Fourier ukuguqula. Fast Fourier ukuguqula. Fourier ezahlukene ukuguqula

inguqu Fourier - yenguqu, ukunxulumana umsebenzi othile variable ngokwenene. Olu sebenziso kwenziwa lonke ixesha siyazi izandi ezahlukeneyo. Indlebe ivelisa oluzenzekelayo "ukubala", ezenza ezingqondweni zethu kuphela emva kokuba iimviwo candelo nemathematika eziphezulu. ukuva sitho inguqu yabantu owakha, apho isandi (isindululo eqhelekileyo vibrational amasuntswana ngendlela ilastiki medium, leyo ukusasaza ekwimo ngumtshangatshangiso kwindawo eqinileyo, engamanzi okanye sukube medium) esikhusela umqolo wamaxabiso ezilandelelanayo umgangatho nomthamo abanalo eziphakamileyo ezahlukahlukeneyo. Emva koku, ingqondo ejika ulwazi lonke isandi eziqhelekileyo.

Mathematical Fourier nokuguqula

Ukuguqulwa amaza isandi okanye iinkqubo enye lokungcangcazela (by wokunciphisa ukukhanya ulwandle isimo kuxhotyiwe obalaseleyo okanye elanga) inokwenziwa yaye kusetyenziswa iindlela zezibalo. Ngoko ke, ngokusebenzisa ezi ndlela, imisebenzi ekwayiyo ngokwazisa iinkqubo vibrational isethi components sinusoidal, ngamanye esimfaneleko ezigoso leyo emke ubuncinane ukuya kuma uze kwakhona ubuncinane, ufana nokulatyuza kolwandle. inguqu Fourier - umsebenzi inguqu leyo echaza isigaba okanye awo of sinusoid ngalinye ukuya kumaza ethile. Yesigaba ibe sisiqalo le kwigophe, kunye namandla awo - yokuphakama kwawo.

Fourier ukuguqula (imizekelo ziboniswa kwi ifoto) sisixhobo esinamandla kakhulu, nto leyo esetyenziswa kwiinkalo ezahlukeneyo zenzululwazi. Kwezinye iimeko, lisetyenziswa njengesisombululo zibalo kunokuba entsonkothileyo echaza kwiinkqubo eziguquguqukayo ezenzeka phantsi kwempembelelo wokukhanya, ubushushu okanye amandla ombane. Kwezinye iimeko, it ikuvumela ukuba ukuchaza amacandelo rhoqo waveforms entsonkothileyo, ngenxa oku yinyaniso ukutolika imigqaliselo ezahlukeneyo zovavanyo chemistry, amayeza kunye neenkwenkwezi.

ulwazi zembali

Umntu wokuqala ukusebenzisa le ndlela sezibalo French Zhan Batist Fure. Ukuguquka, kamva babizwa emva kwakhe, esasifudula ukuchaza indlela ubushushu Uhanjiso. Fourier ubomi bakhe abadala lonke benza ukufunda iimpawu ubushushu. Waba negalelo elikhulu kwithiyori zemathematika ngokokuqingqwa esineengcambu quadratic algebra. Fourier waba uprofesa uhlalutyo kwi Ecole Polytechnique, uNobhala Institute of Egyptology, yaba inkonzo zethu, nto leyo ebangele inkathazeko ngexesha lokwakha kwendlela eya eTurin (phantsi kobunkokheli bakhe amanzi angaphezu kwe 80 ophosa sikwere imigxobhozo nemalariya). Noko ke, konke oku ntliziyo zange ayeke sisazinzulu abandakanyeke uhlalutyo zezibalo. Ngowe-1802 oko kususelwe inxaki echaza andise ubushushu eziqinileyo. Ngowe-1807, Scientist wafumanisa indlela ukuxazulula lalenxaki, leyo yaziwa ngokuba "Fourier ukuguqula".

analysis conductivity thermal

Abaphandi wasebenzisa indlela izibalo ukuchaza indlela ubushushu Uhanjiso. A Umzekelo elula, elingamiwe ubunzima intelekelelo na andise amandla thermal ngumntu umsesane yentsimbi, inxalenye enye antywiliselwa emlilweni. Ukwenza imifuniselo Fourier yinxalenye ezibomvu eshushu ring waza wamngcwaba entlabathini entle. Emva koko, imilinganiselo ubushushu olwenziwe phezu inxalenyeemelene yayo. Ekuqaleni, ukwabiwa ubushushu mgaqweni: yinxalenye ring - kuyabanda, kunye nezinye - shushu, phakathi imimandla nga bagcine lobushushu ithambeka elibukhali. Nangona kunjalo, ngexesha unikezelo ubushushu umphezulu metal liphela, uba iyunifom ngakumbi. Ngoko, kungekudala, le nkqubo ngokohlobo iliza sine. ngcembe igrafu Okokuqala kwandisa kwaye unciphisa kakuhle, ngokuchanileyo imithetho ukwehluka i cosine okanye umsebenzi sine. Wave ngcembe equalized yaye ngenxa yoko iqondo uba iyunifom kwi yonke kumphezulu khonkcweni.

Umbhali yale ndlela kucingelwa ukuba ukwabiwa zokuqala kakhulu engacwangciswanga kuba ibolile kwinani kwamaza nezakhiwo sine. Ngamnye kubo kuya kuba isigaba yayo (indawo lokuqala) kwaye iqondo layo eliphezulu. Ngoko ngalinye tshintsho elilolo hlobo ukusuka ubuncinane ukuya kuma uze ubuyele ukugqiba revolution ngeenxa amaxesha ring elipheleleyo. Component kokuba kwexesha ngokuba harmonic esisiseko, kunye nexabiso kunye iiphiriyodi ezimbini okanye nangaphezulu - yesibini njalo njalo. Umzekelo, umsebenzi lwezibalo ichaza lobushushu eliphezulu, isigaba okanye indawo ebizwa ngokuba Fourier okuguqula lomsebenzi lonikezelo. Scientist wazisa isakhi esinye kunzima inkcazelo izibalo, kuba izixhobo lula-to-ukusetyenziswa - imiqolo sine kunye cosine, xa lo mali enika unikezelo lokuqala.

Kakuhle eli yokuhlaziya

Ukusebenzisa olu ucazululo uguqulo lonikezelo ubushushu phezu into eqinileyo, nokuba imilo thunzi, a yezibalo waqiqa ukuba ukwanda kwamathuba zamacandelo sinusoidal kukhokelela damping yayo ekhawulezileyo. Oku ngokucacileyo kwi harmonics ezingundoqo yesibini. Iqondo lokugqibela ifikelela kabini amaxabiso kwelona qondo kwi yokupasa olunye, kunye lokuqala - kanye kuphela. Kubonakala ukuba umgama ohanjiweyo yi ubushushu kwi harmonic yesibini isiqingatha ongundoqo. Ukongeza, i-ithambeka kwesiqingatha sesibini iya kuba ayanyuka ngakumbi yokuqala. Ngoko ke, ekubeni wampompoza kakhulu ngakumbi thermal udlula umhlolokazi umgama omncinci, ngoko oku kuya damped harmonic izihlandlo ezine ngokukhawuleza ngaphezu engundoqo, njengomsebenzi ixesha. Kule ilandelayo le nkqubo iya kuba ngokukhawuleza. Sezibalo babekholelwa ukuba le ndlela ivumela ukuba ukubala inkqubo lonikezelo lokuqala lobushushu kunye nexesha.

Call ababephila

Fourier ukuguqula algorithm iye umngeni iziseko theoretical component kwimathematika ngelo xesha. Ngenkulungwane yokuqala yeshumi elinesithoba, ezininzi izazinzulu abadumileyo, kuquka आयडेंटीफिकेशन iSazisi, Laplace, Poisson, Legendre kunye Biot akazange ukwamkela lwalubonisa ukuba iqondo lobushushu lonikezelo lokuqala ibolile zibe kumacandelo ngohlobo lencinibakazi angundoqo kunye rhoqo ephakamileyo. Nangona kunjalo, i Academy of Sciences babengalibalanga ukuba iziphumo ezifunyenweyo sezibalo, yaye yanikwa naye Prize for kwithiyori ubushushu Uhanjiso imithetho, ngokunjalo ekuqhubeni uthelekiso yalo imifuniselo emzimbeni. Kule ndlela Fourier, xikhutazo iphambili kukuba umsebenzi discontinuous limelwe lodibaniso imisebenzi eziliqela sinusoidal, leyo gqolo. Ngapha koko, bachaza imigca caphukisayo tye kunye ukugoba. Isazinzulu Contemporary wayengazange wadibana kwimeko enjalo, xa imisebenzi discontinuous Zichazwe ngumyalelo indibanisela ngokuqhubekayo, ezifana equation ', yomgama, sine okanye exhibitor. Kwimeko yokuba yezibalo okulungileyo izityholo zakhe, isiphumo kungcelele engenasiphelo imisebenzi netrigonometri kufuneka kuphela isantya ngqo. Nangona ibango loo kwabonakala kwizinto ezitenxileyo. Noko ke, nangona amathandabuzo abanye abaphandi (umzekelo UClaude Navier, Sofi Zhermen) banzi umda uphando owabakhuphayo ekuhlalutyeni distribution ubushushu. A imathematika, elingumthombo, waqhubeka ubunzima umbuzo wokuba lodibaniso imisebenzi eziliqela sinusoidal kuyehla ukuba wenkqu ingxa-.

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Lo mbono uye savela kwiinkulungwane ezimbini, namhlanje ekugqibeleni kwakhiwa. Ngoncedo imisebenzi lwendawo okanye temporal baphuke zibe kumacandelo sinusoidal ukuba abe rhoqo, lesigaba kunye namandla awo. Le ukuguqulwa efunyanwa iindlela ezimbini zezibalo ezahlukeneyo. Eyokuqala kuzo isetyenziswa kwimeko xa umthombo ngumsebenzi eqhubekayo, kwaye eyesibini - kwimeko apho limelwe ezininzi kuThixo utshintsho ekhethekileyo ngamnye. Ukuba ibinzana ifunyenwe kwi amaxabiso, leyo ichazwe ngamathuba ekhethekileyo, oko lungohlulwa lube eziliqela obubobu ezitsho sinusoidal amabinzana - ukusuka kwanala koko kabini, kathathu, njalo njalo ngaphezu esisiseko. Le mali kuthiwa i Fourier series. Ukuba ibinzana lokuqala icwangcisa ixabiso inani lokwenene ngamnye, oko yaphulwe phantsi zibe sinusoidal zonke ezitsho ezininzi kunokwenzeka. Ibizwa ngokuba Fourier efunekayo, kwaye isigqibo kuthetha ukuba inguqu umsebenzi ezifunekayo. Kungakhathaliseki indlela zokufumana inguqu, ngokuba frequency ngamnye kufuneka uchaze amanani amabini: awo kunye rhoqo. Ezi nqobo ichazwa njenge elinye inani leqela. Nkulumo izinto ezintsonkothileyo theory kunye Fourier inguqu ukwenza izibalo ukuba ukuyilwa iziphaluka ezahlukeneyo zombane, uhlalutyo intlokoma mechanical, ukufunda mechanism wave ekusasazeni kunye nenye.

Fourier uguqule namhlanje

Namhlanje, isifundo yale nkqubo iindidi mbambano nokufumana iindlela ezisebenzayo ukulungiselela inguqu ukusuka umsebenzi ukuguqula ukusibuyisela engqondweni. Esi sisombululo kuthiwa Fourier ngqo kunye eguqulweyo ngokubaxwa i inguqu. Kuthetha ukuthini? Ukuze kumiselwe ebaluleke wenze Fourier ngqo inguqu, unako ukusebenzisa iindlela zemathematika, kodwa unako ahlaziye. Nangona yokuba xa asetyenziswe practice kukho ezinye iingxaki, uninzi integrals sele akubufumana wangena kwiincwadana zemathematika. Ngamazwi Ngoncedo iindlela wokwamanani kungabalelwa, imilo leyo ngokususela kwidatha zovavanyo, umsebenzi ogama integrals kwiitheyibhile ezingekhoyo, yaye kunzima ukuba uyicamngce ifomu yohlalutyo.

Ngaphambi kokufika yobalo zobunjineli computer utshintshwano baye ludinisayo kakhulu, zifuna uphumezo manual iqela elikhulu imisebenzi izibalo ukuthi kuxhomekeka kwinani lamanqaku ezichaza umsebenzi otshangatshangiswayo. Ukwenza lula yokuhlala namhlanje, kukho iinkqubo ezikhethekileyo, uvunyelwe ukusebenzisa ezintsha iindlela analytical. Ngoko ke, ngo-1965, Dzheyms uKuli kunye Dzhon Tyuki wadala software eyaba eyaziwa ngokuba yi "Fast Fourier Ukutshintsha". Oku konga ixesha ukubala ngokunciphisa inani multiplications ekuhlalutyeni ijika. "Fast Fourier Transform" Le ndlela isekelwe ngokwahlula kwigophe ibe inani elikhulu kumaxabiso isampula efanayo. Ngako oko, inani multiplications kuncitshiswa isiqingatha kwi elifanayo ukunciphisa inani lamanqaku.

Ukusebenzisa i Fourier ukuguqula

Le nkqubo isetyenziswa kwiinkalo ezahlukeneyo: In ithiyori Inombolo, physics, processing umqondiso, combinatorics, ithiyori ngokuqinisekileyo, zokufihlakeleyoComment, amanani, eliJongene, ngokukhanya, umsindo, kunye nezinye geometries. amathuba Rich ukusetyenziswa kwayo zisekelwe eziliqela iimpawu luncedo, ezo zibizwa ngokuba "iimpawu inguqu Fourier." Makhe sihlole kubo.

1. Umsebenzi yokuguqula umqhubi yomgama kunye ukuzinza ehambisana qela liphikisayo. Le propati kwaziwa Parseval theorem, okanye kwimeko jikelele, lo theorem Plansherelja okanye Pontrjagin ezimbini.

2. Le yokuguqula ingathityazwa. Ngaphezu koko, isiphumo esahlukileyo imilo kakhulu efanayo njengoko kwi ngqo dilesi.

3. Le amabinzana ezisisiseko sinusoidal imisebenzi yabo ngokwahlukeneyo. Oku kuthetha ukuba ukumelwa enjalo eguqula quadratic linear kunye okuza rhoqo ngendlela aljibra eqhelekileyo.

4. Ngokutsho "ngokwenqanawaComment" theorem, le nkqubo yenza umsebenzi ezintsonkothileyo-phinda aphantsi.

5. ngokuqhubekayo Fourier Transform ingagcinwa kwakhona ngokukhawuleza zenzelwe kwi kwikhompyutha usebenzise indlela "fast".

Ezahlukeneyo Fourier ukuguqula

1. ubukhulu becala eli gama lisetyenziswa ukubhekisela inguqu eqhubekayo, ngokubonelela nayiphi na imbonakalo quadratically integrable njengoko isixa yokuvakalisa ezintsonkothileyo okucacileyo kunye ezitsho ezithile emboxo ngezantya. Olu hlobo iye ngeendlela ezininzi ezahlukeneyo, esenokuba okuza ezahlukeneyo rhoqo. Indlela okuqhubekayo iquka itafile uguqulelo, lexi kumekaka kwiincwadana zemathematika. Ityala nolahlo ngokubanzi kwezintandane iqhezu apho le nkqubo kuvuswa ukuze igunya lokwenene ezinqwenelekayo,.

2. Indlela rhoqo isiphetho of ubuchule sangaphambili Fourier series lochaziweyo nayiphi na imisebenzi ngamaxesha athile okanye amabinzana, sezibona kwindawo limited yaye bamele babe uthotho sinusoids.

3. ngokuqhubekayo Fourier inguqu. Le ndlela isetyenziswa computing ngenxa intelekelelo zenzululwazi processing uphawu digital. Ukwenza olu hlobo yokubala kufuneka ukuba umsebenzi yovandlakanyo kwiisethi ezahlukene ze amanqaku ngamnye, ummandla ngamaxesha ingephi endaweni integrals Fourier rhoqo. ukuguqulwa uphawu kule meko uchazwa lodibaniso sinusoids. Ukusetyenziswa Indlela "fast" ivumela ukusetyenziswa kwezisombululo digital zonke iinjongo ezithile.

4. Le window Fourier ukuguqula imbono jikelele le ndlela zakudala. Ngokungafaniyo izisombululo umgangatho xa spectrum uphawu iyasetyenziswa, osuswe kuluhlu ngokupheleleyo ubukho kwalo mahluko umdla ethile apha kuphela unikezelo frequency lendawo lo gama kugcinwe variable yokuqala (ixesha).

5. Le Fourier okuntlandlo-mbini ukuguqula. Le ndlela iyasetyenziswa ukusebenza kwimiboniso emacala-mabini of data. Kwimeko enjalo, ukuguqulwa lwenziwa kwicala elinye, uze - kwezinye.

isiphelo

Namhlanje, indlela Fourier ngokuqinileyo elikhuselweyo kwiinkalo ezahlukeneyo zenzululwazi. Ngokomzekelo, ngowe-1962 oko kwavula imilo DNA kweleli kabini usebenzisa Uhlalutyo Fourier ngokusebenzisana ne-X-ray diffraction. ziikristale zakutsha swi kwimicu DNA, nto leyo umfanekiso efunyanwa diffraction, ebhalwe ngale filimu. Lo mfanekiso wanika ulwazi malunga ixabiso awo ngokusebenzisa i Fourier ukuguqula kwesi sakhiwo nomkhenkce. data Phase afunyenwe ngokuthelekisa amakhadi DNA diffraction ngoonotsheluza zifunyenwe ekuhlalutyeni izakhiwo kwemichiza efanayo. Ngenxa yoko, zebhayoloji wayibuyisela isakhiwo crystal - umsebenzi yokuqala.

Fourier ukuguqula kudlala indima enkulu ekufundeni emajukujukwini, nokwakheka izinto semiconductor plasma, nemisindo microwave, eliJongene, radar, ngezakhi kunye neemviwo zonyango.