In mathematics, FOURIER ANALYSIS (English: /ˈfʊəriˌeɪ/ ) is the
study of the way general functions may be represented or approximated
by sums of simpler trigonometric functions .
Today, the subject of
The decomposition process itself is called a
CONTENTS * 1 Applications * 1.1 Applications in signal processing * 2 Variants of
* 2.1 (Continuous)
* 3 History * 4 Interpretation in terms of time and frequency * 5 See also * 6 Notes * 7 References * 8 Further reading * 9 External links APPLICATIONS
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
VARIANTS OF FOURIER ANALYSIS A
(CONTINUOUS) FOURIER TRANSFORM Main article:
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: S ( f ) = s ( t ) e 2 i 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 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
* conventions for amplitude normalization and frequency scaling/units * transform properties * tabulated transforms of specific functions * an extension/generalization for functions of multiple dimensions, such as images. FOURIER SERIES Main article:
The
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 e 2 i k P t F k = + S ( f k P ) . {displaystyle s_{P}(t)=sum _{k=-infty }^{infty }Scdot e^{2ipi {frac {k}{P}}t}quad {stackrel {displaystyle {mathcal {F}}}{Longleftrightarrow }}quad sum _{k=-infty }^{+infty }S,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 = 1 P S ( k P ) . {displaystyle S={frac {1}{P}}cdot Sleft({frac {k}{P}}right).} 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
DISCRETE-TIME FOURIER TRANSFORM (DTFT) Main article: Discrete-time
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 e
2 i f n T
which is known as the DTFT. Thus the DTFT of the s sequence is also
the FOURIER TRANSFORM of the modulated
The
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, 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 , the applicable reconstruction
formula is the
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
* normalized frequency units * windowing (finite-length sequences) * transform properties * tabulated transforms of specific functions DISCRETE FOURIER TRANSFORM (DFT) Main article: Discrete
Similar to a Fourier series, the DTFT of a periodic sequence, sN,
with period N, becomes a
where ∑N is the sum over any n-sequence of length N. The S 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 = 1 N N S e 2 i n N k , {displaystyle s_{N}={frac {1}{N}}sum _{N}Scdot e^{2ipi {frac {n}{N}}k},} where ∑N is the sum over any k-sequence of length N. When sN is expressed as a periodic summation of another function: s N = def m = s , {displaystyle s_{N},{stackrel {text{def}}{=}},sum _{m=-infty }^{infty }s,} and s = def s ( n T ) , {displaystyle s,{stackrel {text{def}}{=}},s(nT),} the coefficients are proportional to samples of S1/T( f ) at discrete intervals of 1/P = 1/NT: S = 1 T S 1 T ( k P ) . {displaystyle S={frac {1}{T}}cdot S_{frac {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, S1/T( f ), it
can be done by computing the relatively simple DFT of sN, as defined
above. In most cases, N is chosen equal to the length of non-zero
portion of s. 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
sequence represents a longer sequence that was truncated by the
application of a finite-length window function or
The DFT can be computed using a fast
See Discrete
* transform properties * applications * tabulated transforms of specific functions SUMMARY For periodic functions, both the
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 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 = def 1 P s ( t ) e 2 i k P t d t 1 P P s P ( t ) e 2 i k P t d t {displaystyle overbrace {{frac {1}{P}}cdot Sleft({frac {k}{P}}right)} ^{S},{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 f t d f
{displaystyle s(t)=int _{-infty }^{infty }S(f)cdot e^{2ipi ft},df}
s P ( t ) = k = S e 2 i k P t
s(nT) transforms (discrete-time) CONTINUOUS FREQUENCY DISCRETE FREQUENCIES TRANSFORM
1 T S 1 T ( f ) = def n = s
( n T ) e 2 i f n T
1 T S 1 T ( k N T ) S = def n = s ( n T ) e 2 i k n N N s P ( n T ) e 2 i k n N DFT {displaystyle {begin{aligned}overbrace {{frac {1}{T}}S_{frac {1}{T}}left({frac {k}{NT}}right)} ^{S},&{stackrel {text{def}}{=}},sum _{n=-infty }^{infty }s(nT)cdot e^{-2ipi {frac {kn}{N}}}\ width:38.95ex; height:20.176ex;" alt="{displaystyle {begin{aligned}overbrace {{frac {1}{T}}S_{frac {1}{T}}left({frac {k}{NT}}right)} ^{S},&{stackrel {text{def}}{=}},sum _{n=-infty }^{infty }s(nT)cdot e^{-2ipi {frac {kn}{N}}}\"> s ( n T ) = T 1 T 1 T S 1 T ( f ) e 2 i 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 ) ( t n T ) = 1 T S
1 T ( f ) e 2 i f t d f inverse
s P ( n T ) = 1 N N S e 2 i k n
N inverse DFT = 1 P N S 1 T ( k P
) e 2 i k n N {displaystyle
{begin{aligned}s_{P}(nT)&=overbrace {{frac {1}{N}}sum _{N}Scdot
e^{2ipi {frac {kn}{N}}}} ^{text{inverse DFT}}\ width:34.927ex;
height:16.509ex;" alt="{displaystyle
{begin{aligned}s_{P}(nT)&=overbrace {{frac {1}{N}}sum _{N}Scdot
e^{2ipi {frac {kn}{N}}}} ^{text{inverse DFT}}\ there, the Fourier
transform takes functions on a group to functions on the dual group.
This treatment also allows a general formulation of the convolution
theorem , which relates Fourier transforms and convolutions . See also
the
TIME–FREQUENCY TRANSFORMS For more details on this topic, see
In signal processing terms, a function (of time) is a representation
of a signal with perfect time resolution, but no frequency
information, while the
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
HISTORY See also:
A primitive form of harmonic series dates back to ancient Babylonian mathematics , where they were used to compute ephemerides (tables of astronomical positions). The classical Greek concepts of deferent and epicycle in the
In modern times, variants of the discrete
An early modern development toward
A number of authors, notably Jean le Rond d\'Alembert , and Carl
Friedrich Gauss used trigonometric series to study the heat equation ,
but the breakthrough development was the 1807 paper Mémoire sur la
propagation de la chaleur dans les corps solides by
Historians are divided as to how much to credit Lagrange and others
for the development of Fourier theory:
The subsequent development of the field is known as harmonic analysis , and is also an early instance of representation theory . The first fast
INTERPRETATION IN TERMS OF TIME AND FREQUENCY In signal processing , the
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 * Generalized
NOTES * ^ P ( m = s ( t m P ) ) e 2 i k P t d t = s ( t ) e 2 i 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 ) ( t n T ) = n = + T s ( t ) ( t n T ) = s ( t ) T n = + ( 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)\ width:51.47ex; height:14.509ex;" alt="{displaystyle {begin{aligned}sum _{n=-infty }^{+infty }Tcdot s(nT)delta (t-nT)&=sum _{n=-infty }^{+infty }Tcdot s(t)delta (t-nT)\">T. * ^ N ( m = s ( T ) ) e 2 i k N n = n = s ( n T ) e 2 i k N n = d e f 1 T S 1 T ( k N T ) {displaystyle sum _{N}left(sum _{m=-infty }^{infty }s(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 * ^ 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
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) . The Exact Sciences in Antiquity (2nd
ed.).
FURTHER READING * 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).
EXTERNAL LINKS * Tables of Integral Transforms at EqWorld: The World of Mathematical |