum _ n=-infty ^ infty s(nT)cdot e^ -2ipi frac k N n _ stackrel mathrm def = , frac 1 T S_ frac 1 T left( frac k NT right)

References[edit]

^ Fourier, Collins English Dictionary - Complete & Unabridged 10th Edition, HarperCollins, accessed 5 May 2017 ^ 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.  ^ Prestini, Elena (2004). The Evolution of Applied Harmonic
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 (2nd ed.). Dover Publications. ISBN 978-0-486-22332-2.  ^ Brack-Bernsen, Lis; Brack, Matthias. "Analyzing shell structure from Babylonian and modern times". arXiv:physics/0310126 .  ^ Terras, Audrey (1999). Fourier Analysis on Finite Groups and Applications. Cambridge University Press. p. 30. 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. p. 4. ISBN 978-0-89871-342-8.  ^ a b Briggs, William L.; Henson, Van Emden (1995). The DFT: An Owner's Manual for the Discrete Fourier Transform. SIAM. p. 2. 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.  ^ 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" (PDF). Reviews of Geophysics. New York: John Wiley & Sons. 37 (1): 151–172. doi:10.1029/1998RG900006. ISSN 1944-9208. OCLC 5156426043.  ^ Terras, Audrey (1999). Fourier Analysis on Finite Groups and Applications. Cambridge University Press. p. 31. ISBN 978-0-521-45718-7. 

Further reading[edit]

Conte, S. D.; de Boor, Carl (1980). Elementary Numerical Analysis (Third ed.). New York: McGraw Hill, Inc. ISBN 0-07-066228-2.  Evans, L. (1998). Partial Differential Equations. American Mathematical Society. ISBN 3-540-76124-1.  Howell, Kenneth B. (2001). Principles of Fourier Analysis. CRC Press. ISBN 978-0-8493-8275-8.  Kamen, E. W.; Heck, B. S. (2000-03-02). Fundamentals of Signals and Systems Using the Web and Matlab (2 ed.). Prentiss-Hall. ISBN 0-13-017293-6.  Knuth, Donald E. (1997). The Art of Computer Programming
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 0-201-89684-2.  Müller, Meinard (2015). The Fourier Transform in a Nutshell (PDF). Springer. In Fundamentals of Music Processing, Section 2.1, p. 40–56. doi:10.1007/978-3-319-21945-5. ISBN 978-3-319-21944-8.  Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 0-8493-2876-4.  Rudin, Walter (1990). Fourier Analysis on Groups. Wiley-Interscience. ISBN 0-471-52364-X.  Smith, Steven W. (1999). The Scientist and Engineer's Guide to Digital Signal Processing (Second ed.). San Diego: California Technical Publishing. ISBN 0-9660176-3-3.  Stein, E. M.; Weiss, G. (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press. ISBN 0-691-08078-X. 

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). "∑ Summation (and Fourier Analysis)". Sixty Symbols. Brady Haran
Brady Haran
for the University of

.