Quadrature Signals: Complex, But Not Complicated


 Scarlett Johnson
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1 Quadraur Signals: Complx, Bu No Complicad by Richard Lyons Inroducion Quadraur signals ar basd on h noion of complx numbrs and prhaps no ohr opic causs mor harach for nwcomrs o DSP han hs numbrs and hir srang rminology of jopraor, complx, imaginary, ral, and orhogonal. If you'r a lil unsur of h physical maning of complx numbrs and h j = 1 opraor, don' fl bad bcaus you'r in good company. Why vn Karl Gauss, on h world's gras mahmaicians, calld h jopraor h "shadow of shadows". Hr w'll shin som ligh on ha shadow so you'll nvr hav o call h Quadraur Signal Psychic Holin for hlp. Quadraur signal procssing is usd in many filds of scinc and nginring, and quadraur signals ar ncssary o dscrib h procssing and implmnaion ha aks plac in modrn digial communicaions sysms. In his uorial w'll rviw h fundamnals of complx numbrs and g comforabl wih how hy'r usd o rprsn quadraur signals. Nx w xamin h noion of ngaiv frquncy as i rlas o quadraur signal algbraic noaion, and larn o spak h languag of quadraur procssing. In addiion, w'll us hrdimnsional im and frquncydomain plos o giv som physical maning o quadraur signals. This uorial concluds wih a brif look a how a quadraur signal can b gnrad by mans of quadraursampling. Why Car Abou Quadraur Signals? Quadraur signal formas, also calld complx signals, ar usd in many digial signal procssing applicaions such as:  digial communicaions sysms,  radar sysms,  im diffrnc of arrival procssing in radio dircion finding schms  cohrn puls masurmn sysms,  annna bamforming applicaions,  singl sidband modulaors,  c. Ths applicaions fall in h gnral cagory known as quadraur procssing, and hy provid addiional procssing powr hrough h cohrn masurmn of h phas of sinusoidal signals. A quadraur signal is a wodimnsional signal whos valu a som insan in im can b spcifid by a singl complx numbr having wo pars; wha w call h ral par and h imaginary par. (Th words ral and imaginary, alhough radiional, ar unforuna bcaus hir of manings in our vry day spch. Communicaions nginrs us h rms inphas and quadraur phas. Mor on ha lar.) L's rviw h mahmaical noaion of hs complx numbrs. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
2 Th Dvlopmn and Noaion of Complx Numbrs To sablish our rminology, w dfin a ral numbr o b hos numbrs w us in vry day lif, lik a volag, a mpraur on h Fahrnhi scal, or h balanc of your chcking accoun. Ths ondimnsional numbrs can b ihr posiiv or ngaiv as dpicd in Figur 1(a). In ha figur w show a ondimnsional and say ha a singl ral numbr can b rprsnd by a poin on ha. Ou of radiion, l's call his, h. This poin rprsns h ral numbr a = . (j) This poin rprsns h complx numbr c =.5 + j (a) +j lin (b) lin Figur 1. An graphical inrpraion of a ral numbr and a complx numbr. A complx numbr, c, is shown in Figur 1(b) whr i's also rprsnd as a poin. Howvr, complx numbrs ar no rsricd o li on a ondimnsional lin, bu can rsid anywhr on a wodimnsional plan. Tha plan is calld h complx plan (som mahmaicians lik o call i an Argand diagram), and i nabls us o rprsn complx numbrs having boh ral and imaginary pars. For xampl in Figur 1(b), h complx numbr c =.5 + j is a poin lying on h complx plan on nihr h ral nor h imaginary. W loca poin c by going +.5 unis along h ral and up + unis along h imaginary. Think of hos ral and imaginary axs xacly as you hink of h EasWs and NorhSouh dircions on a road map. W'll us a gomric viwpoin o hlp us undrsand som of h arihmic of complx numbrs. Taking a look a Figur, w can us h rigonomry of righ riangls o dfin svral diffrn ways of rprsning h complx numbr c. b (j) c = a + jb M φ a Figur Th phasor rprsnaion of complx numbr c = a + jb on h complx plan. Our complx numbr c is rprsnd in a numbr of diffrn ways in h liraur, such as: Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
3 Noaion Nam: Rcangular form: Mah Exprssion: c = a + jb Rmarks: Usd for xplanaory purposs. Easis o undrsand. [Also calld h Carsian form.] Trigonomric c = M[cos(φ) + jsin(φ)] Commonly usd o dscrib form: quadraur signals in communicaions sysms. Polar form: c = M jφ Mos puzzling, bu h primary form usd in mah quaions. [Also calld h Exponnial form. Somims wrin as Mxp(jφ).] Magniudangl c = M φ Usd for dscripiv purposs, form: bu oo cumbrsom for us in algbraic quaions. [Essnially a shorhand vrsion of Eq. (3).] (1) () (3) (4) Eqs. (3) and (4) rmind us ha c can also b considrd h ip of a phasor on h complx plan, wih magniud M, in h dircion of φ dgrs rlaiv o h posiiv ral as shown in Figur. Kp in mind ha c is a complx numbr and h variabls a, b, M, and φ ar all ral numbrs. Th magniud of c, somims calld h modulus of c, is M = c = a + b (5) [Trivia qusion: In wha 1939 movi, considrd by many o b h gras movi vr mad, did a main characr amp o quo Eq. (5)?] Back o businss. Th phas angl φ, or argumn, is h arcangn of h raio imaginary par ral par, or φ = an 1 b a (6) If w s Eq. (3) qual o Eq. (), M jφ = M[cos(φ) + jsin(φ)], w can sa wha's namd in his honor and now calld on of Eulr's idniis as: jφ = cos(φ) + jsin(φ) (7) Th suspicious radr should now b asking, "Why is i valid o rprsn a complx numbr using Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
4 ha srang xprssion of h bas of h naural logarihms,, raisd o an imaginary powr?" W can valida Eq. (7) as did h world's gras xpr on infini sris, Hrr Lonard Eulr, by plugging jφ in for z in h sris xpansion dfiniion of z in h op lin of Figur 3. Tha subsiuion is shown on h scond lin. Nx w valua h highr ordrs of j o arriv a h sris in h hird lin in h figur. Thos of you wih lvad mah skills lik Eulr (or hos who chck som mah rfrnc book) will rcogniz ha h alrnaing rms in h hird lin ar h sris xpansion dfiniions of h cosin and sin funcions. z z z 3 z 4 z 5 = 1 + z +! + 3! + 4! + 5! + z 6 6! +... jφ (jφ) = 1 + jφ +! + 3 (jφ) 3! + (jφ) 4! 4 + (jφ) 5! 5 + (jφ) 6! jφ φ = 1 + jφ !  j φ3 3! + φ 4 φ 5 φ 6 4! + j 5!  6! +... = cos(φ) + jsin(φ) Figur 3 On drivaion of Eulr's quaion using sris xpansions for z, cos(φ), and sin(φ). Figur 3 vrifis Eq. (7) and our rprsnaion of a complx numbr using h Eq. (3) polar form: M jφ. If you subsiu jφ for z in h op lin of Figur 3, you'd nd up wih a slighly diffrn, and vry usful, form of Eulr's idniy: Th polar form of Eqs. (7) and (8) bnfis us bcaus: jφ jφ = cos(φ)  jsin(φ) (8)  I simplifis mahmaical drivaions and analysis,  urning rigonomric quaions ino h simpl algbra of xponns, and  mah opraions on complx numbrs follow xacly h sam ruls as ral numbrs.  I maks adding signals mrly h addiion of complx numbrs (vcor addiion),  I's h mos concis noaion,  I' s indicaiv of how digial communicaions sysm ar implmnd, and dscribd in h liraur. W'll b using Eqs. (7) and (8) o s why and how quadraur signals ar usd in digial communicaions applicaions. Bu firs, l s ak a dp brah and nr h Twiligh Zon of ha 'j' opraor. You'v sn h dfiniion j = 1 bfor. Sad in words, w say ha j rprsns a numbr whn muliplid by islf rsuls in a ngaiv on. Wll, his dfiniion causs difficuly for h bginnr bcaus w all know ha any numbr muliplid by islf always rsuls in a posiiv numbr. (Unforunaly DSP xbooks ofn dfin j and hn, wih jusifid has, swifly carry on wih all h ways ha h j opraor ca n b usd o analyz sinusoidal signals. Radrs soon forg abou h qusion: Wha dos j = 1 acually man?) Wll, 1 had bn on h mahmaical scn for som im, bu wasn' akn sriously unil i had o b usd o solv Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
5 cubic quaions in h sixnh cnury. [1], [] Mahmaicians rlucanly bgan o accp h absrac concp of 1, wihou having o visualiz i, bcaus is mahmaical propris wr consisn wih h arihmic of normal ral numbrs. I was Eulr's quaing complx numbrs o ral sins and cosins, and Gauss' brillian inroducion of h complx plan, ha finally lgiimizd h noion of 1 o Europ's mahmaicians in h ighnh cnury. Eulr, going byond h provinc of ral numbrs, showd ha complx numbrs had a clan consisn rlaionship o h wllknown ral rigonomric funcions of sins and cosins. As Einsin showd h quivalnc of mass and nrgy, Eulr showd h quivalnc of ral sins and cosins o complx numbrs. Jus as modrnday physiciss don know wha an lcron is bu hy undrsand is propris, w ll no worry abou wha 'j' is and b saisfid wih undrsanding is bhavior. For our purposs, h jopraor mans roa a complx numbr by 9 o counrclockwis. (For you good folk in h UK, counrclockwis mans aniclockwis.) L's s why. W'll g comforabl wih h complx plan rprsnaion of imaginary numbrs by xamining h mahmaical propris of h j = 1 opraor as shown in Figur 4. j8 = muliply by "j" j8 Figur 4. Wha happns o h ral numbr 8 whn you sar muliplying i by j. Muliplying any numbr on h ral by j rsuls in an imaginary produc ha lis on h imaginary. Th xampl in Figur 4 shows ha if +8 is rprsnd by h do lying on h posiiv ral, muliplying +8 by j rsuls in an imaginary numbr, +j8, whos posiion has bn road 9 o counrclockwis (from +8) puing i on h posiiv imaginary. Similarly, muliplying +j8 by j rsuls in anohr 9 o roaion yilding h 8 lying on h ngaiv ral bcaus j = 1. Muliplying 8 by j rsuls in a furhr 9 o roaion giving h j8 lying on h ngaiv imaginary. Whnvr any numbr rprsnd by a do is muliplid by j h rsul is a counrclockwis roaion of 9 o. (Convrsly, muliplicaion by j rsuls in a clockwis roaion of 9 o on h complx plan.) If w l φ = π/ in Eq. 7, w can say ha jπ/ = cos( π/) + jsin(π/) = + j1, or jπ/ = j (9) Hr's h poin o rmmbr. If you hav a singl complx numbr, rprsnd by a poin on h complx plan, muliplying ha numbr by j or by jπ/ will rsul in a nw complx numbr ha's road 9 o counrclockwis (CCW) on h complx plan. Don' forg his, as i will b Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
6 usful as you bgin rading h liraur of quadraur procssing sysms! L's paus for a momn hr o cach our brah. Don' worry if h idas of imaginary numbrs and h complx plan sm a lil mysrious. I's ha way for vryon a firs you'll g comforabl wih hm h mor you us hm. (Rmmbr, h jopraor puzzld Europ's havywigh mahmaicians for hundrds of yars.) Grand, no only is h mahmaics of complx numbrs a bi srang a firs, bu h rminology is almos bizarr. Whil h rm imaginary is an unforuna on o us, h rm complx is downrigh wird. Whn firs ncounrd, h phras complx numbrs maks us hink 'complicad numbrs'. This is rgrabl bcaus h concp of complx numbrs is no rally all ha complicad. Jus know ha h purpos of h abov mahmaical rigmarol was o valida Eqs. (), (3), (7), and (8). Now, l's (finally!) alk abou imdomain signals. R prsning Signals Using Complx Phasors OK, w now urn our anion o a complx numbr ha is a funcion im. Considr a numbr whos magniud is on, and whos phas angl incrass wih im. Tha complx numbr is h jπfo poin shown in Figur 5(a). (Hr h π rm is frquncy in radians/scond, and i corrsponds o a frquncy of cycls/scond whr is masurd in Hrz.) As im gs largr, h complx numbr's phas angl incrass and our numbr orbis h origin of h complx plan in a CCW dircion. Figur 5(a) shows h numbr, rprsnd by h black do, frozn a som arbirary insan in im. If, say, h frquncy = Hz, hn h do would roa around h circl wo ims pr scond. W can also hink of anohr complx numbr jπfo (h whi do) orbiing in a clockwis dircion bcaus is phas angl gs mor ngaiv as im incrass. j = im in sconds, = frquncy in Hrz j jπ jπ 1 φ = π 1 φ = πf o 1 φ = π 1 φ = πf o jπ jπ j j (a) (b) Figur 5. A snapsho, in im, of wo complx numbrs whos xponns chang wih im. L's now call our wo jπfo and jπfo complx xprssions quadraur signals. Thy ach m. Thos jπfo and jπ hav boh ral and imaginary pars, and hy ar boh funcions of i xprssions ar ofn calld complx xponnials in h liraur. W can also hink of hos wo quadraur signals, jπfo and jπfo, as h ips of wo phasors roaing in opposi dircions as shown in Figur 5(b). W'r going o sick wih his phasor noaion for now bcaus i'll allow us o achiv our goal of rprsning ral sinusoids in h conx of h complx plan. Don' ouch ha dial! Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
7 To nsur ha w undrsand h bhavior of hos phasors, Figur 6(a) shows h hrdimnsional pah of h jπfo phasor as im passs. W'v addd h im, coming ou of h pag, o show h spiral pah of h phasor. Figur 6(b) shows a coninuous vrsion of jus h ip of h jπfo phasor. Tha jπfo complx numbr, or if you wish, h phasor's ip, follows a corkscrw pah spiraling along, and cnrd abou, h im. Th ral and imaginary pars of jπfo ar shown as h sin and cosin projcions in Figur 6(b). ( j ) o o 9 Imag 11 jπf o sin(π ) 18 o o 736 o 1 Tim 1 Tim cos(π ) (a) (b) Figur 6. Th moion of h jπ phasor (a), and phasor 's ip (b). Rurn o Figur 5(b) and ask yourslf: "Slf, wha's h vcor sum of hos wo phasors as hy roa in opposi dircions?" Think abou his for a momn... Tha's righ, h phasors' ral pars will always add consrucivly, and hir imaginary pars will always cancl. This mans ha h summaion of hs jπfo and jπfo phasors will always b a purly ral numbr. Implmnaions of modrnday digial communicaions sysms ar basd on his propry! To mphasiz h imporanc of h ral sum of hs wo complx sinusoids w'll draw y anohr picur. Considr h wavform in h hrdimnsional Figur 7 gnrad by h sum of wo halfmagniud complx phasors, jπfo / and jπfo /, roaing in opposi dircions abou, and moving down along, h im. ( j ) 1 = cos(π ) jπ Tim Figur 7. A cosin rprsnd by h sum of wo roaing complx phasors. Thinking abou hs phasors, i's clar now why h cosin wav can b quad o h sum of jπ Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
8 wo complx xponnials by cos(π ) = jπ + jπ = j o πf + j π. (1) Eq. (1), a wllknown and imporan xprssion, is also calld on of Eulr's idniis. W could hav drivd his idniy by solving Eqs. (7) and (8) for jsin(φ), quaing hos wo xprssions, and solving ha final quaion for cos(φ). Similarly, w could go hrough ha sam algbra xrcis and show ha a ral sinwav is also h sum of wo complx xponnials as sin(π ) = jπ  j jπ = j jπ j jπ. (11) Look a Eqs. (1) and (11) carfully hy ar h sandard xprssions for a cosin wav and a sinwav, using complx noaion, sn hroughou h liraur of quadraur communicaions sysms. To kp h radr's mind from spinning lik hos complx phasors, plas raliz ha h sol purpos of Figurs 5 hrough 7 is o valida h complx xprssions of h cosin and sinwav givn in Eqs. (1) and (11). Thos wo quaions, along wih Eqs. (7) and (8), ar h Rosa Son of quadraur signal procssing. cos(π ) = jπ + jπ W can now asily ransla, back and forh, bwn ral sinusoids and complx xponnials. Again, w ar larning how ral signals, ha can b ransmid down a coax cabl or digiizd and sord in a compur's mmory, can b rprsnd in complx numbr noaion. Ys, h consiun pars of a complx numbr ar ach ral, bu w'r raing hos pars in a spcial way w'r raing hm in quadraur. R prsning Quadraur Signals In h uncy Domain Now ha w know much abou h imdomain naur of quadraur signals, w'r rady o look a hir frquncydomain dscripions. This marial will b asy for you o undrsand bcaus w'll illusra h full hrdimnsional aspcs of h frquncy domain. Tha way non of h phas rlaionships our quadraur signals will b hiddn from viw. Figur 8 lls us h ruls for rprsning complx xponnials in h frquncy domain. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
9 Ngaiv frquncy j jπ Dircion along h imaginary jπ  j Posiiv frquncy Magniud is 1/ Figur 8. Inrpraion of complx xponnials. W'll rprsn a singl complx xponnial as a narrowband impuls locad a h frquncy spcifid in h xponn. In addiion, w'll show h phas rlaionships bwn hos complx xponnials along h ral and imaginary axs. To illusra hos phas rlaionships, a complx frquncy domain rprsnaion is ncssary. Wih all ha said, ak a look a Figur cos(π ) Imag Par... Par Tim cos(π ) = jπ + jπ  Imag Par Par... sin(π ) Imag Par... Par Tim sin(π ) = j jπ  j jπ  Imag Par Par Figur 9. Complx frquncy domain rprsnaion of a cosin wav and sinwav. S how a ral cosin wav and a ral sinwav ar dpicd in our complx frquncy domain rprsnaion on h righ sid of Figur 9. Thos bold arrows on h righ of Figur 9 ar no roaing phasors, bu insad ar frquncydomain impuls symbols indicaing a singl spcral lin for singl a complx xponnial jπ. Th dircions in which h spcral impulss ar poining mrly indica h rlaiv phass of h spcral componns. Th ampliud of hos spcral impulss ar 1/. OK... why ar w bohring wih his 3D frquncydomain rprsnaion? Bcaus i's h ool w'll us o undrsand h gnraion (modulaion) and dcion (dmodulaion) of quadraur signals in digial (and som analog) communicaions sysms, and hos ar wo of h goals of his uorial. Bfor w considr hos procsss howvr, l's valida his frquncydomain rprsnaion wih a lil xampl. Figur 1 is a sraighforward xampl of how w us h complx frquncy domain. Thr w bgin wih a ral sinwav, apply h j opraor o i, and hn add h rsul o a ral cosin wav of h sam frquncy. Th final rsul is h singl complx xponnial jπ illusraing graphically Eulr's idniy ha w sad mahmaically in Eq. (7). Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
10 .5  Imag muliply by j  Imag sin(π ) .5 jsin(π ) Imag add Imag 1  cos(π ) jπ = cos(π ) + jsin(π ) Figur 1. Complx frquncydomain viw of Eulr's: jπ = cos(π ) + jsin(π ). On h frquncy, h noion of ngaiv frquncy is sn as hos spcral impulss locad a π radians/sc on h frquncy. This figur shows h big payoff: Whn w us complx noaion, gnric complx xponnials lik jπf and jπf ar h fundamnal consiuns of h ral sinusoids sin(πf) or cos(πf). Tha's bcaus boh sin(πf) and cos(πf) ar mad up of jπf and jπf componns. If you wr o ak h discr Fourir ransform (DFT) of discr imdomain sampls of a sin(π ) sinwav, a cos(π ) cosin wav, or a jπ complx sinusoid and plo h complx rsuls, you'd g xacly hos narrowband impulss in Figur 1. If you undrsand h noaion and opraions in Figur 1, pa yoursln h back bcaus you know a gra dal abou naur and mahmaics of quadraur signals. Bandpass Quadraur Signals In h uncy Domain In quadraur procssing, by convnion, h ral par of h spcrum is calld h inphas componn and h imaginary par of h spcrum is calld h quadraur componn. Th signals whos complx spcra ar in Figur 11(a), (b), and (c) ar ral, and in h im domain hy can b rprsnd by ampliud valus ha hav nonzro ral pars and zrovalud imaginary pars. W'r no forcd o us complx noaion o rprsn hm in h im domain h signals ar ral. signals always hav posiiv and ngaiv frquncy spcral componns. For any ral signal, h posiiv and ngaiv frquncy componns of is inphas (ral) spcrum always hav vn symmry around h zrofrquncy poin. Tha is, h inphas par's posiiv and ngaiv frquncy componns ar mirror imags of ach ohr. Convrsly, h posiiv and ngaiv frquncy componns of is quadraur (imaginary) spcrum ar always ngaivs of ach ohr. This mans ha h phas angl of any givn posiiv quadraur frquncy componn is h ngaiv of h phas angl of h corrsponding quadraur ngaiv frquncy componn as shown by h hin solid arrows in Figur 11(a). This 'conjuga symmry' is h invarian naur of ral signals whn hir spcra ar rprsnd using complx noaion. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
11 Quadraur phas (Imag.) Inphas () cos(π +φ)  φ (a) Quadraur phas (Imag.) Inphas () Complx xponnial Inphas (ral) par Quadraur phas (imaginary) par 1/ φ Quadraur phas (Imag.) Inphas ()  (b) Quadraur phas (Imag.) Inphas () B B B  (c)  (d) Figur 11. Quadraur rprsnaion of signals: (a) sinusoid cos(π + φ), (b) bandpass signal conaining six sinusoids ovr bandwidh B; (c) bandpass signal conaining an infini numbr of sinusoids ovr bandwidh B Hz; (d) Complx bandpass signal of bandwidh B Hz. L's rmind ourslvs again, hos bold arrows in Figur 11(a) and (b) ar no roaing phasors. Thy'r frquncydomain impuls symbols indicaing a singl complx xponnial jπf. Th dircions in which h impulss ar poining show h rlaiv phass of h spcral componns. Thr's an imporan principl o kp in mind bfor w coninu. Muliplying a im signal by h complx xponnial jπ, wha w can call quadraur mixing (also calld complx mixing), shifs ha signal's spcrum upward in frquncy by Hz as shown in Figur 1 (a) and (b). Likwis, muliplying a im signal by jπ shifs ha signal's spcrum down in frquncy by Hz. Quad. phas Inphas Quad. phas Inphas Quad. phas Inphas  (a)  (b)  (c) Figur 1. Quadraur mixing of a signal: (a) Spcrum of a complx signal x(), (b) Spcrum of x() jπ, (c) Spcrum of x() jπ. A QuadraurSampling Exampl W can us all ha w'v larnd so far abou quadraur signals by xploring h procss of quadraursampling. Quadraursampling is h procss of digiizing a coninuous (analog) bandpass signal and ranslaing is spcrum o b cnrd a zro Hz. L's s how his popular procss works by hinking of a coninuous bandpass signal, of bandwidh B, cnrd abou a carrir frquncy of f c Hz. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
12 Original Coninuous Spcrum f c X(f) B f c Dsird Digiizd "Basband" Spcrum f s X(m) f s (m) Figur 13. Th 'bfor and afr' spcra of a quadraursampld signal. Our goal in quadraursampling is o obain a digiizd vrsion of h analog bandpass signal, bu w wan ha digiizd signal's discr spcrum cnrd abou zro Hz, no f c Hz. Tha is, w wan o mix a im signal wih jπf c o prform complx downconvrsion. Th frquncy f s is h digiizr's sampling ra in sampls/scond. W show rplicad spcra a h boom of Figur 13 jus o rmind ourslvs of his ffc whn A/D convrsion aks plac. OK,... ak a look a h following quadraursampling block diagram known as I/Q dmodulaion (or 'Wavr dmodulaion' for hos folk wih xprinc in communicaions hory) shown a h op of Figur 14. Tha arrangmn of wo sinusoidal oscillaors, wih hir rlaiv 9 o phas, is ofn calld a 'quadrauroscillaor'. Thos jπf c and jπf c rms in ha busy Figur 14 rmind us ha h consiun complx xponnials comprising a ral cosin duplicas ach par of X bp (f) spcrum o produc h X i (f) spcrum. Th Figur shows how w g h filrd coninuous inphas porion our dsird complx quadraur signal. By dfiniion, hos X i (f) and I(f) spcra ar rad as 'ral only'. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
13 Coninuous Discr x () bp Coninuous spcrum Inphas coninuous spcrum cos(πf ) c sin(πf ) c o 9 x () i x q () f c LPF LPF X (f) i i() q() X bp (f) f s A/D A/D par i(n) q(n) f c jπf c jπf c complx squnc is: i(n)  jq(n) f f B/ +B/ c c f c f c B LP filrd inphas coninuous spcrum I(f) B/ B/ Filrd ral par Figur 14. Quadraursampling block diagram and spcra wihin h inphas (uppr) signal pah. Likwis, Figur 15 shows how w g h filrd coninuous quadraur phas porion our complx quadraur signal by mixing x bp () wih sin(πf c ). Coninuous spcrum f c X bp (f) B f c Quadraur coninuous spcrum par X (f) q f c f c f c Ngaiv du o h minus sign of h sin's f c j jπf Q(f) LP filrd quadraur coninuous spcrum B/ B/ Filrd imaginary par Figur 15. Spcra wihin h quadraur phas (lowr) signal pah of h block diagram. Hr's whr w'r going: I(f)  jq(f) is h spcrum of a complx rplica our original bandpass signal x bp (). W show h addiion of hos wo spcra in Figur 16. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
14 I(f) Filrd coninuous inphas (ral only) B/ B/ B/ Q(f) Filrd coninuous quadraur (imaginary only) B/ B/ I(f)  jq(f) Spcrum of coninuous complx signal: i()  jq() B/ Figur 16. Combining h I(f) and Q(f) spcra o obain h dsird 'I(f)  jq(f)' spcra. This ypical dpicion of quadraursampling sms lik mumbo jumbo unil you look a his siuaion from a hrdimnsional sandpoin, as in Figur 17, whr h j facor roas h 'imaginaryonly' Q(f) by 9 o, making i 'ralonly'. This jq(f) is hn addd o I(f). ( j ) ( j ) I(f) A hrdimnsional viw ( j ) Q(f) jq(f) ( j ) I(f)  jq(f) Figur D viw of combining h I(f) and Q(f) spcra o obain h I(f)  jq(f) spcra. Th complx spcrum a h boom Figur 18 shows wha w wand; a digiizd vrsion of h complx bandpass signal cnrd abou zro Hz. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
15 Spcrum of coninuous complx signal: i()  jq() B/ B/ This is wha w wand. A digiizd complx vrsion of h original x bp (), bu cnrd abou zro Hz. A/D convrsion Spcrum of discr complx squnc: i(n)  jq(n) f f B/ s f s B/ s f s Figur 18. Th coninuous complx signal i()  q() is digiizd o obain discr i(n)  jq(n). Som advanags of his quadraursampling schm ar:  Each A/D convrr opras a half h sampling ra of sandard ralsignal sampling,  In many hardwar implmnaions opraing a lowr clock ras sav powr.  For a givn f s sampling ra, w can capur widrband analog signals.  Quadraur squncs mak FFT procssing mor fficin du o a widr frquncy rang covrag.  Bcaus quadraur squncs ar ffcivly ovrsampld by a facor of wo, signal squaring opraions ar possibl wihou h nd for upsampling.  Knowing h phas of signals nabls cohrn procssing, and  Quadraursampling also maks i asir o masur h insananous magniud and phas of a signal during dmodulaion. Rurning o h block diagram rminds us of an imporan characrisic of quadraur signals. W can snd an analog quadraur signal o a rmo locaion; o do so w us wo coax cabls on which h wo ral i() and q() signals ravl. (To ransmi a discr imdomain quadraur squnc, w'd nd wo muliconducor ribbon cabls as indicad by Figur 19.) Coninuous Discr x () i LPF i() A/D i(n) f s x q () LPF q() A/D q(n) Rquirs wo coax cabls o ransmi quadraur analog signals i() and q() Rquirs wo ribbon cabls o ransmi quadraur discr squncs i(n) and q(n) Figur 19. Riraion of how quadraur signals compris wo ral pars. To apprcia h physical maning our discussion hr, l's rmmbr ha a coninuous quadraur signal x c () = i() + jq() is no jus a mahmaical absracion. W can gnra x c () in our laboraory and ransmi i o h lab down h hall. All w nd is wo sinusoidal signal Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
16 gnraors, s o h sam frquncy. (Howvr, somhow w hav o synchroniz hos wo hardwar gnraors so ha hir rlaiv phas shif is fixd a 9 o.) Nx w connc coax cabls o h gnraors' oupu conncors and run hos wo cabls, labld 'i()' for our cosin signal and 'q()' for our sinwav signal, down h hall o hir dsinaion Now for a woqusion pop quiz. In h ohr lab, wha would w s on h scrn of an oscilloscop if h coninuous i() and q() signals wr conncd o h horizonal and vrical inpu channls, rspcivly, of h scop? (Rmmbring, of cours, o s h scop's Horizonal Swp conrol o h 'Exrnal' posiion.) q() = sin(π ) i() = cos(π ) Oscop Vr. In Horiz. In Figur. Displaying a quadraur signal using an oscilloscop. Nx, wha would b sn on h scop's display if h cabls wr mislabld and h wo signals wr inadvrnly swappd? Th answr o h firs qusion is ha w d s a brigh 'spo' roaing counrclockwis in a circl on h scop's display. If h cabls wr swappd, w'd s anohr circl, bu his im i would b orbiing in a clockwis dircion. This would b a na lil dmonsraion if w s h signal gnraors' frquncis o, say, 1 Hz. This oscilloscop xampl hlps us answr h imporan qusion, "Whn w work wih quadraur signals, how is h jopraor implmnd in hardwar?" Th answr is ha h j opraor is implmnd by how w ra h wo signals rlaiv o ach ohr. W hav o ra hm orhogonally such ha h inphas i() signal rprsns an EasWs valu, and h quadraur phas q() signal rprsns an orhogonal NorhSouh valu. (By orhogonal, I man ha h NorhSouh dircion is orind xacly 9 o rlaiv o h EasWs dircion.) So in our oscilloscop xampl h jopraor is implmnd mrly by how h conncions ar mad o h scop. Th inphas i() signal conrols horizonal dflcion and h quadraur phas q() signal conrols vrical dflcion. Th rsul is a wodimnsional quadraur signal rprsnd by h insananous posiion of h do on h scop's display. Th prson in h lab down h hall who's rciving, say, h discr squncs i(n) and q(n) has h abiliy o conrol h orinaion of h final complx spcra by adding or subracing h jq(n) squnc as shown in Figur 1. i(n) q(n) 1 i(n)  jq(n) B/ B/ i(n) + jq(n) B/ B/ Figur 1. Using h sign of q(n) o conrol spcral orinaion. Th op pah in Figur 1 is quivaln o muliplying h original x bp () by jπf c, and h boom pah is quivaln o muliplying h x bp () by jπf c. Thrfor, had h quadraur porion our Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
17 quadrauroscillaor a h op of Figur 14 bn ngaiv, sin(πf c ), h rsulan complx spcra would b flippd (abou Hz) from hos spcra shown in Figur 1. Whil w r hinking abou flipping complx spcra, l s rmind ourslvs ha hr ar wo simpl ways o rvrs (invr) an x(n) = i(n) + jq(n) squnc s spcral magniud. As shown in Figur 1, w can prform conjugaion o obain an x'(n) = i(n)  jq(n) wih an invrd magniud spcrum. Th scond mhod is o swap x(n) s individual i(n) and q(n) sampl valus o cra a nw squnc y(n) = q(n) + ji(n) whos spcral magniud is invrd from x(n) s spcral magniud. (No, whil x'(n) s and y(n) s spcral magniuds ar qual, hir spcral phass ar no qual.) Conclusions This nds our lil quadraur signals uorial. W larnd ha using h complx plan o visualiz h mahmaical dscripions of complx numbrs nabld us o s how quadraur and ral signals ar rlad. W saw how hrdimnsional frquncydomain dpicions hlp us undrsand how quadraur signals ar gnrad, ranslad in frquncy, combind, and sparad. Finally w rviwd an xampl of quadraursampling and wo schms for invring h spcrum of a quadraur squnc. Rfrncs [1] D. Sruik, A Concis Hisory of Mahmaics, Dovr Publicaions, NY, [] D. Brgamini, Mahmaics, Lif Scinc Library, Tim Inc., Nw York, [3] N. Bouin, "Complx Signals," RF Dsign, Dcmbr Answr o rivia qusion jus following Eq. (5) is: Th scarcrow in Wizard of Oz. Also, I say Thanks o Gran Griffin whos suggsions improvd h valu of his uorial. Hav you hard his lil sory? Whil in Brlin, Lonhard Eulr was ofn involvd in philosophical dbas, spcially wih Volair. Unforunaly, Eulr's philosophical abiliy was limid and h ofn blundrd o h amusmn of all involvd. Howvr, whn h rurnd o Russia, h go his rvng. Cahrin h Gra had invid o hr cour h famous Frnch philosophr Didro, who o h chagrin of h czarina, ampd o convr hr subjcs o ahism. Sh askd Eulr o qui him. On day in h cour, h Frnch philosophr, who had no mahmaical knowldg, was informd ha somon had a mahmaical proof h xisnc of God. H askd o har i. Eulr hn sppd forward and sad: "Sir, a + bn n = x, hnc God xiss; rply!" Didro had no ida wha Eulr was alking abou. Howvr, h did undrsand h chorus of laughr ha followd and soon afr rurnd o Franc. (Abov paragraph was found on a rrific wbsi dailing h hisory of mahmaics and mahmaicians: hp://www.shu.du/acadmic/ars_sci/undrgradua/mah_cs/sis/mah/rals/hisory/ulr.hml) Alhough i's a cu sory, srious mah hisorians don' bliv i. Thy know ha Didro did hav som mahmaical knowldg and hy jus can imagin Eulr clowning around in ha way. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd
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