Fourier transform. Fast Fourier transform. Discrete Fourier transform

Fourier transformation - transformation, tham ib tug tej yam muaj nuj nqi ntawm ib tug tiag tiag nce mus nce los. Qhov no lub lag luam yog ua txhua txhua lub sij hawm peb tau perceive suab sib txawv. Pob ntseg ua tsis siv neeg "xam", uas txaus siab rau peb nco qab ua tau tsuas yog tom qab qhov kev xeem ntawm lub seem ntawm kev kawm leb siab. lub rooj sib hais hloov nyob rau hauv ib tug tib neeg transformation constructs, nyob rau hauv uas lub suab (cov pa vibrational tsab ntawv tsa suab hais nyob rau hauv ib tug elastic nruab nrab, uas propagate nyob rau hauv yoj daim ntawv nyob rau hauv cov khoom, kua los yog gaseous nruab nrab) yog muab nyob rau hauv ib tug ntau ntawm sib law liag tseem ceeb ntawm cov volume theem ntawm tones ntawm ua heights. Tom qab no, lub paj hlwb puv cov ntaub ntawv mus rau hauv tag nrho cov paub lub suab.

Lej Fourier transform

Conversion ntawm lub suab tsis los yog lwm yam kev co dab (los ntawm lub teeb emission thiab dej hiav txwv tom tsim thiab stellar los yog hnub ci mus) yuav tsum tau ua thiab los ntawm txoj kev xyuam xim txoj kev. Yog li, siv cov kev kawm, cov kev khiav dej num yuav tsum txhab los ntawm plaug vibrational dab teem ntawm sinusoidal Cheebtsam, i.e. laim ntoom curves uas mus los ntawm ib tug tsawg kawg nkaus rau ib tug tshaj plaws thiab ces rov qab mus rau ib tug tsawg kawg nkaus, zoo ib yam li nthwv dej ntawm lub hiav txwv. Fourier transformation - transformation muaj nuj nqi uas qhia txog cov theem los yog amplitude ntawm txhua sinusoid coj mus rau ib tug kev zaus. Theem yog ib tug starting taw tes rau ntawm tus nkhaus, thiab cov amplitude - ntawm nws qhov siab.

Fourier transform (piv txwv yog muaj nyob rau hauv cov kev yees duab) yog ib tug heev haib cuab tam, uas yog siv nyob rau hauv ntau yam teb ntawm science. Nyob rau hauv tej rooj plaub, nws yog siv raws li ib tug tshuaj es complex sib npaug uas piav qhia txog tus dynamic dab tshwm sim nyob rau hauv tus ntawm lub teeb, Thaum tshav kub kub los yog hluav taws xob lub zog. Nyob rau hauv lwm yam mob, nws tso cai rau koj txhais kom meej cov kev cheebtsam hauv complex waveforms, vim qhov no yuav yog muaj tseeb tiag los txhais lus ntau yam kev sim pom nyob rau hauv Science News for KIDS, tshuaj thiab astronomy.

keeb kwm ntaub ntawv

Thawj thawj tus mus thov cov qauv no yog lub Fabkis mathematician Zhan Batist Fure. Conversion, tom qab tis npe hu ua tom qab nws, yog Ameslikas siv los piav cov tshav kub conduction mechanism. Fourier nws tag nrho neeg laus lub neej koom nyob rau hauv kev kawm rau hauv lub thaj chaw ntawm lub tshav kub. Nws ua ib tug enormous pab mus rau cov zauv hom phiaj ntawm qhov kev txiav txim ntawm cov keeb kwm ntawm algebraic equations. Fourier yog ib tug xib fwb ntawm tsom xam ntawm lub Ecole Polytechnique, tus Secretary ntawm lub koom haum ntawm Egyptology, yog tus imperial kev pab cuam, uas tshwm sim los ib tug stir thaum lub sij hawm ntawm kev tsim kho ntawm txoj kev mus rau Turin (nyob rau hauv nws coj noj coj ua twb drained los ntawm ntau tshaj 80 txhiab square kilometers ntawm malarial swamps). Txawm li cas los, tag nrho cov no activism tsis tau nres tus paub txog koom nyob rau hauv lej tsom xam. Nyob rau hauv 1802 nws twb muab tau ib qho kev uas piav txog cov propagation ntawm tshav kub nyob rau hauv cov khib nyiab. Nyob rau hauv 1807, tus paub txog sab ib txoj kev los daws qhov no kab zauv, uas lub npe hu ua "Fourier transform".

thermal conductivity tsom xam

Soj ntsuam ntawm siv ib tug lej txoj kev los piav qhia txog lub tshav kub conduction mechanism. Ib tug yooj yim piv txwv li, nyob rau tsis muaj teeb meem nyob rau hauv xam yog cov hais tawm ntawm thermal zog los ntawm ib tug hlau nplhaib, ib feem raus nyob rau hauv ib tug tua hluav taws. Mus nqa tawm thwmsim Fourier liab kub ib feem ntawm lub nplhaib thiab faus nws nyob rau hauv lub zoo xuab zeb. Tom qab ntawd, qhov kub thiab txias ntsuam nqa tawm nyob rau hauv opposite ib feem ntawd. Chiv, lub tshav kub tis yog lus: ib feem ntawm lub nplhaib - txias, thiab lwm yam - kub, nruab nrab ntawm cov aav yuav soj ntsuam ib tug ntse kub gradient. Txawm li cas los, thaum lub sij hawm lub tshav kub faib thoob plaws hauv lub hlau nto, nws yuav ntau yam. Yog li ntawd, tsis ntev, txoj kev no yuav siv sij hawm rau hauv daim ntawv ntawm ib tug sine yoj. Thawj teeb maj tsub kom thiab kuj txo ntseeg nkaws, meej cov kev cai ntawm variation ntawm lub cosine los yog sine muaj nuj nqi. Wave maj equalized thiab raws li ib tug tshwm sim rau hauv lub kub yuav uas teeb rau ntawm tag nrho saum npoo ntawm lub nplhaib.

Tus sau ntawm no txoj kev lam xav hais tias cov thawj zaug tis yog heev lus yuav decomposed rau hauv ib tug xov tooj ntawm elementary sine tsis. Txhua yam ntawm lawv yuav muaj nws cov theem (xub pib txoj hauj lwm) thiab nws cov nyiaj pab ntau tshaj kub. Yog li txhua xws tivthaiv kev hloov los ntawm ib tug tsawg kawg nkaus rau ib tug tshaj plaws thiab rov qab mus ua kom tiav kiv puag ncig nyob ib ncig ntawm lub nplhaib integer lub sij hawm. Tivthaiv muaj ib lub sij hawm uas tau hu ua lub tsum muaj harmonic, thiab tus nqi nrog ob los yog ntau tshaj sij hawm - lub thib ob thiab thiaj li nyob. Piv txwv li, ib tug zauv muaj nuj nqi uas piav txog cov nyiaj pab ntau tshaj kub, cov theem los yog txoj hauj lwm hu ua lub Fourier transform ntawm lub tsev muaj nuj nqi. Tus paub txog coj ib tug hluas tivthaiv uas yog ib qhov nyuaj rau zauv hauj lwm, rau ib qho yooj yim-rau-siv cov cuab yeej - kab ntawm sine thiab cosine, nyob rau hauv tus nqi ntawm cov muab thawj zaug tis.

Lub essence ntawm tus tsom xam

Thov no tsom xam rau cov hloov dua siab tshiab ntawm tshav kub tis rau cov khoom kwv, muaj ib tug annular zoo, ib tug mathematician piav rau lawv tias ua lub sij hawm ntawm sinusoidal Cheebtsam ua rau nws ceev ceev damping. Qhov no yog kom meej meej pom nyob rau lub ntsiab thiab ob harmonics. Qhov kawg kub nce mus txog ob zaug cov nyiaj pab ntau tshaj thiab yam tsawg kawg nkaus qhov tseem ceeb nyob rau hauv ib pass, thiab nyob rau hauv thawj - ib zaug xwb. Nws hloov tawm hais tias qhov kev ncua deb mus ncig los ntawm tshav kub nyob rau hauv lub thib ob harmonic yog ib nrab ntawm cov tub ntxhais. Nyob rau hauv tas li ntawd, lub gradient ntawm lub thib ob ib nrab kuj yuav steeper dua thawj zaug. Yog li ntawd, txij li thaum ib tug ntau khaus thermal flux kis poj ntsuam tsawg heev deb, ces qhov no yuav tsum damped harmonic plaub lub sij hawm sai tshaj li lub ntsiab, raws li ib tug muaj nuj nqi ntawm lub sij hawm. Nyob rau hauv cov nram no tus txheej txheem yuav tsum tau sai. Mathematician ntseeg hais tias qhov no txoj kev tso cai rau peb kom peb xam tus txheej txheem ntawm lub thawj zaug muab faib rau kub nrog lub sij hawm.

Hu contemporaries

Fourier transform algorithm tau los ua ib tug kev sib tw rau lub theoretical lub hauv paus ntawm kev kawm txog zauv rau lub sij hawm. Nyob rau hauv thaum ntxov thib kaum cuaj xyoo pua, feem ntau mas zaum, xws li Lagrange, Laplace, Poisson, Legendre thiab Biot tsis kam txais nws assertion uas tus kub ntawm cov thawj zaug faib yog decomposed rau hauv cheebtsam hauv daim ntawv ntawm qhov siv yoj thiab ntau dua zaus. Txawm li cas los, lub Academy ntawm Sciences yuav tsis quav ntsej txog kev tshwm sim tau mathematician, thiab muab nws lub nqi zog rau lub hom phiaj ntawm tshav kub conduction ntawm cov kev cai, raws li tau zoo raws li kev nws kev sib piv nrog rau lub cev thwmsim. Nyob rau hauv lub Fourier mus kom ze, lub ntsiab pom yog qhov tseeb hais tias ib tug discontinuous muaj nuj nqi yog sawv cev los ntawm ib tug sum ntawm ob peb sinusoidal zog, uas yog tas mus li. Tom qab tag nrho, lawv piav qhia txog lub bursting ncaj thiab nkhaus kab. Kawm paub txog tsis tau ces yuav tsum xws li ib tug teeb meem no, thaum lub discontinuous zog piav los ntawm ib tug ua ke ntawm tas mus li, xws li quadratic, linear, sine los yog exhibitor. Nyob rau hauv cov kev tshwm sim hais tias ib tug mathematician yog txoj cai nyob rau hauv nws assertions, lub sum ntawm ib tug infinite series ntawm trigonometric functions yuav tsum muaj rau lub caij nyoog ceev. Thaum xws li ib tug thov nyiaj ciali absurd. Txawm li cas los, txawm lub xav phem cais tsis ntawm ib co kev soj ntsuam (xws li Claude Navier, Sofi Zhermen) broadened lub Scope ntawm kev tshawb fawb thiab coj lawv tawm ntawm lub tsom xam ntawm tshav kub tis. Ib tug lej, lub caij no, txuas ntxiv mus raug kev txom nyem rau lo lus nug ntawm seb ib tug sum ntawm ob peb sinusoidal zog yog txo mus rau ib qho txaus sawv cev ntawm bursting.

200-xyoo yav dhau los

Qhov no kev tshawb xav tau hloov zuj zuj tshaj li ob centuries, niaj hnub no nws yog thaum kawg tsim. Nrog kev pab los ntawm lub spatial los yog sab nqaij daim tawv zog yog tawg rau hauv sinusoidal cheebtsam uas muaj ib zaus, theem thiab amplitude. Qhov no hloov dua siab tshiab yog tau los ntawm ob tug sib txawv zauv txoj kev. Cov Thawj zaug ntawm lawv yog siv nyob rau hauv lub rooj plaub thaum lub qhov chaw yog ib tug tas mus muaj nuj nqi, thiab lub thib ob - nyob rau hauv lub rooj plaub uas nws yog sawv cev los ntawm ib tug plurality ntawm discrete ib tug neeg hloov. Yog hais tias cov kev qhia yog tau los ntawm qhov tseem ceeb, uas txhais tau hais tias ntawm discrete Team sib, nws yuav tsum tau muab faib mus rau hauv ob peb discrete sinusoidal frequencies kab zauv - los ntawm tus nqi qis tshaj thiab ces ob npaug, peb npaug, thiab thiaj li tawm saum toj no yuav tsum muaj. Cov nyiaj no yog hu ua tus Fourier series. Yog hais tias cov thawj zaug qhia teev tus nqi ntawm txhua tiag tiag tooj, nws muaj peev xwm yuav tawg mus rau hauv ntau yam sinusoidal tag nrho cov tau frequencies. Nws yog hu ua ib tug Fourier integral, thiab qhov kev txiav txim implies ib tug transformation ntawm lub ib muaj nuj nqi. Tsis hais txog ntawm txoj kev muab transformation, rau txhua zaus yuav tsum qhia tias ob tus xov tooj: amplitude thiab zaus. Cov qhov tseem ceeb yog qhia raws li ib tug tib complex tooj. Qhia complex variables kev tshawb xav ua ke nrog Fourier transformation ua suav tso cai rau tsim ntawm ntau yam hluav taws xob circuits, tus tsom xam ntawm cov neeg kho tshuab deeg, txoj kev tshawb ntawm yoj propagation mechanism thiab lwm.

Fourier transform hnub no

Niaj hnub no, txoj kev tshawb ntawm tus txheej txheem no yeej boils down mus nrhiav zoo txoj kev yuav hloov txoj kev muaj nuj nqi rau hloov nws rov qab mus rau lub siab. Cov tshuaj no yog hu ua tus tib neeg thiab rov qab Fourier transform. Ua li cas nws txhais li cas? Nyob rau hauv thiaj li yuav txiav txim seb lub ib thiab ua ib tug ncaj qha Fourier transform, koj muaj peev xwm siv mathematical txoj kev, tab sis koj muaj peev xwm analytic. Txawm tias muaj tseeb hais tias thaum lawv yog siv nyob rau hauv kev xyaum muaj ib co teeb meem, feem ntau integrals twb tau pom thiab nkag mus rau hauv lej phau ntawv. Nrog kev pab los ntawm hais txog zauv txoj kev muaj peev xwm yuav xam zauv, qhov zoo ntawm uas yog raws li nyob rau hauv lub sim cov ntaub ntawv, ib tug muaj nuj nqi uas integrals rau hauv lub rooj uas ploj lawm, thiab lawv tsis yooj yim xav txog tej yam nyob rau hauv ib tug analytical daim ntawv.

Ua ntej lub advent ntawm lub computer engineering suav xws transformations tau heev tsho, lawv yuav tsum tau phau ntawv tua ntawm ib tug loj tus naj npawb ntawm xam hauj lwm uas yog nyob rau hauv lub xov tooj ntawm cov ntsiab lus uas piav txog cov yoj muaj nuj nqi. Yuav kom pab txhawb lub zos niaj hnub no, muaj cov kev kawm tshwj xeeb, tso cai rau siv tshiab analytical txoj kev. Yog li ntawd, nyob rau hauv 1965, Dzheyms Kuli thiab Dzhon Tyuki tsim software uas hu ua "Fast Fourier txia". Nws cawm lub sij hawm ntawm lub xam los ntawm kev txo tus naj npawb ntawm multiplications nyob rau hauv lub tsom xam ntawm tus nkhaus. "Fast Fourier txia" Cov kev yog raws li nyob rau hauv faib lub nkhaus rau hauv ib tug loj tus naj npawb ntawm niaj hnub qauv tseem ceeb. Raws li, qhov tooj ntawm multiplications yog txo los ntawm ib nrab nyob rau tib lub txo tus naj npawb ntawm cov ntsiab lus.

Thov lub Fourier transform

Qhov no tus txheej txheem no siv nyob rau hauv ntau yam teb: Nyob rau hauv kev tshawb xav tooj, physics, lub teeb liab ua, combinatorics, muaj tseeb kev tshawb xav, cryptography, statistics, oceanography, optics, acoustics, thiab lwm yam geometries. Rich possibilities rau nws siv yog raws li nyob rau hauv ib tug xov tooj ntawm pab nta, uas yog hu ua "thaj chaw ntawm lub Fourier transformation." Cia peb kawm lawv.

1. hloov dua siab tshiab muaj nuj nqi yog ib tug linear neeg teb xov tooj thiab ib tug coj normalization yog unitary. Qhov no tej khoom vaj tse yog hu ua tus Parseval theorem, los yog nyob rau hauv lub dav dav, lub theorem Plansherelja los yog Pontrjagin dualism.

2. hloov dua siab tshiab yog reversible. Ntxiv mus, lub opposite no tshuam zoo xws li cov zoo li nyob rau hauv qhov ncaj hais.

3. Cov sinusoidal yooj yim kab zauv uas lawv tus kheej differentiated zog. Qhov no txhais tau tias li sawv cev hloov linear equations nrog rau qhov coefficients nyob rau hauv ib tug pa algebraic.

4. Raws li cov "convolution" theorem, cov txheej txheem yuav ua rau ib tug complex lag luam nyob rau hauv elementary npaug.

5. Discrete Fourier txia yuav ua sai sai tsim ib lub computer siv lub "ceev" method.

Variations ntawm lub Fourier transform

1. Feem ntau cov feem ntau lub sij hawm yog siv los xa mus rau ib tug tas mus transformation, muab tej quadratically integrable qhia raws li lub sum ntawm complex exponential qhia nrog kev angular frequencies thiab amplitudes. Qhov no tsiaj nws muaj ob peb hom sib txawv, uas tej zaum yuav sib txawv qhov coefficients. Cov nruam txoj kev xws li ib tug hloov dua siab tshiab lub rooj, uas muaj peev xwm yuav nrhiav tau nyob rau hauv lej phau ntawv. Ib tug generalized cov ntaub ntawv yog lub fractional hloov dua siab tshiab, uas tus txheej txheem no yuav raug tsa mus rau lub kev kawm lub hwj chim.

2. Cov nruam txoj kev yog ib tug generalization ntawm ua ntej lawm cov txheej txheem ntawm Fourier series txhais rau tej periodic zog los yog kab zauv, uas muaj nyob rau hauv ib tug tsawg cheeb tsam thiab sawv cev rau lawv raws li ib tug series ntawm sinusoids.

3. Discrete Fourier transform. Qhov no yog siv nyob rau hauv koj laij rau scientific caag thiab cov teeb liab ua. Mus nqa tawm no hom ntawm xam yuav tsum tau muaj ib tug muaj nuj nqi ntawm kev txiav txim rau ib tug discrete txheej ntawm ib tug neeg cov ntsiab lus, periodic los sis cheeb tsam es tsis txhob ntawm tas mus li Fourier integrals. Teeb liab hloov dua siab tshiab nyob rau hauv cov ntaub ntawv no yog sawv cev raws li ib tug zaum ntawm sinusoids. Txoj kev siv "ceev" txoj kev tso cai rau kev siv ntawm cov kev daws teeb meem rau tag nrho cov tswv yim lub hom phiaj.

4. Lub qhov rai Fourier transform yog ib tug generalized saib ntawm lub classic txoj kev. Tsis zoo li tus txheej txheem kev daws teeb meem thaum lub teeb liab spectrum yog siv, uas yog coj nyob rau hauv tag nrho cov ntau yam ntawm cov hav zoov ntawm no nce mus nce los yog ntawm kev txaus siab ntawm no yog tsuas lub zos zaus faib thaum tswj cov thawj nce mus nce los (lub sij hawm).

5. Lub ob-dimensional Fourier transform. Qhov no yog siv los mus ua hauj lwm nrog ob tug-dimensional arrays ntawm cov ntaub ntawv. Nyob rau hauv xws li ib tug cov ntaub ntawv, cov conversion yog ua nyob rau hauv ib tug coj, thiab ces - nyob rau hauv lwm.

xaus

Niaj hnub no, cov Fourier txoj kev yog nrees entrenched nyob rau hauv lub ntau yam teb ntawm science. Piv txwv li, nyob rau hauv 1962 nws qhib qhov zoo ntawm cov DNA ob helix siv Fourier tsom xam nyob rau hauv nrog X-hluav taws xob diffraction. Tsis ntev los no muaju teem rau DNA fibers, uas ua rau ib cov duab uas yog tau ua los ntawm diffraction, kaw rau cov zaj duab xis. Qhov no daim duab muab cov lus qhia txog tus nqi ntawm cov amplitude los ntawm kev siv lub Fourier transform mus no siv lead ua qauv. Theem cov ntaub ntawv tau los ntawm muab piv rau cov DNA diffraction cards nrog daim ntawv uas yog tau nyob rau hauv lub tsom xam ntawm zoo xws li cov tshuaj lug. Raws li ib tug tshwm sim, biologists rov qab los siv lead ua qauv - tus thawj muaj nuj nqi.

Fourier transform ua si ib tug lossis loj lub luag hauj lwm nyob rau hauv txoj kev tshawb no ntawm txheej chaw, lub physics ntawm semiconductor cov ntaub ntawv thiab ntshav, microwave acoustics, oceanography, radar, seismology thiab kev kho mob kuaj.