Fourier uguqule. Fast Fourier uguqule. Fourier leyehlukene uguqule

Fourier ukuguqulwa - ushintsho, ukuzihlanganisa umsebenzi othile kwenombolo yangempela. Lo msebenzi wenziwa njalo siyaqonda imisindo ehlukene. Indlebe ukhiqiza okuzenzakalelayo "ukubala", okuyinto ukufeza ukuqaphela kwethu can kuphela emva ukuhlolwa kwesigaba wezibalo ephakeme. ukuzwa isitho kwammangaza womuntu lolwakha, lapho umsindo (ezivamile ezinyakazayo vibrational izinhlayiya e i osebenzelana nokunwebeka, okuyinto asakaze ngefomu wave in the medium okuqinile, uketshezi noma gaseous) inikezwa a ububanzi bamanani elandelanayo ivolumu ezingeni amathoni of obuhlukahlukene eziphakeme. Ngemva kwalokhu, ubuchopho uphendulela ulwazi kuwo wonke umsindo ajwayelekile.

Mathematical Fourier uguqule

Ukuguqulwa amaza omsindo noma ezinye vibration izinqubo (by ekushayweni ukukhanya ocean tide futhi emjikelezweni lezinkanyezi noma welanga) engenziwa futhi esebenzisa izindlela zezibalo. Ngakho, esebenzisa lezi amasu, imisebenzi kungaba ukunwetshwa ngokwethula izinqubo vibrational setha izingxenye sinusoidal, isb mihlobiso Curves okuyinto hamba ukusuka ebuncaneni kuze kufike bese futhi esiphansi, efana negagasi lolwandle. Fourier ukuguqulwa - ukuguqulwa umsebenzi echaza esigabeni noma amplitude sinusoid ngasinye sihambisane imvamisa othize. Isigaba kuyinto esizoqala kuso ukubala ijika, futhi lo amplitude - yokuphakama kwawo.

Fourier uguqule (izibonelo ziboniswe esithombeni) iyithuluzi elinamandla kakhulu, okuyinto esetshenziswa ehlukahlukene. Kwezinye izimo, isetshenziswa njengoba isixazululo zibalo kunalokho eziyinkimbinkimbi ezichaza izinqubo ashukumisayo ezenzeka ngaphansi kwethonya ukukhanya, ukushisa noma amandla kagesi. Kwezinye izimo, it ikuvumela ukuba ichaze izingxenye ovamile waveforms eziyinkimbinkimbi, ngenxa kungaba njalo ukuze ahumushe kokuma ezihlukahlukene zokuhlola kwamakhemikhali, imithi nesayensi yokuhlolwa kwezinkanyezi.

Imininingwane zomlando

Umuntu wokuqala ukusebenzisa le ndlela isazi sezibalo French Zhan Batist Fure. Ukuguqulwa, kamuva aqanjwa ngemva kwakhe, ekuqaleni esasisetshenziselwa ukuchaza indlela ukushisa conduction. Fourier konke ukuphila kwakhe omdala wahlanganyela ukutadisha izakhiwo yokushisa. Yena negalelo elikhulu tiyori nezibalo ukuzimisela izimpande zibalo Aljibhra. Fourier waba uprofesa sokuhlaziya at the ecole Polytechnique, uNobhala we-Institute of Egyptology, enkonzweni yasebukhosini, okuyinto abangela isiyaluyalu ngesikhathi ukwakhiwa komgwaqo ukuze eTurin (ngaphansi kwesandla sakhe akhanywa engaphezu kuka-80 eziyinkulungwane amakhilomitha-skwele emaxhaphozini malarial). Nokho, konke lokhu yobushoshovu akuzange kuwumise usosayensi wahlanganyela izingabunjalo. Ngo 1802 elisuselwa i equation ukuthi uchaza esiphelweni ezifudumeza ukudla okuqinile. Ngo 1807, usosayensi wathola indlela ukuxazulula le equation, okuyinto yaziwa ngokuthi "Fourier uguqule".

ezishisayo conductivity ukuhlaziywa

Abacwaningi wasebenzisa indlela zezibalo ukuchaza indlela ukushisa conduction. Isibonelo elula, lapho kungekho nobunzima kathisha iyona Lokusakaza energy ezishisayo i indandatho yensimbi, enye ingxenye egxilile umlilo. Ukuze afeze ucwaningo Fourier obomvu ingxenye eshisayo esiyingini bamngcwaba esihlabathini esihle. Ngemva kwalokho, izilinganiso lokushisa eyenziwa engxenyeni okuphambene yalo. Ekuqaleni, ukusatshalaliswa ukushisa kuyinto okungajwayelekile: ingxenye ring - abandayo, kanye nezinye - ashisayo, phakathi esishintshashintshayo bangase bazibone ebukhali lokushisa umyela. Nokho, ngesikhathi ukusatshalaliswa ukushisa ngaphesheya kwamanzi metal, kuba iyunifomu ngaphezulu. Ngakho, ngokushesha, le nqubo ithatha ifomu igagasi sine. igrafu Okokuqala kancane kancane kwandisa futhi encipha ngokushelela, ngokunembile imithetho ukushintshashintsha cosine noma umsebenzi sine. Wave kancane kancane equalized futhi ngenxa yalokho izinga lokushisa iba iyunifomu kubo bonke ubuso esiyingini.

Umbhali le ndlela bazitshela ukuthi ukusatshalaliswa sokuqala ngempela ngendlela engafanele ukubola zibe eziningi amagagasi aphansi sine. Ngamunye wabo uyoba isigaba salo (isikhundla sokuqala) kanye lokushisa walo omkhulu. Ngakho ngalinye izinguquko ingxenye enjalo ukusuka ebuncaneni kuze kufike futhi emuva ukuqedela revolution emhlabeni izikhathi indandatho inamba. Uphiko kokuba isikhathi elibizwa ngokuthi harmonic ayisisekelo, kanye nenani nge nezikhathi ezimbili noma ngaphezulu - yesibili nokunye. Ngokwesibonelo, umsebenzi zezibalo ukuthi uchaza lokushisa esiphezulu, esigabeni noma isikhundla ebizwa ngokuthi Fourier uguqule umsebenzi wokusabalalisa. Usosayensi baletha ingxenye eyodwa okunzima incazelo etibalweni, okulula amathuluzi asebenziseka - imigqa sine futhi cosine, ngenani akunika ukusatshalaliswa kokuqala.

Ingqikithi yalo ukuhlaziywa

Ukusebenzisa le ukuhlaziywa ekuguqukeni ukushisa wokusabalalisa ku-eqinile, nokuba ukuma annular, isazi sezibalo babecabanga ukuthi okwandisa izinkathi izingxenye sinusoidal kuholele Damping yayo okusheshayo. Lokhu kubonakala ngokucacile phezu Harmonics main nelesibili. Amazinga yokugcina ifinyelela kabili amagugu benombolo kanye eliphansi pass olulodwa, futhi owokuqala - kanye kuphela. It kuvela ukuthi ibanga by ukushisa harmonic yesibili uhhafu ukuthi core. Ngaphezu kwalokho, umyela Engxenyeni yesibili beyoba steeper kulokuqala. Ngakho-ke, njengoba sisekhona ecucuza bazolokhu ezishisayo sihamba umfelokazi ibanga okuncane, bese lokhu uzobe damped harmonic ngokushesha kane kuka main, njengoba umsebenzi isikhathi. Esikhathini esilandelayo inqubo kuyoba ngisho ngokushesha. Sezibalo wayekholelwa ukuthi le ndlela kusivumela ukuba abale inqubo ukusatshalaliswa kokuqala lokushisa nge sikhathi.

Abantu besikhathi Call

Fourier uguqule algorithm uye ziba inselele izisekelo ezicatshangelwayo wezibalo ngaleso sikhathi. Ekuqaleni kwekhulu nesishiyagalolunye, ososayensi abaningi abavelele, kuhlanganise Lagrange, Laplace, Poisson, Legendre futhi Biot bengalamukeli nokugomela kwakhe ukuthi izinga lokushisa kwe-ukusatshalaliswa wokuqala ukubola zibe izingxenye ngesimo igagasi lokubalulekile imvamisa ephakeme. Nokho, Academy of Sciences awuzibe imiphumela eyatholakala sezibalo, futhi ukhishwa kuye Umklomelo yokuziphendukela ukushisa conduction imithetho, kanye kokuqhuba ukuqhathanisa sayo ucwaningo ngokomzimba. Esikhathini Fourier ndlela, ukuphikisana eyinhloko yokuthi umsebenzi discontinuous imelelwa isamba imisebenzi eminingana sinusoidal, okuyizinto okuqhubekayo. Phela, nabo basuke ukuchaza okuncane iqonde futhi lwalugobile imigqa. Usosayensi Contemporary wayengakaze ihlangabezane isimo, ezifana lapho imisebenzi discontinuous esachazwa inhlanganisela okuqhubekayo, ezifana quadratic, komugqa, sine noma Exhibitor. Esimweni lapho isazi sezibalo tye eziqinile zakhe, isamba i uchungechunge esingapheliyo imisebenzi Trigonometric kufanele igcine ijubane ngqo. Nakuba nxephezelo kwakubonakala aluhlaza cwe. Nokho, naphezu ukungabaza abanye abacwaningi (isib Claude Navier, Sofi Zhermen) titfutfukiswe kwenkululeko ucwaningo wasikhipha wokuhlaziywa ukushisa ukusatshalaliswa. A izibalo, Okwamanje, aqhubeka nokuhlukumezeka umbuzo wokuthi isizumbulu imisebenzi eminingana sinusoidal kuyehla kube i ingukubengezela kwenkazimulo yakhe Ekhala.

emlandweni weminyaka engu-200

Lo mbono uye savela eminyakeni engamakhulu amabili edlule, namhlanje it is ekugcineni kwakhiwa. Ngosizo imisebenzi ngekwendzawo noma yesikhashana ayahlaselwa izingxenye sinusoidal ukuthi abe imvamisa, isigaba kanye kwawo namandla awo. Lokhu ukuguqulwa etholwe izindlela ezimbili ezahlukene zezibalo. Eyokuqala wabo 'eyasetshenziswa endabeni lapho umthombo umsebenzi okuqhubekayo, kanti eyesibili - in the icala lapho emelelwa sebuningini izinguquko ngokuqhubekayo ngabanye. Uma inkulumo ethi is etholakala amagugu, okuyinto kuchazwa ngezikhawu ezehlukene, kungashiwo ihlukaniswe eziningana ezehlukene amaza sinusoidal izinkulumo - kusukela aphansi bese kabili, kathathu, njalo njalo ngaphezu eliyisisekelo. Le mali ibizwa ngokuthi i-Fourier chungechunge. Uma inkulumo ethi kokuqala usetha ukubaluleka inombolo ngayinye real, kungaba kuphihlizwe zibe amaningi sinusoidal zonke amaza kungenzeka. Ubizwa ngokuthi i-Fourier ebalulekile, futhi isinqumo kusho noshintsho umsebenzi ebalulekile. Kungakhathaliseki indlela yokuthola ukuguqulwa, ngoba imvamisa ngamunye kufanele bakhombise izinombolo ezimbili: amplitude futhi imvamisa. Lamagugu kuvetwa ngokuqhubekayo njengenkathi eyodwa inombolo eziyinkimbinkimbi. Isengezo eziguquguqukayo eziyinkimbinkimbi theory kanye Fourier ukuguqulwa ukwenza izibalo avunyelwe ukuklanywa ezifundeni ezihlukahlukene kagesi, ukuhlaziywa kunyakaza mechanical, ukutadisha wave esiphelweni indlela nomunye.

Fourier uguqule namuhla

Namuhla, lolu cwaningo yale nqubo ngokuyisisekelo yangempela ekutholeni izindlela eziphumelelayo ngoba aphuma umsebenzi ukuguqula it emuva engqondweni. Lolu hlelo olubizwa ngokuthi Fourier eqondile ephambene uguqule. Kusho ukuthini? Ukuze sithole ebalulekile futhi wenze Fourier oqondile uguqule, ungasebenzisa izindlela zezibalo, kodwa ungakwazi ahlaziye. Naphezu kweqiniso lokuthi lapho asetshenziswa umkhuba kukhona ubunzima obuthile, integrals kakhulu kakade ziye zatholakala futhi afakwe wabahlinzeki zezibalo. Ngamazwi ngosizo izindlela zezinombolo ingabalwa, ukuma ogxile idatha zokuhlola, umsebenzi kabani integrals kumathebula angekho, futhi kunzima ukucabanga ngendlela kohlaziyo.

Ngaphambi kokufika izibalo computer ubunjiniyela kugucuka abanjalo baye eyisicefe kakhulu, zidinga ukubulawa ibhukwana inani elikhulu izibalo imisebenzi encike inani lamaphuzu achaza umsebenzi wave. Ukuze iwenze ube lula settlement namuhla, zikhona izinhlelo ezikhethekile, bavunyelwe nokuqalisa amasha nezindlela kohlaziyo. Ngakho, ngo-1965, Dzheyms uKuli futhi Dzhon Tyuki wadala software waziwa ngokuthi "Fast Fourier Ukuguqula". It kusindisa isikhathi ukubala ngokunciphisa isibalo multiplications e ukuhlaziywa ijika. "Fast Fourier Ukuguqula" Indlela kusekelwe sokuhlukanisa ijika ibe inombolo enkulu iyunifomu isampula amagugu. Ngakho, inani multiplications kuyehla ngesigamu ngesikhathi esifanayo sinciphisa inani lamaphuzu.

Ukusebenzisa Fourier uguqule

Le nqubo isetshenziswa emikhakheni ehlukahlukene: Ngo inombolo theory, i-physics, signal processing, combinatorics, okungenzeka theory, indlela yokubhala efihla incazelo, izibalo, Oceanography, yokukhanya, umsindo, nezinye amajiyomethri. amathuba Rich ukusetshenziswa kwaso zisekelwe eziningi izici ewusizo, ezibizwa ngokuthi "izindawo ushintsho-Fourier." Ake sihlole kubo.

1. Umsebenzi ukuguqulwa kuyinto opharetha lwento kanye ukuzinza okuhambisanayo kuyinto ebumbeneko. Le mpahla is njengoba Parseval theorem ezaziwayo, noma esimweni jikelele, theorem Plansherelja noma Pontrjagin kwenza.

2. Ukuguqula esethola. Ngaphezu kwalokho, umphumela obuhlukile iyona ukuma esifana njengasezinsukwini oqondile ekhuluma.

3. sinusoidal izinkulumo eziyisisekelo zingezakho imisebenzi yabo umahluko. Lokhu kusho ukuthi ukumelwa ezinjengalezi ziyashintsha zibalo lwento ne okuza njalo endaweni algebraic ezivamile.

4. Ngokomfanekiso "convolution" theorem, inqubo kwenza ukusebenza eyinkimbinkimbi ukubuyabuyelela aphansi.

5. leyehlukene Fourier Ukuguqula kungenziwa ngokushesha yakhelwe kukhompyutha usebenzisa "fast" indlela.

Ukwehlukana the Fourier uguqule

1. Ngokuvamile leli gama lisetshenziswa ukuze libhekisele kumuntu ukuguqulwa okuqhubekayo, ukuhlinzeka noma yiliphi quadratically integrable njengoba isamba eziyinkimbinkimbi Inkulumo lomchazi nge amaza ethize emincane futhi amplitudes. Lolu hlobo has izinhlobo eziningana ezahlukene, okungenzeka okuza ezahlukene njalo. Indlela okuqhubekayo kuhlanganisa ithebula ukuguqulwa, okuyinto ingatholakala wabahlinzeki zezibalo. Ikesi generalized yilona ukuguqulwa wamaqhuzu, lapho le nqubo bangavuselwa wayefisa bamandla wangempela.

2. Indlela okuqhubekayo kuwukuqhathaniswa nje we amasu angaphambili Fourier uchungechunge esichaziwe iyiphi imisebenzi gqwa noma izinkulumo, okuyinto akhona endaweni elinganiselwe futhi zimelela nabo njengoba uchungechunge sinusoids.

3. leyehlukene Fourier uguqule. Le ndlela isetshenziswa Computing ngenxa kathisha yezesayensi nasezinzuzweni digital signal processing. Ukuze afeze lokhu uhlobo ukubala liyadingeka abe umsebenzi yokunquma ku isethi ngokuqhubekayo amaphuzu ngabanye, esifundeni ezikhathini ezithile noma elinganiselwe esikhundleni okuqhubekayo Fourier integrals. Esiginali ukuguqulwa kulokhu umelwe isamba sinusoids. Ukusetshenziswa "fast" indlela evumela ukusetshenziswa izixazululo digital ima-.

4. Iwindi Fourier uguqule umbono generalized indlela yakudala. Ngokungafani izixazululo ejwayelekile lapho bohlu isignali isetshenziswa, okuyinto esithathwayo uhla eligcwele ubufakazi bokuba khona kwalesi variable athakazelisa ngokukhethekile lapha kuphela wendawo imvamisa ukusatshalaliswa ebe egcina variable yokuqala (isikhathi).

5. I-Fourier mgudumbili uguqule. Le ndlela isetshenziswa ukuba asebenze afanayo iyimidwebo idatha. Esimweni esinjalo, ukuguqulwa wenziwa kwelinye direction, bese - kwesinye.

isiphetho

Namuhla, indlela Fourier lizimelele ku ehlukahlukene. Ngokwesibonelo, ngo-1962 wavula ukuma DNA Helix double usebenzisa Fourier ukuhlaziywa ngokuhlanganyela-X-ray umumo womsebenzi. zinhlayiya kwakamuva igxile Imicu yotshani DNA, okuholela isithombe okuyinto etholwe umumo womsebenzi, aqoshiwe eyayiphathelene ifilimu. Lesi sithombe wanika ulwazi mayelana ukubaluleka amplitude ngokusebenzisa Fourier uguqule kule isakhiwo crystal. idatha Isigaba etholwe ngokuqhathanisa amakhadi DNA umumo womsebenzi ngemakhadi zisemaphepheni ukuhlaziywa efanayo amakhemikhali izakhiwo. Ngenxa yalokho, izazi zezinto eziphilayo ziye abuyiselwe crystal nesakhiwo - umsebenzi yokuqala.

Fourier uguqule ukudlala indima enkulu ekuhloleni Uma uwubuka usemkhathini, yemvelo ka Semiconductor izinto ne-plasma, microwave umsindo, Oceanography, radar, seismology neluhlolo wezokwelapha.