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]
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^ We may also note that:
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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.
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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