
apurnank fourier rupantaran

ganitamadhye, swarmelatmak vishleshanachya kshetrat, apurnank fourier rupantaran (eff. aar. eff. tii.) heey forier parivartan samanya karnare resheey parivartananche eka kutumb aahe. yacha vichar 'aine'-th powermadhye forier transform mhanoon keela jaau shakato, jithe 'aine' haah purnank asanyachi garaj nahee-asha prakare, teey vel aanee varanvaratedarmyan konatyahi 'madhyavarti' domenmadhye functionche rupantar karoo shakate. tyache anuprayog filter rachana aanee signal vishleshanapasun teey tappa punarprapti aanee namuna olakhanyaparyant aahet.
eff. aar. eff. tii. chaa vapar fractional convolution, parasparsambandh aanee itar operations paribhashit karnyasathi keela jaau shakato aanee pudhe resheey canonical transformationmadhye (ell. see. tii.) samanyikrut dekhil kele jaau shakate. eff. aar. eff. tii. chee prarambhik vyakhya kandonidware, fages-space rotationsathi greenchya karyasathi nirakaran karoon aanee namiyasdware dekhil, hermite bahupadanvaril winerchya karyala samanya karoon sadar kelee gelee.
tathapi, 1993 chya sumaras anek gatanni swatantrapane punha sadar kareparyant signal prosesingmadhye yala mothya pramanat manyata midali navhati. tevhapasoon, fractional fourier domenmadhye band-maryadit asalelya sanketansathi shannonche sampling pramey vadhavanyat swarasya vadhale aahe.
fractional fourier transform" sathee eka purnapane vegala arth bellie aanee swartztrober yanni zed-transformsathi dusare naav mhanoon sadar keela hota aanee visheshtah frequency spacemadhil fractional rakmene (inputla resheey kilbilane gunakar karane) aanee frequency pointschya fractional sanchavar moolyankan (udaa.jee. spectrumcha fakt eka lahansa bhag vicharat ghya). (asha parivartananche moolyankan bluesteinchya eff. eff. tii. algoridamdware karyakshamatene kele jaau shakate.) hii sandnya bahutansh tantrik sahityat vaparat nahee, tathapi, eff. aar. eff. tii. chya tulanet. yaa lekhacha urvarit bhag eff. aar. eff. tii. chey varnan karato.
parichay.
function formula _ 2 chey satatche fourier transform formula _ 1 heey formula _ 3 spaceche ekatmak operator aahe jeey function formula _ 4 laa tyachya varanvar aavrutti formula _ 5 madhye mapp karate (serve abhivyakti binduvruttiaivaji formula _ 3 arthane ghetalya jaatat):
sootra _ 7
aanee sootra 4 heey inverse transform sootra 10 dware sootra 5 dware nirdharit kele jaate.
sootra _ 11
chala tyachya 'aine'-vya punaravrutti sutracha abhyas karuya _ 12 dware paribhashit 
sootra _ 13 aanee sootra _ 14 jevha "n" haah eka nakaratmak nasalela purnank asato aanee sootra _ 15. tyancha kram maryadit asato kaaran sootra _ 1 heey 4-aavadhik automorphism aaheh pratyek karya sootra _ 4 sathee, sootra _ 18.
adhik tantotant, aapan parity operator sootra _ 19 sadar karuya jeey sootra _ 20, sootra _ 21 ulate karate. mug khaleel gunadharm dharan karatatah
sootra _ 22
sootra _ 23
eff. aar. eff. tii. resheey parivartananche eka kutumb pradan karate jeey footchya 24 yaa purnank nasalelya shaktinche sootra hatalanyasathi yaa vyakhyecha vistar karate.
vyakhya.
teepah kahee lekhak 'coone' aivaji 'kramanusar' parivartan lihitat, asha paristhitit sahasa vel asate. heey done prakar jaree samatulya asale taree lekhak konati vyakhya vaparato yabaddal savadhgiri balgane aavashyak aahe.
konatyahi vastavik karyasathi, f yaa karyache-coone apurnank fourier rupantaran sootra _ 25 dware darshavile jaate aanee paribhashit kele jaate
aupcharikpane, heey sootra tevhach vaidh asate jevha input karya puresha chhan jaaget (jase qee l1 kinwa schwartz jaaga) asate aanee samanya babatit, samanya fourier transformpramanech (lekh pahaa), ghanata yuktivadadvare paribhashit kele jaate.
jarr π chaa purnank gunank asel, tarr vareel coatenjunt aanee cosenut functions bhinn hotat. tathapi, heey maryada gheuun hatalale jaau shakate aanee integrandmadhye diarac delta karya hote. adhik thet, sootra _ 26 pasoon 
teey anukrame saral kinwa summ kinwa visham gunankasathi asane aavashyak aahe.
kaaran, hii tantotant satatchya forier parivartanachi vyakhya banate aanee tyasathi ulat satatchya forier parivartanachi vyakhya aahe.
eff. aar. eff. tii. yuktivad haah avkashiy kinwa varanvarta nahee. donhi nirdeshankanche resheey sanyojan mhanoon yacha arth qaa lavala jaau shakato heey aapan pahoo. jevha aaplyala-angler fractional domenmadhye farak karayacha asel, tevha aapan sootra _ 27 laa sootra _ 28 chaa yuktivad darshavu.
tippanih frequency eka aivaji koneey frequency ω conventions, eff. aar. eff. tii. sootra mhanje mehar curnell,
sootra _ 29
malmatta.
- th aurder fractional fourier transform operator, sootra _ 28, madhye gunadharm aaheth
jodani.
konatyahi vastavik konansathi,
sootra _ 31
reshatmakata.
sootra _ 32
purnank aadesh.
jarr sootra _ 33 chaa purnank gunank asel, tarah
sootra _ 34
shivay, tyacha khaleel sambandh aaheh
sootra _ 35
ulat.
sootra _ 36
parasparsanvad.
sootra _ 37
sahsambandh.
sootra _ 38
ekatmata.
sootra _ 39
vel ulatat jaate.
sootra _ 40
sootra _ 41
shift kelelya karyache rupantar.
shift aanee fages shift operatorschi vyakhya khalilpramane karah
sootra _ 42
mug
sootra _ 43
mhanje,
sootra _ 44
scale kelelya karyache rupantar.
scaling aanee charp gunakar prachalakanchi vyakhya khalilpramane karah
sootra _ 45
mug, 
sootra _ 46
lakshat ghya qee sootra _ 47 chey apurnank fourier rupantaran sootra _ 48 chee scale keleli aavrutti mhanoon vyakt kele jaau shakat nahee. tyaeivaji, sootra _ 47 chey apurnank fourier rupantaran sootra _ 50 chee scale keleli aanee kilbill keleli aavrutti asalyache disun yete jithe sootra _ 51 hii eka vegali kramvari aahe.
apurnank curnell.
eff. aar. eff. tii. heey eka avibhajya parivartan aahe
sootra _ 52
jethe α-coone curnell aahe
sootra _ 53
yethe punha jevha gunankakade jaate tevha vishesh prakarane maryada vartanashi susangat asatat.
eff. aar. eff. tii. madhye tyachya guthalyapramanech gunadharm aaheth
sambandhit parivartan.
swatantra fourier transformsarkhya samaan transformchi sambandhit apurnank samanyikaran dekhil astitvat aahe.
samanyikaran.
forier transform moolatah bosonic aahe; teey karya karate kaaran teey superposition tattva aanee sambandhit hastakshep namunyanshi susangat aahe. eka fermionic fourier transform dekhil aahe. hyanche supersymmetric eff. aar. eff. tii. aanee supersymmetric redon transformmadhe samanyikaran karnyat aale aahe. eka apurnank redon rupantaran, eka symplectic eff. aar. eff. tii. aanee eka symplectic wavelet rupantaran dekhil aahe. quantum circuits ekatmak karyanvar aadharit asalyamule, teey avibhajya parivartananchya gananesathi upayukt aahet kaaran nantarche karya jaaget ekatmak prachalak aahet. eka quantum circuit tayar karnyat aale aahe jeey eff. aar. eff. tii. lagoo karate.
arth lavane.
forier transformcha nehmicha arth, time domain signalche frequency domain signalmadhye rupantar karane assa aahe. dusarikade, inverse fourier transformcha arth frequency domain signalche time domain signalmadhye rupantar karane assa aahe. fractional fourier transform signalche (ektar time domain kinwa frequency domenmadhye) time aanee frequency darmyanchya domenmadhye rupantar karateh heey time-frequency domenmadhil rotation aahe. haah drushtikon resheey canonical transformationdware samanyikrut keela jaato, joou fractional fourier transformla samanya karato aanee rotation vyatirikt vel-frequency domenchya resheey transformsla anumati deto.
udaharan mhanoon khaleel aakruti ghya. jarr vel kshetramadhil sanket aayatakruti asel (khalee dilyapramane), tarr teey varanvarta kshetramadhye sink karya banate. parantu jarr konee aayatakruti signalvar apurnank fourier transform lagoo kele tarr transformation output vel aanee varanvartechya darmyanchya domenmadhye asel.
apurnank fourier rupantaran heey vel-varanvarta vitarnavaril eka paribhraman karya aahe. vareel vyakhyenusar, "α" = 0 sathee, apurnank fourier rupantaran lagoo kelyanantar konatahi badal honar nahee, tarr "α" = "π"/2 sathee, apurnank fourier rupantaran eka sadhe fourier rupantaran banate, jeey vel-varanvarta vitaran "π"/2 neey firate. "α" chya itar mulyansathi, apurnank fourier rupantaran α nusar vel-varanvarta vitaran firavate. khaleel aakruti "α" chya vegvegalya moolyansah apurnank fourier parivartanache parinam darshavate.
arj.
apurnank fourier rupantaran velechya varanvartechya vishleshanat aanee dii. ess. pii. madhye vaparale jaau shakate. aavaj filter karane upayukt aahe, parantu too vel-varanvarta domenmadhil ichchit signalshi overlap honar nahee yaa attis. khaleel udaharan vicharat ghya. aavaj door karnyasathi aapan thet filter lavu shakat nahee, parantu fractional fourier transformchya madatine aapan pratham signal (ichchit signal aanee noiges) firavu shakato. tyanantar aapan eka vishisht filter lavto, joou keval ichchit signalla paas karoo deil. tyamule aavaj purnapane door hoil. mug aapan signal parat firavanyasathi punha fractional fourier transform vaparato aanee aaplyala ichchit signal miloo shakato.
asha prakare, time domenmadhye fakt kapanicha vapar karoon kinwa frequency domenmadhye samatulya loow-paas filter vaparoon, vel-frequency spacemadhil konatahi bahirmukh sanch kapla jaau shakato. yaulat, apurnank fourier rupantaranashivay time domain kinwa frequency domain sadhane vaparane keval akshanna samantar aayatakruti kapoon taknyas anumati deil.
apurnank fourier rupantarananche quantum bhautikshastratahi upayog aahet. udaharanarth, tyancha vapar entropic anishchitata sambandh tayar karnyasathi, ekal photons uchch-aayami quantum qee vitaran yojnanmadhye aanee photon jodyanche avkashiy guntagunt pahanyasathi keela jaato.
teey optical pranalinchya rachnet aanee holographic sathavan karyakshamta anukool karnyasathi dekhil upayukt aahet.
hehee pahaa.
itar vel-varanvarta badalah