Fourier canza. Fast Fourier canza. Mai hankali Fourier fasalin

Fourier canji - canji, shirkin wani aiki na wani real m. Wannan aiki da aka yi a duk lokacin da muka gane daban-daban sautuna. Kunnen samar atomatik "lissafi", wanda cika mu sani iya kawai bayan jarrabawa daga cikin sashe na mafi girma lissafi. ji sashin jiki a wani mutum canji gina, a cikin abin da sauti (na al'ada vibrational motsi na barbashi a wani na roba matsakaici, wanda propagate a kalaman form a cikin m, ruwa ko gaseous matsakaici) da aka bayar a cikin kewayon jere dabi'u na girma matakin na sautunan mai sãɓãwar launukansa a Heights. Bayan wannan, a cikin kwakwalwa ya jũya da bayanai zuwa duk da saba sauti.

Ilmin Lissafi Fourier fasalin

Hira da sauti tãguwar ruwa ko wasu vibration matakai (da haske watsi da teku komowar ruwa da kuma gagarumi ko hasken rana hawan keke) za a iya yi, kuma ta hanyar ilmin lissafi hanyoyin. Saboda haka, yin amfani da wadannan dabaru, da ayyuka za a iya fadada ta hanyar gabatar vibrational matakai kafa na sinusoidal aka gyara, Ina nufin wavy masu lankwasa da tafi daga m to a kalla sa'an nan kuma zuwa m, kamar kalaman na teku. Fourier canji - canji aiki wanda ya bayyana cikin lokaci ko mawadãta daga kowane sinusoid m zuwa wani mita. Phase ne masomin da kwana, da kuma mawadãta - na da tsawo.

Fourier fasalin (misalai suna nuna a cikin photo) ne mai matukar iko kayan aiki, wanda aka yi amfani a fannoni daban daban na kimiyya. A wasu lokuta, shi ne amfani da wani bayani m hadaddun lissafai wanda bayyana tsauri matakai abin da ke faruwa a ƙarƙashin rinjayar haske, zafi ko na'urar samar da makamashi. A wasu lokuta, shi ba ka damar ayyana yau da kullum da aka gyara a hadaddun waveforms, saboda wannan zai iya zama gaskiya ya fassara daban-daban gwaji lura a cikin sunadarai, magani da kuma ilmin taurari.

tarihi bayanai

Na farko mutum a yi amfani da wannan hanya ya Faransa lissafi Zhan Batist Fure. Chanza, baya mai suna bayan shi, aka asali amfani da su bayyana kadan madugu inji. Fourier dukan balagaggun rai tsunduma a nazarin da kaddarorin cikin zãfi. Ya sanya wani babban taimako ga ilmin lissafi ka'idar tabbatar da dalilin da tushen da algebraic lissafai. Fourier ya farfesa bincike a Ecole Polytechnique, Sakataren Cibiyar Egyptology, shi ne na sarki sabis, wanda ya sa a dama a lokacin da gine na hanya zuwa Turin (a karkashin jagorancin aka drained na fiye da 80 dubu murabba'in kilomita na zazzabin cizon sauro swamps). Duk da haka, duk da wannan fafatukar bai dakatar da masanin kimiyyar tsunduma a ilmin lissafi analysis. A 1802 an samu wani lissafi da ya bayyana da yaduwa da zafi a cikin daskararru. A 1807, masanin kimiyya ya gano wani hanya domin magance wannan lissafi, wanda ya zama sananne a matsayin "Fourier canza".

thermal watsin analysis

Masu bincike amfani da wani ilmin lissafi Hanyar bayyana kadan madugu inji. A m misali, a cikinta babu wata wahala a ƙidãyar ne yaduwa daga thermal makamashi da wani baƙin ƙarfe zobe, wani sashi na nutsa cikin wata wuta. Domin gudanar da gwaje-gwajen da Fourier ja zafi ɓangare na zobe da kuma rufe shi a cikin lafiya yashi. Bayan nan, da zazzabi ma'aunai da za'ayi a kan kishiyar sãshensa. Da farko, da zafi rarraba ne sababbu: ɓangare na zobe - sanyi, da kuma sauran - zafi, tsakanin bangarori iya tsayar kaifi zazzabi dan tudu. Duk da haka, a lokacin zafi rarraba a fadin karfe surface, shi ya zama mafi uniform. Saboda haka, nan da nan, wannan tsari daukan wani nau'i na ba tare da kalaman. Da farko jadawali hankali qara da kuma rage-rage smoothly, daidai da dokokin bambancin da cosine ko ba tare da aiki. Kalaman hankali ya farke da kuma a sakamakon da zazzabi zama uniform a kan dukan surface na zobe.

Marubucin wannan Hanyar zaci cewa da farko rarraba ne quite sababbu za a iya bazu zuwa cikin wani adadin na farko ba tare da taguwar ruwa. Kowace daga cikinsu za su yi ta zamani (na farko matsayi) da matsakaicin yawan zafin jiki. Kamar haka kowane irin bangaren canje-canje daga m to a kalla, kuma baya don kammala juyin juya halin a kusa da zobe lamba sau. Bangaren ciwon lokaci wanda aka kira da muhimman hakkokin masu jituwa, da darajar da biyu ko fiye lokaci - na biyu da sauransu. Alal misali, a ilmin lissafi aiki da ya bayyana matsakaicin yawan zafin jiki, da lokaci ko wuri da ake kira Fourier fasalin na rarraba aiki. Scientist kawo guda bangaren cewa da wuya a ilmin lissafi description, domin sauki-to-amfani da kayayyakin aiki - layuka na ba tare da kuma cosine, a cikin adadin ba da farko rarraba.

Jigon da bincike

Bin wannan bincike da hira da zafi rarraba a kan m abu, da ciwon wani annular siffar, wani lissafi muhawwara cewa, kara lokaci na sinusoidal aka gyara kai ta zuwa ga m damping. Wannan a fili gani a kan babban da na biyu harmonics. A karshe zazzabi ya kai sau biyu, matsakaicin kuma m dabi'u a cikin guda izinin tafiya, kuma a cikin na farko - sau ɗaya kawai. Sai dai itace cewa da nesa tafiya da zafi a karo na biyu masu jituwa rabin abin da na core. Bugu da kari, da dan tudu na biyu da rabi zai zama steeper fiye da na farko. Saboda haka, tun a mafi tsanani thermal juyi wuce bazawara kadan nesa, to, wannan za a damped masu jituwa hudu sau da sauri fiye da main, kamar yadda wani aiki na lokaci. A cikin wadannan da tsari zai zama ko da sauri. Lissafi yi imani da cewa wannan hanya yale mu mu lissafi da tsari na farko rarraba zafin jiki da lokaci.

kira Sahaban

Fourier fasalin algorithm ya zama kalubale ga msar tambayar tushe na lissafi a lokacin. A farkon karni na sha tara, mafi shahararren masana kimiyya, ciki har da Lagrange, Laplace, Poisson, Legendre da Biot bai yarda da hujjarsa da cewa yawan zafin jiki na farko da rarraba ne bazu zuwa cikin gyara a cikin nau'i daga cikin muhimman hakkokin kalaman kuma mafi girma mita. Duk da haka, da Academy of Sciences iya ba watsi da sakamakon samu lissafi, da kuma bayar da shi da Prize ga ka'idar zafi a aiwatar da dokokin, kazalika da gudanar da ta kwatanta da jiki gwaje-gwajen. A Fourier m, babban ƙin yarda ne da cewa wani aiki ne discontinuous wakilta wani ware Naira Miliyan Xari da dama sinusoidal ayyuka, wanda suke ci gaba da. Bayan duk, suka bayyana da yadda nake alfaharin fa mike kuma mai lankwasa Lines. Littafi masanin kimiyyar ya taba ci karo da irin wannan halin da ake ciki, a lokacin da discontinuous ayyuka aka bayyana ta a hade da m, kamar quadratic, mikakke, ba tare da ko exhibitor. A cikin taron cewa wani lissafi ya dama a cikin da'awoyinsu, Naira Miliyan Xari da wani iyaka jerin trigonometric ayyuka da ya kamata a iyakance zuwa daidai gudun. Duk da yake irin wannan iƙirari da jũna m. Duk da haka, duk da wannan shakku daga wasu masu bincike (misali Claude Navier, Sofi Zhermen) ta fadada ta da ikon yinsa, na gudanar da bincike da kuma fitar da su daga bincike na zafi rarraba. A lissafi, A halin yanzu, ci gaba da fama da tambayar ko a ware Naira Miliyan Xari da dama sinusoidal ayyuka an rage wa wani ainihin kamanni na bursting.

200-shekara tarihi

Wannan ka'idar ya samo asali kan ƙarni biyu, a yau shi ne a karshe kafa. Tare da taimakon da na sarari ko na boko ayyuka suna karya cikin sinusoidal aka gyara da cewa suna da wani mita, lokaci da kuma mawadãta. Wannan hira da aka samu ta hanyar biyu daban-daban ilmin lissafi hanyoyin. A farko daga gare su da ake amfani a yanayin saukan lokacin da tushen ne a ci gaba da aiki, da kuma na biyu - a cikin akwati inda aka wakilta wani jam'i na mai hankali mutum canje-canje. Idan magana da aka samu daga dabi'u, wanda aka tsare a hankali jinkiri da shi za a iya raba da dama mai hankali sinusoidal mitoci maganganu - daga cikin mafi ƙasƙanci sa'an nan ninki biyu, tripled, da sauransu sama da muhimman hakkokin. Wannan adadin da aka kira da Fourier jerin. Idan da farko magana buga darajar kowane real lambar, za a iya karya saukar zuwa mahara sinusoidal duk yiwu mitoci. Shi ne ake kira mai Fourier integral, da kuma yanke shawara yakan haifar da wani canji na na game aiki. Ko da kuwa da hanya domin samun canji, domin kowane mita kamata nuna biyu lambobi: amplitude da mita. Wadannan dabi'u suna bayyana a matsayin guda hadaddun lambar. Zancen hadaddun canji ka'idar tare da Fourier canji yin lissafin yarda da zane na daban-daban da wutar lantarki haihuwarka, da bincike na inji vibrations, nazarin kalaman yaduwa inji da wani.

Fourier canza a yau

A zamanin yau, nazarin wannan tsari m tafasa saukar zuwa gano tasiri hanyoyin da miƙa mulki daga aiki don maida shi baya zuwa hankali. Wannan bayani ne da ake kira kai tsaye da kuma kishiya Fourier canza. Abin da ake nufi? Domin sanin da na game da yin wani kai tsaye Fourier canza, ba za ka iya amfani da ilmin lissafi hanyoyin, amma za ka iya nazari. Duk da cewa a lokacin da suke amfani da su a yi akwai wasu matsaloli, mafi integrals an riga an samu da kuma shiga cikin ilmin lissafi handbooks. Da taimakon lamba hanyoyin za a iya lasafta maganganu, da siffar wanda dogara ne a kan gwaji data, wani aiki wanda integrals a cikin alluna suna bace, kuma sun kasance wuya su yi tunanin a cikin wani analytical form.

Kafin zuwan kwamfuta injiniya lissafin irin canje sun sosai tedious, suka bukatar manual kisan wani babban yawan ilmin lissafi ayyukan da suke dogara a kan yawan maki cewa bayyana kalaman aiki. Don sauƙaƙe shiri a yau, akwai na musamman shirye-shirye, a yarda da su aiwatar da sabon hanyoyin bincikar su. Saboda haka, a shekarar 1965, Dzheyms Kuli da Dzhon Tyuki halitta software cewa zama da aka sani a matsayin "Fast Fourier canza". Yana ceton lokacin da lissafi da rage yawan multiplications a cikin bincike na kwana. "Fast Fourier canza" The hanyar dogara ne a kan rarraba kwana a cikin wani babban yawan uniform samfurin dabi'u. Haka kuma, yawan multiplications an rage zuwa rabin at guda rage yawan maki.

Da ake ji da Fourier fasalin

Wannan tsari ne da amfani a fannoni daban daban: A lambar ka'idar, kimiyyar lissafi, siginar aiki, combinatorics, Yiwuwar ka'idar, cryptography, statistics, oceanography, kimiyyan gani da hasken wuta, acoustics, da kuma sauran geometries. Rich yiwuwa ga ta yin amfani da su ne bisa wani yawan amfani fasali, wanda ake kira "kaddarorin da Fourier canji." Bari mu auna su.

1. A yi hira aiki ne mikakke sadarwarka kuma a daidai daidaita ne unitary. Wannan dukiyar da aka sani a matsayin Parseval Theorem, ko a cikin general yanayin, da Theorem Plansherelja ko Pontrjagin dualism.

2. A hira ne reversible. Bugu da ƙari, m sakamakon haka ne ma irin wannan siffar kamar yadda a cikin kai tsaye magance.

3. A sinusoidal asali maganganu ne nasu bambancin ayyuka. Wannan yana nufin cewa, irin wannan misali ya canjãwa mikakke lissafai tare da m coefficients a wani al'ada algebraic.

4. Bisa ga "convolution" Theorem, da aiwatar da ke sa wani hadadden aiki a yan multiplication.

5. hankali Fourier canza za a iya sauri tsara a kan kwamfuta ta yin amfani da "sauri" Hanyar.

Bambance-bambance na Fourier fasalin

1. Mafi sau da yawa ajalin da aka amfani da su koma zuwa ga ci gaba da canji, samar da wani quadratically integrable magana a matsayin Naira Miliyan Xari da hadaddun karuwa da sauri magana da takamaiman kusurwa mitoci, kuma amplitudes. Wannan nau'in na da dama daban-daban siffofin, wanda zai iya zama daban-daban akai coefficients. A ci gaba da hanyar hada da wani hira tebur, wanda za a iya samu a ilmin lissafi handbooks. A jimlace hali ne fractional hira, inda wannan tsari za a iya tayar da su da ake so real iko.

2. A ci gaba da hanya ne hakan na baya dabara na Fourier jerin tsare domin wani lokaci-lokaci ayyuka ko maganganu, wanda ya wanzu a wata iyaka yankin da kuma wakiltar su a jerin sinusoids.

3. hankali Fourier canza. Wannan hanya da aka yi amfani da sarrafa kwamfuta for kimiyya ƙidãyar da sigina na dijital aiki. Domin gudanar da irin wannan lissafi ne da ake bukata domin samun aiki na kayyade a kan wani mai hankali sa na mutum maki, lokaci-lokaci, ko iyakance yankin maimakon ci gaba da Fourier integrals. Signal hira a wannan harka da aka wakilta a matsayin mai Naira Miliyan Xari sinusoids. Da amfani da "sauri" Hanyar damar yin amfani da dijital da mafita ga duk m dalilai.

4. The window Fourier fasalin ne kowa yana da ra'ayi na classic Hanyar. Ba kamar misali mafita sa'ad da alamar hayaƙin bakan da ake amfani, wanda aka dauka a cikin cikakken kewayon wanzuwar wannan m ne na musamman sha'awa a nan shi ne kawai da na gida mita rarraba yayin da rike asali m (lokaci).

5. The biyu-girma Fourier canza. Wannan hanya da ake amfani da su yi aiki tare da biyu-girma iri-iri na bayanai. A irin wannan yanayin, da hira da aka yi a daya shugabanci, sa'an nan - a cikin wasu.

ƙarshe

Yau, da Fourier Hanyar Yã daidaitu kunshe a fannoni daban daban na kimiyya. Alal misali, a shekarar 1962 shi ya bude siffar na DNA biyu Helix amfani Fourier bincike a tare da X-ray diffraction. Recent lu'ulu'u mayar da hankali a kan DNA zaruruwa, sakamakon wani image wanda aka samu ta hanyar diffraction, da ke rubuce a kan film. Wannan hoto ya bayani game da darajar da mawadãta ta yin amfani da Fourier canza zuwa wannan crystal tsarin. Phase data samu ta hanyar kwatanta DNA diffraction katunan tare da katunan cewa an samu a cikin bincike na kama sinadaran Tsarin. A sakamakon haka, masana kimiyyar mayar crystal tsarin - asali aiki.

Fourier fasalin wasa mai babbar rawa a cikin binciken na m sarari, da kimiyyar lissafi na semiconductor kayan da jini, obin na lantarki acoustics, oceanography, radar, seismology kuma likita Nazarin.