Hauv kev ua lej thiab kev ua haujlwm, lub tswv yim ntawm kev ntsuas kev ntsuas (luv luv - C, AC) yog qhov ua haujlwm nyuaj uas tsis muaj qhov cuam tshuam tsis zoo. Qhov tseeb thiab qhov xav txog ntawm qhov tshwm sim no yog cov haujlwm tiag tiag cuam tshuam rau ib leeg los ntawm Hilbert hloov pauv. Cov teeb liab tsom xam yog ib qho tshwm sim zoo sib xws hauv chemistry, qhov tseem ceeb ntawm qhov zoo ib yam li lej txhais ntawm lub tswv yim no.
Kev ua haujlwm
Analytic sawv cev ntawm lub luag haujlwm tiag tiag yog lub teeb liab analytic uas muaj cov haujlwm qub thiab nws qhov hloov pauv Hilbert. Qhov kev sawv cev no pab txhawb ntau yam kev ua lej. Lub tswv yim tseem ceeb yog tias qhov tsis zoo zaus ntawm Fourier hloov pauv (lossis spectrum) ntawm kev ua haujlwm tiag tiag yog qhov tsis txaus ntseeg vim Hermitian symmetry ntawm xws li spectrum. Cov khoom siv tsis zoo no tuaj yeem muab pov tseg yam tsis muajpoob ntawm cov ntaub ntawv, muab hais tias koj xav mus nrog ib tug complex muaj nuj nqi xwb. Qhov no ua rau qee tus cwj pwm muaj txiaj ntsig ntau dua thiab ua rau nws yooj yim dua los muab cov kev hloov pauv thiab cov txheej txheem demodulation xws li SSB.
Neegative Cheebtsam
Tsuav qhov ua haujlwm raug tswj tsis muaj qhov cuam tshuam tsis zoo (piv txwv li nws tseem yog analytic), hloov los ntawm txoj kev rov qab mus rau qhov tseeb tsuas yog qhov teeb meem ntawm kev pov tseg qhov kev xav. Tus sawv cev analytic yog ib tug generalization ntawm lub tswv yim ntawm ib tug vector: thaum lub vector yog txwv rau lub sij hawm-invariant amplitude, theem, thiab zaus, ib tug zoo soj ntsuam ntawm ib tug analytic teeb liab tso cai rau lub sij hawm-varying parameters.
Instantaneous amplitude, instantaneous theem thiab zaus yog siv nyob rau hauv qee daim ntawv thov los ntsuas thiab kuaj xyuas cov yam ntxwv hauv zos ntawm C. Lwm daim ntawv thov ntawm tus sawv cev analytical muaj feem xyuam rau demodulation ntawm modulated signals. Polar tswj tau yooj yim cais qhov cuam tshuam ntawm AM thiab theem (lossis zaus) kev hloov kho thiab ua kom zoo demodulate qee yam.
Tom qab ntawd qhov yooj yim-kawg lim lim nrog cov coefficients tiag tiag tuaj yeem txiav tawm qhov kev txaus siab. Lwm qhov laj thawj yog txo qis qhov siab tshaj plaws, uas txo qis qhov tsawg zaus rau kev kuaj tsis yog npe. Kev hloov zaus tsis ua rau kev ua lej muaj txiaj ntsig ntawm tus sawv cev. Yog li, nyob rau hauv qhov kev txiav txim no, downconverted tseem analytic. Txawm li cas los xij, kev kho dua tshiab ntawm tus sawv cev tiagtsis yog ib qho teeb meem yooj yim ntawm kev tshem tawm cov khoom tiag tiag. Upconversion tej zaum yuav tsum tau, thiab yog hais tias lub teeb liab yog sampled (discrete time), interpolation (upsampling) kuj yuav tsum tau kom tsis txhob aliasing.
Variables
Lub tswv yim tau txhais tau zoo rau ib qho xwm txheej sib txawv, uas feem ntau yog ib ntus. Lub sijhawm no ua rau ntau tus lej pib ua lej. Rau ob lossis ntau qhov sib txawv, analytic C tuaj yeem txhais tau ntau txoj hauv kev, thiab ob txoj hauv kev tau nthuav tawm hauv qab no.
Qhov tiag thiab xav txog ntawm qhov tshwm sim no sib haum rau ob lub ntsiab lus ntawm vector-valued monogenic teeb liab, raws li tau hais tseg rau cov xwm txheej zoo sib xws nrog ib qho sib txawv. Txawm li cas los xij, monogenic tuaj yeem txuas ntxiv mus rau ntau qhov sib txawv ntawm qhov sib txawv hauv txoj kev yooj yim, tsim kom muaj (n + 1)-dimensional vector function rau rooj plaub ntawm n-variable signals.
Signal conversion
Koj tuaj yeem hloov lub teeb liab tiag tiag rau qhov kev tshuaj ntsuam xyuas los ntawm kev ntxiv qhov kev xav (Q) tivthaiv, uas yog Hilbert hloov pauv ntawm cov khoom tiag.
Los ntawm txoj kev, qhov no tsis yog qhov tshiab rau nws cov kev ua digital. Ib txoj hauv kev los tsim ib leeg sideband (SSB) AM, txoj kev phasing, suav nrog kev tsim cov teeb liab los ntawm kev tsim Hilbert hloov pauv ntawm lub suab teeb liab hauv analog resistor-capacitor network. Txij li thaum nws tsuas muaj qhov zoo zaus, nws yog ib qho yooj yim los hloov nws mus rau lub teeb liab RF modulated nrog tsuas yog ib sab.
Definition formulas
Kev ntsuas cov teeb liab qhia yog qhov ua haujlwm ntawm qhov kev ua haujlwm holomorphic uas tau hais tseg rau ntawm tus ciam teb ntawm lub dav hlau sab sauv ib nrab. Lub ciam teb ntawm lub sab sauv ib nrab ntawm lub dav hlau coincides nrog lub random, yog li C yog muab los ntawm daim ntawv qhia fa: R → C. Txij li thaum nruab nrab ntawm lub xyoo pua xeem, thaum Denis Gabor tau npaj nyob rau hauv 1946 los siv qhov tshwm sim los kawm tas li amplitude thiab theem., lub teeb liab tau pom ntau daim ntawv thov. Qhov peculiarity ntawm qhov tshwm sim no tau hais txog [Vak96], qhov uas nws tau pom tias tsuas yog ib qho kev soj ntsuam zoo ntawm cov teeb liab tsom xam raws li lub cev lub cev rau amplitude, theem thiab zaus.
Kev ua tiav dhau los
Lub sijhawm ob peb xyoos dhau los, tau muaj kev txaus siab rau kev kawm ntawm cov teeb liab hauv ntau qhov ntev, txhawb nqa los ntawm cov teeb meem tshwm sim hauv thaj chaw xws li kev ua duab / yees duab mus rau cov txheej txheem oscillatory multidimensional hauv physics, xws li seismic, electromagnetic thiab gravitational nthwv dej. Nws feem ntau tau lees paub tias, txhawm rau txhawm rau txheeb xyuas qhov tseeb C (kev soj ntsuam zoo) rau cov ntaub ntawv ntawm ob peb qhov ntev, ib tus yuav tsum tso siab rau kev tsim kho algebraic uas txuas ntxiv cov lej nyuaj hauv txoj hauv kev yooj yim. Cov kev tsim kho no feem ntau hu ua tus lej hypercomplex [SKE].
Thaum kawg, nws yuav tsum tsim kom muaj lub teeb liab hypercomplex analytic fh: Rd → S, qhov twg qee qhov kev qhia txog kev ua lej hypercomplex sawv cev, uas ib txwm txuas ntxiv tag nrho cov khoom xav tau kom tau txais ib qho amplitude instantaneous thiabPhase.
Study
Ntau cov ntaub ntawv tau mob siab rau ntau yam teeb meem ntsig txog kev xaiv kom raug ntawm tus lej hypercomplex, lub ntsiab lus ntawm hypercomplex Fourier transforms thiab fractional Hilbert transforms rau kev kawm tam sim no amplitude thiab theem. Feem ntau ntawm txoj haujlwm no yog ua raws cov khoom ntawm ntau qhov chaw xws li Cd, quaternions, Clearon algebras, thiab Cayley-Dixon kev tsim kho.
Tom ntej no, peb yuav sau tsuas yog qee qhov haujlwm uas tau mob siab rau kev kawm txog lub teeb liab hauv ntau qhov ntev. Raws li peb paub, thawj cov haujlwm ntawm txoj kev sib txawv tau txais thaum xyoo 1990s. Cov no suav nrog Ell txoj haujlwm [Ell92] ntawm hypercomplex transformations; Bulow txoj haujlwm ntawm kev ua kom dav dav ntawm txoj kev tshuaj ntsuam xyuas cov tshuaj tiv thaiv (kev ntsuas teeb liab) rau ntau qhov kev ntsuas [BS01] thiab kev ua haujlwm ntawm Felsberg thiab Sommer ntawm monogenic signals.
Kev cia siab ntxiv
Lub teeb liab hypercomplex yuav tsum tau txuas ntxiv tag nrho cov khoom muaj txiaj ntsig peb muaj nyob rau hauv 1D rooj plaub. Ua ntej ntawm tag nrho cov, peb yuav tsum muaj peev xwm rho tawm thiab generalize lub instantaneous amplitude thiab theem rau kev ntsuas. Qhov thib ob, Fourier spectrum ntawm cov teeb meem kev soj ntsuam nyuaj tsuas yog khaws cia ntawm qhov zoo zaus xwb, yog li peb cia siab tias qhov kev hloov pauv ntawm Fourier muaj nws tus kheej hypervalued spectrum, uas tsuas yog khaws cia hauv qee qhov zoo ntawm qhov chaw hypercomplex. Vim nws tseem ceeb heev.
Thib peb, sib txuas qhov chaw ntawm lub tswv yim nyuajntawm cov teeb liab analytic muaj feem xyuam rau Hilbert transforms, thiab peb tuaj yeem cia siab tias cov khoom sib txuas hauv qhov chaw hypercomplex kuj tseem cuam tshuam nrog qee qhov sib xyaw ntawm Hilbert transforms. Thiab thaum kawg, qhov tseeb, lub teeb liab hypercomplex yuav tsum raug txhais raws li qhov txuas ntxiv ntawm qee qhov kev ua haujlwm hypercomplex holomorphic ntawm ntau qhov sib txawv ntawm hypercomplex uas tau teev tseg ntawm ciam teb ntawm qee daim ntawv hauv qhov chaw hypercomplex.
Peb tab tom daws cov teeb meem no raws li kev txiav txim. Ua ntej tshaj plaws, peb pib los ntawm kev saib Fourier integral formula thiab qhia tias Hilbert hloov mus rau 1-D muaj feem xyuam rau cov qauv Fourier integral hloov. Qhov tseeb no tso cai rau peb txheeb xyuas qhov tam sim no amplitude, theem thiab zaus yam tsis muaj kev siv rau hypercomplex tooj systems thiab holomorphic functions.
Kev hloov pauv ntawm kev sib koom ua ke
Peb txuas ntxiv los ntawm kev txuas ntxiv qhov hloov pauv Fourier integral formula rau ntau qhov ntev, thiab txiav txim siab tag nrho cov theem tsim nyog hloov pauv uas peb tuaj yeem sau rau hauv qhov tam sim no amplitude thiab theem. Qhov thib ob, peb tig mus rau lo lus nug ntawm lub hav zoov ntawm holomorphic functions ntawm ob peb hypercomplex variables. Tom qab [Sch93] nws hloov tawm hais tias cov commutative thiab associative hypercomplex algebra generated los ntawm ib txheej ntawm elliptic (e2i=−1) generators yog ib qhov chaw haum rau ib tug hypercomplex analytic teeb liab nyob, peb hu xws li ib tug hypercomplex algebra lub Schaefers qhov chaw thiab denote. nwsSd.
Yog li ntawd, qhov hypercomplex ntawm kev ntsuas kev ntsuas tau txhais tau tias yog qhov ua haujlwm holomorphic ntawm tus ciam teb ntawm polydisk / sab sauv ib nrab ntawm lub dav hlau hauv qee qhov chaw hypercomplex, uas peb hu ua Schaefers qhov chaw dav dav, thiab txhais los ntawm Sd. Peb mam li saib xyuas qhov siv tau ntawm Cauchy ib qho qauv mis rau cov dej num Sd → Sd, uas yog xam hla ib qho hypersurface hauv ib lub polydisk hauv Sd thiab muab cov feem sib thooj Hilbert transforms uas cuam tshuam cov hypercomplex conjugate Cheebtsam. Thaum kawg, nws hloov tawm tias Fourier hloov pauv nrog cov txiaj ntsig hauv Schaefers qhov chaw tau txais kev txhawb nqa tsuas yog ntawm cov zaus tsis zoo. Ua tsaug rau cov lus no, koj tau kawm dab tsi yog qhov kev ntsuas ntsuas.
