Fourier hoʻololi. Fast Fourier hoʻololi. Discrete Fourier hoʻololi

Fourier hou '- loli, ÷ ino i kekahi hana o ka maoli ee iaaanu aey. Keia hana ua hana kela manawa kekahi mau mea kani-ke ike. Pepeiao produces akomi "i ka ho omaulia", i hoʻokō mākou ike paka i ke hope o ka ninaninau ana i ka māhele o ke kiekie makemakika wale. lohe hui, i loko o ke kanaka hou '? aiey iauaeoia, ma i ka leo (ʻike mau vibrational noi o (particles) i loko o ka laholio meakino, a hoʻopulapula ma ka palapala nalu i loko o ka paa, kinowai ole gaseous meakino) ua hōʻike' ia ma ka laulā o ka moekahi 'aiee o ka buke kiʻekiʻe o ka nanahe o ka loli iki ana kiekie. Mahope iho o keia, o ka lolo huli mai i ka 'ike i loko o nā mea a pau o ka ninau kani.

Makemakika Fourier hoʻololi

Huli ana o nä hawewe kani a me nā kamakawiwoʻole keʻano o ka hanaʻana (ma ka malamalama ea puka a me ka moana i ke kai, a me stellar a me ka ikehu lā pōʻaiapuni) hiki ke hana, a ma ka mea o ka makemakika ki ina hana like. Penei, ka hoʻohana 'ana i kēia mau' ana i kūpono, no ka mana hiki ke hoomaka ia e vibrational keʻano o ka hanaʻana i ka sinusoidal eiiiiiaiou, a laila wavy curves i hele, mai ka palena iki a hiki i ka i kā mākou, a laila, hou i ka palena iki, e like me ka ale o ke kai i hoʻolaunaʻia ma. Fourier hou '- hou' papa a wehewehe i ka pae a me ka ikaika hawewe o kela a me keia sinusoid e like me ka kekahi alapine (frequency). Pae mea he wahi ho omaka o ka hālawa, a me ka ikaika hawewe - o kona kiekie.

Fourier hoʻololi (examples i hōʻike i loko o ka Hawaii photo) mea he nui loa aku kou mana mea paahana, a ua hoʻohana 'ia i loko o kekahi mau kihapai o ka' epekema. Ma kekahi mau hihia, ka mea, ua hoʻohana 'ia e like me ka pāʻoihanaʻano luna' nā helu kaulike a wehewehe i ka hōʻeuʻeu keʻano o ka hanaʻana 'ia ma lalo o ka aoao o ka malamalama, ka wela a me ka uila ikehu. Ma nā hihia, ka mea e leie aku oe, e hoakaka regular eiiiiiaiou ma luna 'waveforms, ma muli o ia i kēia hiki eʻoiaʻiʻo, e hoomaopopo i kela mana loina ma kemika, lāʻau a me ka kilohōkū.

mōʻaukala 'ike

Ke kanaka mua e pili i kēia iaoia i ka Palani mathematician Zhan Batist Fure. Huli ana, a laila inoa ma hope ona, i hoʻohana mua, e wehewehe i ka wela conductionʻano papa hana. Fourier i kona holoʻokoʻa makua ola hana i loko o këia o ka waiwai o ka wela. He i kekahi, alahula na makana no ka makemakika pähola 'ana o ka manao paa o na aa o algebraic nā helu kaulike. Fourier ua he Kumu o Ka Ikepili ma ka Ecole Polytechnique, i ke Kakauolelo o ka Institute o Egyptology, ua ka imperial hana, a kena ae la i ka pioloke ma ka manawa o ke kūkulu o ka alanui a hiki i Turin (ma lalo o kona alakaʻiʻana i moni loa o ka oi mamua o 80 tausani kapuaʻi kuea kilomika o malarial Swamp). Naʻe, i kēia activism a pau i ole ana i ke nä känaka 'epekema komo i loko o makemakika Ka Ikepili. I ka 1802 ka mea, i loko o ka helu kaulike e wehewehe i ka propagation o ka wela i loko o ka'ā. I ka 1807, nä känaka 'epekema loaa i ka papa hana no ka hoʻoponopono i kēia helu kaulike, a i kapa ia "Fourier hoʻololi".

thermal conductivity Ka Ikepili

Noiʻi hoʻohana i ka makemakika iaoia e wehewehe i ka wela conductionʻano papa hana. A kupono laʻana, kahi i pilikia ma ka computation o ka propagation o thermal ikehu ma ka hao ke komo lima, kekahi hapa bapetizo i loko o ke ahi. E lawe mai ia mea Fourier ulaula wela loa o ke apo no ka ihu, a kanu aku la ia ia i loko o ke olona one. Mahope, lawe mai i ka wela ana ma luna o ka mea e ku pono ona. Initially, i ka wela o ka makana mea ole: kekahi hapa o ka apo - anu, a me ka mea 'ē aʻe - wela, ma waena o nā cii ke malama pono i ka pahi ana welaʻano piʻina. Naʻe, i ka wela o ka makana ma ka mea hiki ke hooheheeia luna, ka mea i oiʻaʻahu. No laila, koke, i kēia kaʻina i ke ano o ka sine nalu. First pakuhi i nā maoli a me emi smoothly, pololei na kanawai o ka oko o ka cosine a sine kuleana pili i. Wave i nā equalized, a like me ka hopena o ka mahana iʻaʻahu ma luna o ka holoʻokoʻa ili o ke apo hookahi.

I ka mea kākau o kēia iaoia hilahila i ka loiloi mua hoi o ka makana mea he ole hiki ke decomposed i loko o ka helu o ke kula haʻahaʻaa sine nalu. Kēlā me kēia no ia e i kona pae (loiloi mua wahi) a me kona i kā mākou ana wela. Pela kēlā me neia ke keʻena loli mai ka palena iki a hiki i ka i kā mākou, a hoʻi e hoʻokō 'kipi a puni ke apo helu manawa. Ke keʻena me ka manawa i i kapaia ka kahua harmonic, a me ka waiwai io me elua a oi manawa - i ka lua, a pela aku no. No ka laʻana, he makemakika papa e wehewehe i ka i kā mākou ana wela, i ka pae paha kulana i kapaʻia ka Fourier hoʻololi o ke kuleana pili i ka makana. Nä känaka 'epekema lawe mai i ka hookahi ke keʻena i ka paʻakikī i ka makemakika ano, no ka oluolu-i-ano hana - lalani o ka sine a me ka cosine, ma ka nui o ka haawi ana i ka loiloi mua hoi o ka makana.

Ke ae pā o ke Ka Ikepili

Ana i kēia Ka Ikepili a hiki i ka huli ana o ka wela o ka makana ma luna o ka mea paa e hoopii mai ai, me ka annular kinona, he mathematician kamailio aku la ia e hoʻoulu ai manawa o sinusoidal eiiiiiaiou alakai i kona kokololio damping. Keia ua akaka ai ma luna o ka papa a me ka lua o ka harmonics. Ka hope loa ana wela kīnā palua ka i kā mākou a me ka palena iki Hawaiʻi i loko o ka hookahi alahele, a ma ka mua - wale kekahi manawa. Ua huli mai i ka loihi ma ka wela i loko o ka lua o ka harmonic ka hapalua ia o ka Kora naʻEsau. Eia hou, i keʻano piʻina o ka lua o ka hapalua, e hoi e steeper ma mua o ka mea mua. Nolaila, mahope mai o ka hou, pili thermal koko māʻalo wahine kāne make ka palena haʻahaʻa mamao, a laila, i kēia, e ke pulu ka harmonic eha manawa ma mua o ka papa, e like me ka hana o ka manawa. Ma ka hahai ana i ke kaʻina hana e ia e wikiwiki. Mathematician manaoio i keia papa hana e leie aku mākou, e hoʻomaulia i ka kaʻina hana o ka loiloi mua hoi o ka makana o ke ana wela me ka manawa.

Call wā

Fourier hoʻololi algorithm ua lilo ia he hana nui loa i kaʻimi naʻauao kumu o makemakika i ka manawa. I ke kakahiaka nui umikumamaiwa kenekulia, koʻikoʻi loa,ʻepekema, a me Lagrange, Laplace, Poisson, Legendre a me Biot i oleʻae i kona assertion i ka wela o ka loiloi mua hoi o ka makana ua decomposed i loko o eiiiiiaiou i loko o ka palapala o ke kahua nalu, a kiekie ae alapine (frequency). Naʻe, i ka Academy o Sciences hiki ole kāpae'ōlelo i nā hualoaʻa loaa mathematician, a i hookoia aku la ia ia i ka lua i ka kumumanaʻo o ka wela conduction o nā kānāwai, e like me ka pono me ka hoʻokō 'kona e like me ka physical ho okolohua. Ma ka Fourier hoea mai, i ka papa kuhikuhiE kue o ka mea ana i discontinuous kuleana pili i ua poe ma ka puu dala o kekahi mau sinusoidal oihana, a mea ana. Ma hope o nā mea a pau, e hōʻike i ka hikiʻana mai pololei a me ka uma laina. Contemporary nä känaka 'epekema i ole pilikia ia ke kulana, i ka wa a ka discontinuous oihana i ho'ākāka' ia ma kekahi hui malu pu ana o ke ana, e like me quadratic, laina, sine paha exhibitor. I ka hanana i kekahi mathematician ua pololei ia i loko o kona mau assertions, i ka huina o ka mana loa moʻo o trigonometric oihana e e kaupalena 'ia ka kiko'ī māmā. Ia ia i kekahi kuleana mea absurd. Naʻe, hoʻowahāwahā i nā mea pohihihi o kekahi noiʻi (e like Claude Navier, Sofi Zhermen) broadened ka laulā o ka noiʻi, a lawe mai iā lākou mai loko mai o ka Ikepili o ka wela o ka makana. A makemakika, ia manawa, hoomau iho la oia e ae aku i ka ninau o ina paha i ka huina o kekahi mau sinusoidal oihana ua hoemi ia i ka mau e hoohalike ai o ka hikiʻana mai.

200-makahiki mōʻaukala

Kēia kumumanaʻo i Hawaiʻi ma nā kenekulia, i kēia lā ka mea, ua hope i hanaʻia. Me ke kōkua o nā māhele a me ke kino oihana e uhaʻiʻia i loko o sinusoidal eiiiiiaiou i i ke alapine (frequency), pae a me ka ikaika hawewe. Ua huli ana ua loaa ia ma nā okoa makemakika ki ina hana like. ua i ka mua o ia ia i loko o ka hihia i ka wa a ke kumu o ke ana papa, a me ka lua - i loko o ka hihia ma kahi e ua poe ma ka ai iiiaanoaa o discrete kanaka hou. Inā ka 'aoʻao o ua loaa mai ka waiwai, a i ho'ākāka' ia ma discreteʻanuʻu, ka mea hiki ke maheleia i loko o kekahi mau discrete sinusoidal nä alapine, ka 'aoʻao - mai ka lalo, a laila, papalua, pākolu, a pela aku ma luna o ka paepae. He nui ua kapaia ka Fourier moʻo. Inā i ka loiloi mua 'aoʻao o nā pūʻulu i ka waiwai o kēlā me kēia maoli helu, ka mea hiki ke wāwahi iho i mau sinusoidal a pau e hiki ai nä alapine. Ua ua kapaʻia he Fourier integral, a me ka olelo hooholo hoʻohuʻu i ka hou 'ana o ka integral kuleana pili i. Nānā 'ole o ka papa hana no ka loaa hou, no kela a me keia alapine (frequency) e wehewehe mau huahelu: ikaika hawewe a me ka alapine (frequency). Mau loina hoʻohikiʻia me he hookahi luna 'helu. 'Aoʻao o eiiieaena aiaiiuo kumumanaʻo pu me Fourier loli, e ho ae i ka manao o na uila i puhi ai, o ka Ikepili o ÿano kuekueni, ka study o ka nalu propagationʻano papa hana a me kekahi.

Fourier hoʻololi i keia la

Kēia manawa, ke study o kēia kaʻina holo nōhie maʻi hēhē ia i loaa ia ka hoʻokō epekema no ka hoʻololi mai ka hana, e hoohuli ia hoʻi i ka manaʻo. Ua kākoʻo pāʻoihana ua kapaia ka pololei a me ka inverse Fourier hoʻololi. He aha aʻole ia ke ano o? I mea e ho oholo i ka integral , a e ma ke ala Fourier hoʻololi, e hiki ke hoʻohana makemakika ki ina hana, akā, e hiki analytic. I loko nō o ka mea i ka wā e e hoʻohana i ka hoʻomaʻamaʻa, aia nō kekahi pilikia, ka hapanui integrals i mua, ua loaʻa, a komo aku la i loko o makemakika Akanahe. Me ke kōkua o laulā ano hiki ke pōpilikia, ka 'aoʻao, ke kinona o ka i ua ma muli o ka mana ikepili, he kuleana pili i kona integrals i loko o ka papa i nalowale, a me ka mea, ua hiki ole i ka manao wale ana i ka mākau palapala.

Ma mua o ka Hekikai o ke kamepiula 'enekinia, ho mau loli' i Ua loa hooluhi, ka mea e noi mai ana e manual hooko ana o ka helu nui o ka makemakika hana e hilinaʻi ana ma ka helu o nā kumu e wehewehe i ke kuleana pili i ka nalu. E kōkua i ka hooponopono i kēia lā, he mea kūikawā papa hana, 'ae' ia? Hou mākau ki ina hana like. No laila, i ka 1965, Dzheyms Kuli a me Dzhon Tyuki hana lako polokalamu i lilo i kapa ia "Fast Fourier hoʻololi". Ua hoopakele i ka manawa o ka i ka ho omaulia ma ka hoemi i ka helu ana o multiplications i loko o ka Ikepili o ka hālawa. "Fast Fourier hoʻololi" ka papa hana ua nānā 'ana i ka mahele ana i ka hālawa i loko o ka helu nui o nāʻaʻahu hāpana loina. Alaila, ua hoemiia ka helu ana o multiplications ma ka hapalua ma ka ia hoemi i ka helu o nā wahi.

Ana i ka Fourier hoʻololi

Kēia kaʻina ua hoʻohana 'ia i loko o nā mahinaʻai: Ma ka helu kumumanaʻo, ke kālaikūlohea, hoailona aaioee, combinatorics, pahiki kumumanaʻo, cryptography,' ikepili helu, oceanography, optics, Acoustic, a me nā geometries. Rich nā papaha no kona hoʻohana i ka nānā 'ana ma luna o ka helu o ka mea pono keia mau hiʻona, i ka poe i kaheaia "waiwai o ka Fourier loli." E mākou huli ia.

1. Ka papa huli mea he laina Aʻole a me ka AYEIE normalization mea unitary. Keia waiwai ua ikeia e like me ka Parseval theorem, a ma ka mea mau hihia, o ka theorem Plansherelja a Pontrjagin dualism.

2. Ka huli ana Ua reversible. A, ka mea ku pono hopena ua 'ano like kinona like i loko o ka pololei kamaʻilio.

3. Na sinusoidal walaʻauʻana, ka 'aoʻao nō kā lākou mau differentiated hana. Kēia 'o ia hoʻi i ka mea e hoohalike ai hoʻololi laina nā helu kaulike me ka mālama coefficients i loko o kaʻike mau algebraic.

4. E like me ka "convolution" theorem, i ke kaʻina hana o ka luna 'ana i loko o kula haʻahaʻaa multiplication.

5. Discrete Fourier hoʻololi hiki ke koke papahana ma ka lolouila e hoʻohana nei i ka "hookeai" hana.

Likeʻole o ka Fourier hoʻololi

1. Oiaio, he pinepine ka manawa ua hoʻohana 'ia e hoʻohuli i i ke ana hou', ina i kekahi quadratically integrable olelo e like me ka huina o ka luna 'exponentialʻana me kekahi angular nä alapine a me amplitudes. Kēia? Ueo loaʻa paha kekahi mau kekahi ano, a hiki e okoa ikaika coefficients. Ke ana iaoiaeii ka loaʻa o ka huli ana papaaina, a e loaʻa i loko o makemakika Akanahe hiki. A generalized hihia o ka fractional huli ana, hiki ia hiki ke hoala mai i kēia kaʻina hana i ka makemake maoli mana.

2.ʻO ana iaoiaeii mea he generalization o keʻano hana mua o Fourier puke i ho'ākāka 'ia no kekahi papa'āweʻaweʻa kemika oihana paha, ka' aoʻao, a nei i loko o ka kaupalena wahi, a kū no ia me he moʻo o sinusoids.

3. Discrete Fourier hoʻololi. Kēia papa hana ua hoʻohana 'ia i loko o Me no ka' epekema computation a kikohoʻe neaiae aaioee. E lawe mai i kēia 'ano o ka ho omaulia ua koi' ia i ka hana o ka hoomaopopo ana i kekahi discrete lākou o kanaka kumu, papa'āweʻaweʻa kemika paha i kaupalena 'māhele kahi o ke ana Fourier integrals. Neaiae huli ana i loko o keia hihia ua poe like me ka huina o nā sinusoids. Ka hoʻohana o ka "hookeai" iaoia e leie aku ka hoʻohana 'ana o kikohoʻe pāʻoihana no nā mea a pau he mea hiki mau hana.

4. Na puka makani Fourier hoʻololi mea he generalized nānaina o ka Classic ina hana like. Haʻalele maʻamau pāʻoihana ka wā o ka hōʻailona pae ua hoʻohana 'ia, a ua laweʻia i loko o ka piha huahelu o ke ola o keia ee iaaanu aey mea a pau hoihoi' aneʻi mea wale nō i ka kūloko alapine puʻunaue ana ka malama ana i ka palapala ee iaaanu aey (manawa).

5. Na mau-dimensional Fourier hoʻololi. Kēia papa hana ua hoʻohana 'ia no ka hana me ka mau-dimensional huihui ikehu lā o kaʻikepili. Ma ia i ka hihia, ai ka huli ana ua hana i kekahi kauoha, a laila, - i loko o ka 'ē aʻe.

ka hopena

I keia la, i ka Fourier iaoiaeii ua paa entrenched i loko o nā kula likeʻole o ka 'epekema. No ka laʻana, i ka 1962 ia wehe i ke kinona o ka DNA papalua helix ka hoʻohana 'ana Fourier Ka Ikepili i loko o ka huipūʻana me ka X-Ray diffraction. Nā nā pōakaʻeleu ma DNA olona, kūpono ana ma ke kiʻi a ua loaa mai ma ka diffraction, kākau ma o ka ho film. Kēia kiʻi i ike e pili ana i ka waiwai o ka ikaika hawewe ma ka hoʻohana 'ana i ka Fourier hoʻololi i kēia aniani' ole. Paeʻikepili loaa ma ke kapakai a ka DNA diffraction pepa me ka pepa i i loaa i loko o ka Ikepili o like Nine Inch hale. E like me ka hopena, biologists hoihoi aniani 'ole - ka palapala kuleana pili i.

Fourier hoʻololi pāʻani i ka nui kūlana i loko o ke ao o waho kahua, na ke kālaikūlohea o ka mea uholo uila mea a me ka wai koko, hawewe Acoustic, oceanography, Radar, seismology a me ka 'inikua examinations.