Analisis Fourier

Analisis Fourier, anu ngaranna nyokot tina ngaran Joseph Fourier anu ngawanohkeun Deret Fourier, nyaéta dékomposisi (pamisahan unsur-unsur) hiji sinyal kana sababaraha unsur panyusunna anu boga wangun fungsi-fungsi sinusoida (anu disebut fungsi dasar) nu masing-masing boga frékuénsi séwang-séwangan. Dina kalimah lain, unggal sinyal bisa ditulis salaku hasil pajumlahan teu kawates tina sinyal-sinyal fungsi sinus jeung kosinus nu miboga rupa-rupa frékuénsi.

Sabalikna, sinyal-sinyal sinusoida kasebut bisa digabungkeun atawa dijumlahkeun nepikeun ka ngahasilkeun sinyal asal. Prosés ngagabungkeun sinyal-sinyal dasar ieu disebutna sintésis Fourier. Dina gambar animasi di gigir bisa dititénan yén pajumlahan 25 sinyal sinusoida geus cukup pikeun ngaharib (ngahasilkeun aproksimasi) sinyal atawa gelombang kotak.

Balik deui kana analisis Fourier, hasil tina dékomposisi, nyaéta satiap amplitudo katut fase sinyal-sinyal sinusoida bisa digambarkeun dina doméin frékuénsi di mana ukuran amplitudona dinyatakeun dina sumbu y sarta frékuénsina dina sumbu x. Panggambaran ieu ngahasilkeun hiji Spéktrum Frékuénsi. Analogi pikeun leuwih ngajelaskeun hal ieu nyaéta sinyal cahaya bodas bisa dipisah-pisah kana sakumpulan spéktrum gelombang cahaya. Analogi lainna nyaéta sinyal sora harmoni (wirahma) musik anu dususun ku sakumpulan not balok musik (komponén-komponén frékuénsi) anu nyusunna.

Panerapan

Loba pisan panerapan ilmiah tina Analisis Fourier, utamana dina widang — dina fisika, téori bilangan, kombinatorika, pamrosésan sinyal, téori probabilitas, statistika, option pricing, kriptograpi, akustik, oséanograpi, optika sarta difraksi, geométri, katut widang-widang lianna.

Lobana panerapan ieu nyumber tina mangfaat sifat éta transformasi:

Sinyal mangrupa fungsi tina waktu. Dina itung-itungan, sinyal dina fungsi waktu kasebut bisa dimanipulasi jeung dinyatakeun (disajikeun) dina fungsi frékuénsi. Itung-itungan anu rumit dina doméin waktu bisa dilakukeun kalawan leuwih gampang dina doméin frékuénsi.
Hiji sistem mibanda kamampuhan pikeun ngarobah sinyal anu ngaliwat kana éta sistem. Dina kalimah lain, sistem miboga fungsi transfer. Lamun hiji sinyal asup kana hiji sistem mangka sinyal anu kaluar tina sistem kasebut bisa diitung atawa ditingali karakteristikna ku cara operasi aljabar saderhana antara sinyal asup (input) dina doméin frékuénsi jeung fungsi transfer dina doméin frékuénsi. Hasilna disebut réspon frékuénsi (frequency response). Saterusna, karakteristik sinyal anu dinyatakeun dina réspon frékuénsi bisa dikonvérsi balik kana fungsi waktu.
Transformasi ieu mangrupa operator linier sarta, ku cara maké normalisasi anu bener, mangrupa unitér ogé (hiji sifat anu dipikawanoh salaku teoréma Parseval atawa, leuwih umum, salaku teoréma Plancherel, jeung paling umum ngaliwatan dualitas Pontryagin).
Transformasi ieu bisa dibalik, jeung dina kanyataannya transformasi balikan éta miboga wangunan ampir sarua siga transformasi majuna.
Fungsi-fungsi basis eksponensial mangrupa fungsi eigen tina diférénsiasi, nu hartina nyaéta pidangan ieu ngarobah rumus diférénsial liniér kalayan koéfisién konstanta kana fungsi aljabar biasa. (Contona, dina sistim fisika waktu tetep anu liniér, frékuénsi mangrupa hiji kuantitas langgeng, ku kituna kalakuan unggal frékuénsi bisa diitung sacara indépénden.)
Ngaliwatan téoréma konvolusi, transformasi Fourier ngarobah operasi konvolusi anu ruwet kana pakalian basajan, anu hartina transformasi Fourier nyadiakeun hiji cara nu éfisién pikeun ngitung operasi dumasar konvolusi saperti pakalian polinomial sarta pakalian bilangan gedé.
 Vérsi diskrit tina transformasi Fourier (tempo di handap) bisa diitung sacara gancang dina komputer ku cara ngagunakeun algoritma fast Fourier transform (FFT).

Ragem tina analisis Fourier

Analisis Fourier boga rupa-rupa wangun, malahan sabagiannana boga ngaran anu béda.

Dérét Fourier 

Analisis Fourier didasarkeun kana konsép yén sinyal dina dunya nyata bisa diharib (diaproksimasi) ku pajumlahan sinusoida-sinusoida dina rupa-rupa frékuénsi. Leuwih loba sinusoida anu dijumlahkeun, mangka leuwih hadé aproksimasina.

Contona, pikeun ngaharib gelombang kotak anu ngulang dina périodeu 1 detik atawa frékuénsi 1 Hz, tingali hiji sinyal sinusoida 

Ayeuna, urang bakal nambah hiji suku sinusoida kadua kana suku sinus kahiji. Sinus anu ditambahkeun frékuénsina tilu kali frekeunsi suku sinusoida kahiji sarta gedéna ngan sapertilu éta sinyal. antukna sinyalna bisa dinyatakeun dina 

Wangun sinyal anu dihasilkeun kaciri masih béda jeung gelombang kotak. Ku cara neruskeun nambahkeun sinyal suku sinusoida katilu, anu frékuénsina lima kali sarta ukuranna saperlima kali suku kahiji, mangka dihasilkeun gelombang anu dinyatakeun ku rumus 

 
 
Hasil pajumlahan suku-suku sinusoida ieu buki narik ati. Urang ngan nambahkeun suku-suku anu frékuénsina kalipetan ganjil tina frékeuénsi suku sinusoida kahiji. Lamun pajumlahan diteruskeun tepi ka suku kagenep anu frékuénsina sabelas kali suku kahiji, hasilna saperti kieu:

 
 
Hasil pajumlahan suku-suku sinusoida ieu katémbong geus ngahasilkeun sinyal anu ngaharib kana gelombang kotak ngan wangunanna rada goréng.

Lamun pajumlahan sinyal dilakukeun ti suku kahiji tepi ka suku ka-25 anu frékuénsina kalipetan 49, mangka sinyal anu dihasilkeun geus ngaharib kana gelombang kotak tapi masih kurang nyeplés.

 
 
Di dieu katingali yén prosés ieu ngahasilkeun sinyal anu beuki deukeut kana sinyal gelombang kotak. Sanajan kitu, ieu katingalina saperti gelombang kotak anu kurang alus. Hayu urang tambahan deui suku-suku anu leuwih loba sarta tingali naon anu baris kajadian.

Ieu sinyal anu dihasilkeun tina pajumlahan suku-suku tepi ka suku anu boga kalipetan 79. Ayeuna urang beuki meunang indikasi anu jelas ngeunaan gelombang kotak kalawan amplitude sauetik di handap 0,8.

 
 
Tina prosés ieu, kanyahoan yén gelombang kotak boga rumus anu mangrupa pajumlahan dérét suku-suku sinusoida saperti kieu:

di mana, pikeun satiap bilangan ganjil non-negatif, n:

Pikeun ngabeunangkeun hasil panyajian anu akurat pisan, anggap n tanpa hingga atawa tanpa kawates.

Pajumlahan suku-suku sinusoida saperti di luhur disebut Dérét Fourier Trigonométrik. Frékuénsi unggal sinusoida dina ieu dérét mangrupa bilangan buleud kalipetan tina frékuénsi sinyal anu keur diaproksimasi. Frékuénsi-frékuénsi éta disebut harmonik tina gelombang kotak. Dina conto di luhur, frekeunsi dasarna nyaéta 1 Hz (frékuénsi gelombang kotak), sedengkeun harmonikna nyaéta 3 Hz, 5 Hz, 7 Hz, 9 Hz, 11 Hz, jst. Katiténan yén dina gelombang kotak ieu kabéh harmonik genep mibanda amplitudo enol; hal ieu henteu lumaku umum, aya gelombang anu mibanda harmoni dina bilangan buleud, aya ogé anu ngan mibanda harmonik dina bilangan ganjil hungkul.

Spektrum frékuénsi

Masing-masing frékuénsi harmonik didefinisikeun ku ukuran (amplitudo) sarta fasena. Ngagambar amplitudo-amplitudo harmonik dina sumbu y sarta frékuénsi dina sumbu x ngahasilkeun hiji Spéktrum Frékuénsi. Spéktrumna mangrupa sakumpulan garis atawa balok vertikal lantaran ngan boga frékuénsi dina kalipetan frékuénsi sinyal asal. Dina téori, spéktrum ngawengku frékuénsi-frékuénsi anu tanpa wates, tapi dina praktékna mplitudo harmonik-harmonik anu frékuénsina luhur pisan biasana euweuh hartina.

Tempo Dérét Fourier pikeun neuleuman leuwih jero, kaasup:
 transformasi kabalikan
 sifat transformasi
 kamekaran nurutkeun sajarah
 kasus husus ngeunaan s(t) nu boga harga-nyata

(Continuous) Fourier transform 

Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, such as time ().   The amplitude and phase of a sinusoidal component of function s(t) depends on the component's frequency.  In terms of ordinary frequency (), it is the complex number:

Evaluating this quantity for all values of  produces the frequency-domain function.

Also see How it works, below.  And see Continuous Fourier transform for even more information, including:
 the inverse transform
 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

Discrete-time Fourier transform (DTFT) 

For use on computers, a useful "discrete-time" function can be obtained by sampling a "continuous-time" function, s(t), which produces a sequence, s(nT), for integer values of n.  The DTFT is equivalent to the Fourier transform of a "continuous" function that is constructed by using the sequence    to modulate a Dirac comb.  In that case, the integral formula above simplifies to a summation:

which is a periodic function, with period   An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded (or finite), with span 

The DTFT can be applied to any discrete sequence.  But in the particular case where s[n] are samples of s(t),    and  are closely related.  See Discrete-time Fourier transform for more information on this and other topics, including:
 the inverse transform
 normalized frequency units
 windowing (finite-length sequences)
 transform properties
 tabulated transforms of specific functions

Discrete Fourier transform (DFT) 

Since the DTFT is also a continuous Fourier transform (of a comb function), the Fourier series also applies to it.  Thus, when  is periodic, with period N,   is another Dirac comb function, modulated by a Fourier series.   And the integral formula for the series simplifies to:

     for all integer values of k.

Since the DTFT is periodic, so is .  And it has the same period (N) as the input function.  This transform is also called DFT, particularly when only one period of the output sequence is computed from one period of the input sequence.

When  is not periodic, but its non-zero portion has finite duration (N),   is continuous and finite-valued.  But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of  that was analyzed.  The same discrete set is obtained by tréating N as if it is the period of a periodic function and computing the Fourier series / DFT.
 The inverse transform of  does not produce the finite-length sequence,   (It takes the inverse of , to do that.)  The inverse DFT can only reproduce the entire time-domain if the input happens to be periodic (forever). Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain, discrete-time functions.   An alternative viewpoint is that the periodicity is the time-domain consequence of approximating the continuous-domain function, , with the discrete subset, .   N can be larger than the actual non-zero portion of .   The larger it is, the better the approximation (also known as zero-padding).

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

See Discrete Fourier transform for much more information, including:
 the inverse transform
 transform properties
 applications
 tabulated transforms of specific functions

The following table recaps the four basic forms discussed above, highlighting the duality of the properties of discreteness and periodicity.  I.e., if the signal representation in one domain has either (or both) of those properties, then its transform representation to the other domain has the other property (or both).

Fourier transforms on arbitrary locally compact abelian topological groups

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 takes functions on a group to functions on the dual group. This tréatment 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
Time-frequency transforms such as the short-time Fourier transform, wavelet transforms, chirplet transforms, and the fractional Fourier transform try to obtain frequency information from a signal as a function of time (or whatever the independent variable is), although the ability to simultanéously resolve frequency and time is limited by an (mathematical) uncertainty principle.

Interpretation in terms of time and frequency

In terms of signal processing, the transform takes a time series  representation of a signal function and maps it into a frequency spectrum, where ω is angular frequency.  That is, it takes a function in the time domain into the frequency domain; it is a decomposition of a function into harmonics of different frequencies.

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:  arctan (imaginary part/real part).

However, it is important to réalize that Fourier transforms are not limited to functions of time, and temporal frequencies.  They can equally be applied to analyze spatial frequencies, and indeed for néarly any function domain.

Applications in signal processing

When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for éasier detection and/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.

Some examples include:

 Telephone dialing;  the touch-tone signals for éach telephone key, when pressed, are éach a sum of two separate tones (frequencies).  Fourier analysis can be used to separate (or analyze) the telephone signal, to revéal the two component tones and therefore which button was pressed.
 Removal of unwanted frequencies from an audio recording (used to eliminate hum from léakage of AC power into the signal, to eliminate the stereo subcarrier from FM radio recordings, or to créate karaoke tracks with the vocals removed);
 Noise gating of audio recordings to remove quiet background noise by eliminating Fourier components that do not exceed a preset amplitude;
 Equalization of audio recordings with a series of bandpass filters;
 Digital radio reception with no superheterodyne circuit, as in a modérn cell phone or radio scanner;
 Image processing to remove periodic or anisotropic artifacts such as jaggies from interlaced vidéo, stripe artifacts from strip aerial photography, or wave patterns from radio frequency interference in a digital camera;
 Cross correlation of similar images for co-alignment;
 X-ray crystallography to reconstruct a protein's 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.

Fourier transformation is also useful as a compact representation of a signal.  For example, JPEG compression uses Fourier transformation of small square pieces of a digital image.  The Fourier components of éach square are rounded to lower arithmetic precision, and wéak components are eliminated entirely, so that the remaining components can be stored very compactly.  In image reconstruction, éach Fourier-transformed image square is réassembled from the preserved approximate components, and then inverse-transformed to produce an approximation of the original image.

How it works (a basic explanation) 

To méasure the amplitude and phase of a particular frequency component, the transform process multiplies the original function (the one being analyzed) by a sinusoid with the same frequency (called a basis function).  If the original function contains a component with the same shape (i.e. same frequency), its shape (but not its amplitude) is effectively squared.
Squaring implies that at every point on the product waveform, the contribution of the matching component to that product is a positive contribution, even though the product might be negative.
Squaring describes the case where the phases happen to match.  What happens more generally is that a constant phase difference produces vectors at every point that are all aimed in the same direction, which is determined by the difference between the two phases.  To maké that happen actually requires two sinusoidal basis functions, cosine and sine, which are combined into a basis function that is complex-valued (see Complex exponential).  The vector analogy refers to the polar coordinate representation.
The complex numbers produced by the product of the original function and the basis function are subsequently summed into a single result.
Note that if the functions are continuous, rather than sets of discrete points, this step requires integral calculus or numerical integration.  But the basic concept is just addition.  
The contributions from the component that matches the basis function all have the same sign (or vector direction).  The other components contribute values that alternate in sign (or vectors that rotate in direction) and tend to cancel out of the summation. The final value is therefore dominated by the component that matches the basis function.  The stronger it is, the larger is the méasurement.  Repéating this méasurement for all the basis functions produces the frequency-domain representation.

Tempo oge
 Bispectrum
 Characteristic function (probability theory)
 Fractional Fourier transform
 Laplace transform
 Least-squares spectral analysis 
 Mellin transform
 Number-theoretic transform
 Orthogonal functions
 Pontryagin duality
 Schwartz space
 Spectral density
 Spectral density estimation
 Two-sided Laplace transform
 Wavelet

Catetan

Rujukan
 Edward W. Kamen, Bonnie S. Heck, "Fundamentals of Signals and Systems Using the Web and Matlab", ISBN 0-13-017293-6 
 E. M. Stein, G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton University Press, 1971. ISBN 0-691-08078-X 
 A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4

Tumbu luar
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

Analisis Fourier
Transformasi integral
Pamrosésan sinyal digital
Téori éstimasi
Pamrosésan sinyal
Algoritma
Transformasi
Élmu komputasi
Analisis matematik
Matematika terapan
Matematika komputasi
Statistika