A Quadrature Signals Tutorial: Complex, But Not Complicated. by Richard Lyons

Transcription

1 A Quadratur Signals Tutorial: Complx, But Not Complicatd by Richard Lyons Introduction Quadratur signals ar basd on th notion of complx numbrs and prhaps no othr topic causs mor hartach for nwcomrs to DSP than ths numbrs and thir strang trminology of j-oprator, complx, imaginary, ral, and orthogonal. If you'r a littl unsur of th physical maning of complx numbrs and th j = -1 oprator, don't fl bad bcaus you'r in good company. Why vn Karl Gauss, on th world's gratst mathmaticians, calld th j-oprator th "shadow of shadows". Hr w'll shin som light on that shadow so you'll nvr hav to call th Quadratur Signal Psychic Hotlin for hlp. Quadratur signal procssing is usd in many filds of scinc and nginring, and quadratur signals ar ncssary to dscrib th procssing and implmntation that taks plac in modrn digital communications systms. In this tutorial w'll rviw th fundamntals of complx numbrs and gt comfortabl with how thy'r usd to rprsnt quadratur signals. Nxt w xamin th notion of ngativ frquncy as it rlats to quadratur signal algbraic notation, and larn to spak th languag of quadratur procssing. In addition, w'll us thr-dimnsional tim and frquncy-domain plots to giv som physical maning to quadratur signals. This tutorial concluds with a brif look at how a quadratur signal can b gnratd by mans of quadratur-sampling. Why Car About Quadratur Signals? Quadratur signal formats, also calld complx signals, ar usd in many digital signal procssing applications such as: - digital communications systms, - radar systms, - tim diffrnc of arrival procssing in radio dirction finding schms - cohrnt puls masurmnt systms, - antnna bamforming applications, - singl sidband modulators, - tc. Ths applications fall in th gnral catgory known as quadratur procssing, and thy provid additional procssing powr through th cohrnt masurmnt of th phas of sinusoidal signals. A quadratur signal is a two-dimnsional signal whos valu at som instant in tim can b spcifid by a singl complx numbr having two parts; what w call th ral part and th imaginary part. (Th words ral and imaginary, although traditional, ar unfortunat bcaus of thir manings in our vry day spch. Communications nginrs us th trms in-phas and quadratur phas. Mor on that latr.) Lt's rviw th mathmatical notation of ths complx numbrs. Th Dvlopmnt and Notation of Complx Numbrs To stablish our trminology, w dfin a ral numbr to b thos numbrs w us in vry day lif, lik a voltag, a tmpratur on th Fahrnhit scal, or th balanc of your chcking account. Ths on-dimnsional numbrs can b ithr positiv or ngativ as dpictd in Figur 1(a). In that figur w show a on-dimnsional and say that a singl ral numbr can b rprsntd by a point on that. Out of tradition, lt's call this, th. Copyright April 13, Richard Lyons, All Rights Rsrvd

2 This point rprsnts th ral numbr a =. (j) This point rprsnts th complx numbr c =.5 + j (a) +j lin (b) lin Figur 1. An graphical intrprtation of a ral numbr and a complx numbr. A complx numbr, c, is shown in Figur 1(b) whr it's also rprsntd as a point. Howvr, complx numbrs ar not rstrictd to li on a ondimnsional lin, but can rsid anywhr on a two-dimnsional plan. That plan is calld th complx plan (som mathmaticians lik to call it an Argand diagram), and it nabls us to rprsnt complx numbrs having both ral and imaginary parts. For xampl in Figur 1(b), th complx numbr c =.5 + j is a point lying on th complx plan on nithr th ral nor th imaginary. W locat point c by going +.5 units along th ral and up + units along th imaginary. Think of thos ral and imaginary axs xactly as you think of th East-Wst and North-South dirctions on a road map. W'll us a gomtric viwpoint to hlp us undrstand som of th arithmtic of complx numbrs. Taking a look at Figur, w can us th trigonomtry of right triangls to dfin svral diffrnt ways of rprsnting th complx numbr c. b (j) c = a + jb M a Figur Th phasor rprsntation of complx numbr c = a + jb on th complx plan. Our complx numbr c is rprsntd in a numbr of diffrnt ways in th litratur, such as: Notation Nam: Rctangular form: Trigonomtric form: Polar form: Magnitudangl form: Math Exprssion: c = a + jb c = M[cos( ) + jsin( )] c = M j c = M Rmarks: Usd for xplanatory purposs. Easist to undrstand. [Also calld th Cartsian form.] Commonly usd to dscrib quadratur signals in communications systms. Most puzzling, but th primary form usd in math quations. [Also calld th Exponntial form. Somtims writtn as Mxp(j ).] Usd for dscriptiv purposs, but too cumbrsom for us in algbraic quations. [Essntially a shorthand vrsion of Eq. (3).] (1) () (3) (4) Eqs. (3) and (4) rmind us that c can also b considrd th tip of a phasor on th complx plan, with magnitud M, in th dirction of dgrs rlativ Copyright April 13, Richard Lyons, All Rights Rsrvd

3 to th positiv ral as shown in Figur. Kp in mind that c is a complx numbr and th variabls a, b, M, and ar all ral numbrs. Th magnitud of c, somtims calld th modulus of c, is M = c = a + b (5) [Trivia qustion: In what 1939 movi, considrd by many to b th gratst movi vr mad, did a main charactr attmpt to quot Eq. (5)?] OK, back to businss. Th phas angl, or argumnt, is th arctangnt of imaginary part th ratio ral part, or = tan -1 b a (6) If w st Eq. (3) qual to Eq. (), M j = M[cos( ) + jsin( )], w can stat what's namd in his honor and now calld on of Eulr's idntitis as: j = cos( ) + jsin( ) (7) Th suspicious radr should now b asking, "Why is it valid to rprsnt a complx numbr using that strang xprssion of th bas of th natural logarithms,, raisd to an imaginary powr?" W can validat Eq. (7) as did th world's gratst xprt on infinit sris, Hrr Lonard Eulr, by plugging j in for z in th sris xpansion dfinition of z in th top lin of Figur 3. That substitution is shown on th scond lin. Nxt w valuat th highr ordrs of j to arriv at th sris in th third lin in th figur. Thos of you with lvatd math skills lik Eulr (or thos who chck som math rfrnc book) will rcogniz that th altrnating trms in th third lin ar th sris xpansion dfinitions of th cosin and sin functions. z = 1 + z + z! + z3 3! + z4 4! + z5 5! + z6 6! +... j = 1 + j + (j )! + (j )3 3! + (j )4 4! + (j )5 5! + (j )6 6! +... = 1 + j -! - j 3 3! + 4 4! + j 5 5! - 6 6! +... j = cos( ) + jsin( ) Figur 3 On drivation of Eulr's quation using sris xpansions for z, cos( ), and sin( ). Copyright April 13, Richard Lyons, All Rights Rsrvd

4 Figur 3 vrifis Eq. (7) and our rprsntation of a complx numbr using th Eq. (3) polar form: M j. If you substitut -j for z in th top lin of Figur 3, you'd nd up with a slightly diffrnt, and vry usful, form of Eulr's idntity: -j = cos( ) - jsin( ) (8) Th polar form of Eqs. (7) and (8) bnfits us bcaus: - It simplifis mathmatical drivations and analysis, -- turning trigonomtric quations into th simpl algbra of xponnts, -- math oprations on complx numbrs follow xactly th sam ruls as ral numbrs. - It maks adding signals mrly th addition of complx numbrs (vctor addition), - It's th most concis notation, - It's indicativ of how digital communications systm ar implmntd, and dscribd in th litratur. W'll b using Eqs. (7) and (8) to s why and how quadratur signals ar usd in digital communications applications. But first, lt s tak a dp brath and ntr th Twilight Zon of that 'j' oprator. You'v sn th dfinition j = -1 bfor. Statd in words, w say that j rprsnts a numbr whn multiplid by itslf rsults in a ngativ on. Wll, this dfinition causs difficulty for th bginnr bcaus w all know that any numbr multiplid by itslf always rsults in a positiv numbr. Unfortunatly DSP txtbooks oftn dfin th symbol j and thn, with justifid hast, swiftly carry on with all th ways that th j oprator can b usd to analyz sinusoidal signals. Radrs soon forgt about th qustion: What dos j = -1 actually man? Wll, -1 had bn on th mathmatical scn for som tim, but wasn't takn sriously until it had to b usd to solv cubic quations in th sixtnth cntury. [1], [] Mathmaticians rluctantly bgan to accpt th abstract concpt of -1, without having to visualiz it, bcaus its mathmatical proprtis wr consistnt with th arithmtic of normal ral numbrs. It was Eulr's quating complx numbrs to ral sins and cosins, and Gauss' brilliant introduction of th complx plan, that finally lgitimizd th notion of -1 to Europ's mathmaticians in th ightnth cntury. Eulr, going byond th provinc of ral numbrs, showd that complx numbrs had a clan consistnt rlationship to th wll-known ral trigonomtric functions of sins and cosins. As Einstin showd th quivalnc of mass and nrgy, Eulr showd th quivalnc of ral sins and cosins to complx numbrs. Just as modrn-day physicists don t know what an lctron is but thy undrstand its proprtis, w ll not worry about what 'j' is and b satisfid with undrstanding its bhavior. For our purposs, th j-oprator mans rotat a complx numbr by 9 o countrclockwis. (For you good folk in th UK, countrclockwis mans anti-clockwis.) Lt's s why. W'll gt comfortabl with th complx plan rprsntation of imaginary numbrs by xamining th mathmatical proprtis of th j = -1 oprator as shown in Figur 4. Copyright April 13, Richard Lyons, All Rights Rsrvd

5 j8 = multiply by "j" 8 8 -j8 Figur 4. What happns to th ral numbr 8 whn you start multiplying it by j. Multiplying any numbr on th ral by j rsults in an imaginary product that lis on th imaginary. Th xampl in Figur 4 shows that if +8 is rprsntd by th dot lying on th positiv ral, multiplying +8 by j rsults in an imaginary numbr, +j8, whos position has bn rotatd 9 o countrclockwis (from +8) putting it on th positiv imaginary. Similarly, multiplying +j8 by j rsults in anothr 9 o rotation yilding th - 8 lying on th ngativ ral bcaus j = -1. Multiplying -8 by j rsults in a furthr 9 o rotation giving th -j8 lying on th ngativ imaginary. Whnvr any numbr rprsntd by a dot is multiplid by j, th rsult is a countrclockwis rotation of 9 o. (Convrsly, multiplication by -j rsults in a clockwis rotation of -9 o on th complx plan.) If w lt = / in Eq. 7, w can say that j / = cos( /) + jsin( /) = + j1, or j / = j (9) Hr's th point to rmmbr. If you hav a singl complx numbr, rprsntd by a point on th complx plan, multiplying that numbr by j or by j / will rsult in a nw complx numbr that's rotatd 9 o countrclockwis (CCW) on th complx plan. Don't forgt this, as it will b usful as you bgin rading th litratur of quadratur procssing systms! Lt's paus for a momnt hr to catch our brath. Don't worry if th idas of imaginary numbrs and th complx plan sm a littl mystrious. It's that way for vryon at first you'll gt comfortabl with thm th mor you us thm. (Rmmbr, th j-oprator puzzld Europ's havywight mathmaticians for hundrds of yars.) Grantd, not only is th mathmatics of complx numbrs a bit strang at first, but th trminology is almost bizarr. Whil th trm imaginary is an unfortunat on to us, th trm complx is downright wird. Whn first ncountrd, th phras complx numbrs maks us think 'complicatd numbrs'. This is rgrttabl bcaus th concpt of complx numbrs is not rally all that complicatd. Just know that th purpos of th abov mathmatical rigmarol was to validat Eqs. (), (3), (7), and (8). Now, lt's (finally!) talk about tim-domain signals. Rprsnting Signals Using Complx Phasors OK, w now turn our attntion to a complx numbr that is a function tim. Considr a numbr whos magnitud is on, and whos phas angl incrass with tim. That complx numbr is th j t point shown in Figur 5(a). (Hr th trm is frquncy in radians/scond, and it corrsponds to a frquncy of cycls/scond whr is masurd in Hrtz.) As tim t gts largr, th complx numbr's phas angl incrass and our numbr orbits th origin of th complx plan in a CCW dirction. Figur 5(a) shows th numbr, Copyright April 13, Richard Lyons, All Rights Rsrvd

6 rprsntd by th black dot, frozn at som arbitrary instant in tim. If, say, th frquncy = Hz, thn th dot would rotat around th circl two tims pr scond. W can also think of anothr complx numbr -j t (th whit dot) orbiting in a clockwis dirction bcaus its phas angl gts mor ngativ as tim incrass. j t = tim in sconds, = frquncy in Hrtz j j fot j fot 1 = t 1 = t 1 = t 1 = f o t j fot j fot j j (a) (b) Figur 5. A snapshot, in tim, of two complx numbrs whos xponnts chang with tim. Lt's now call our two j t and -j t complx xprssions quadratur signals. Thy ach hav quadratur ral and imaginary parts, and thy ar both functions of tim. Thos j t and -j t xprssions ar oftn calld complx xponntials in th litratur. W can also think of thos two quadratur signals, j t and -j t, as th tips of two phasors rotating in opposit dirctions as shown in Figur 5(b). W'r going to stick with this phasor notation for now bcaus it'll allow us to achiv our goal of rprsnting ral sinusoids in th contxt of th complx plan. Don't touch that dial! To nsur that w undrstand th bhavior of thos phasors, Figur 6(a) shows th thr-dimnsional path of th j t phasor as tim passs. W'v addd th tim, coming out of th pag, to show th spiral path of th phasor. Figur 6(b) shows a continuous vrsion of just th tip of th j t phasor. That j t complx numbr, or if you wish, th phasor's tip, follows a corkscrw path spiraling along, and cntrd about, th tim. Th ral and imaginary parts of j t ar shown as th sin and cosin projctions in Figur 6(b). Copyright April 13, Richard Lyons, All Rights Rsrvd

7 ( j ) o o 9 Imag 1 1 j t sin( t) 18 o o 7 36 o Tim 1 1 Tim cos( t) (a) (b) Figur 6. Th motion of th j fot phasor (a), and phasor 's tip (b). Rturn to Figur 5(b) and ask yourslf: "Slf, what's th vctor sum of thos two phasors as thy rotat in opposit dirctions?" Think about this for a momnt... That's right, th phasors' ral parts will always add constructivly, and thir imaginary parts will always cancl. This mans that th summation of ths j t and -j t phasors will always b a purly ral numbr. Implmntations of modrn-day digital communications systms ar basd on this proprty! To mphasiz th importanc of th ral sum of ths two complx sinusoids w'll draw yt anothr pictur. Considr th wavform in th thrdimnsional Figur 7 gnratd by th sum of two half-magnitud complx phasors, j t / and -j t /, rotating in opposit dirctions around, and moving down along, th tim. ( j ) 1 t = cos( t) j fot Tim j fot Figur 7. A cosin rprsntd by th sum of two rotating complx phasors. Thinking about ths phasors, it's clar now why th cosin wav can b quatd to th sum of two complx xponntials by cos( t) = j t + -j t = j t + -j t. (1) Eq. (1), a wll-known and important xprssion, is also calld on of Eulr's idntitis. W could hav drivd this idntity by solving Eqs. (7) and (8) for jsin( ), quating thos two xprssions, and solving that final quation Copyright April 13, Richard Lyons, All Rights Rsrvd

8 for cos( ). Similarly, w could go through that sam algbra xrcis and show that a ral sin wav is also th sum of two complx xponntials as sin( t) = j t - -j t j = j -j t j j t. (11) Look at Eqs. (1) and (11) carfully. Thy ar th standard xprssions for a cosin wav and a sin wav, using complx notation, sn throughout th litratur of quadratur communications systms. To kp th radr's mind from spinning lik thos complx phasors, plas raliz that th sol purpos of Figurs 5 through 7 is to validat th complx xprssions of th cosin and sin wav givn in Eqs. (1) and (11). Thos two quations, along with Eqs. (7) and (8), ar th Rostta Ston of quadratur signal procssing. cos( t) = j t + -j t W can now asily translat, back and forth, btwn ral sinusoids and complx xponntials. Again, w ar larning how ral signals, that can b transmittd down a coax cabl or digitizd and stord in a computr's mmory, can b rprsntd in complx numbr notation. Ys, th constitunt parts of a complx numbr ar ach ral, but w'r trating thos parts in a spcial way w'r trating thm in quadratur. Rprsnting Quadratur Signals In th uncy Domain Now that w know somthing about th tim-domain natur of quadratur signals, w'r rady to look at thir frquncy-domain dscriptions. This matrial is critical bcaus w ll add a third dimnsion, tim, to our normal twodimnsional frquncy domain plots. That way non of th phas rlationships our quadratur signals will b hiddn from viw. Figur 8 tlls us th ruls for rprsnting complx xponntials in th frquncy domain. Ngativ frquncy j fot Dirction along th imaginary j j fot j Positiv frquncy Magnitud is 1/ Figur 8. Intrprtation of complx xponntials. W'll rprsnt a singl complx xponntial as a narrowband impuls locatd at th frquncy spcifid in th xponnt. In addition, w'll show th phas rlationships btwn th spctra of thos complx xponntials along th ral and imaginary axs our complx frquncy domain rprsntation. With all that said, tak a look at Figur 9. Copyright April 13, Richard Lyons, All Rights Rsrvd

9 ... cos( fot) Imag Part... Part Tim cos( fot) = j fot + j fot Imag Part Part... sin( fot) Imag Part... Part Tim sin( fot) = j j fot j j fot Imag Part Part Figur 9. Complx frquncy domain rprsntation of a cosin wav and sin wav. S how a ral cosin wav and a ral sin wav ar dpictd in our complx frquncy domain rprsntation on th right sid of Figur 9. Thos bold arrows on th right of Figur 9 ar not rotating phasors, but instad ar frquncy-domain impuls symbols indicating a singl spctral lin for a singl complx xponntial j t. Th dirctions in which th spctral impulss ar pointing mrly indicat th rlativ phass of th spctral componnts. Th amplitud of thos spctral impulss ar 1/. OK... why ar w bothring with this 3-D frquncy-domain rprsntation? Bcaus it's th tool w'll us to undrstand th gnration (modulation) and dtction (dmodulation) of quadratur signals in digital (and som analog) communications systms, and thos ar two of th goals of this tutorial. Howvr, bfor w considr thos procsss lt's validat this frquncydomain rprsntation with a littl xampl. Figur 1 is a straightforward xampl of how w us th complx frquncy domain. Thr w bgin with a ral sin wav, apply th j oprator to it, and thn add th rsult to a ral cosin wav of th sam frquncy. Th final rsult is th singl complx xponntial j t illustrating graphically Eulr's idntity that w statd mathmatically in Eq. (7)..5 Imag multiply by j Imag sin( t).5 jsin( t) add Imag cos( t) Imag j t = cos( t) + jsin( t) 1 Figur 1. Complx frquncy-domain viw of Eulr's: j fot = cos( t) + jsin( t). On th frquncy, th notion of ngativ frquncy is sn as thos spctral impulss locatd at - radians/sc on th frquncy. This figur shows th big payoff: Whn w us complx notation, gnric complx xponntials lik j ft and -j ft ar th fundamntal constitunts of th ral sinusoids sin( ft) or cos( ft). That's bcaus both sin( ft) and Copyright April 13, Richard Lyons, All Rights Rsrvd

10 cos( ft) ar mad up of j ft and -j ft componnts. If you wr to tak th discrt Fourir transform (DFT) of discrt tim-domain sampls of a sin( t) sin wav, a cos( t) cosin wav, or a j fot complx sinusoid and plot th complx rsults, you'd obtain xactly thos narrowband impulss in Figur 1. If you undrstand th notation and oprations in Figur 1, pat yoursln th back bcaus you know a grat dal about th natur and mathmatics of quadratur signals. Bandpass Quadratur Signals In th uncy Domain In quadratur procssing, by convntion, th ral part of th spctrum is calld th in-phas componnt and th imaginary part of th spctrum is calld th quadratur componnt. Th signals whos complx spctra ar in Figur 11(a), (b), and (c) ar ral, and in th tim domain thy can b rprsntd by amplitud valus that hav nonzro ral parts and zro-valud imaginary parts. W'r not forcd to us complx notation to rprsnt thm in th tim domain th signals ar ral. signals always hav positiv and ngativ frquncy spctral componnts. For any ral signal, th positiv and ngativ frquncy componnts of its inphas (ral) spctrum always hav vn symmtry around th zro-frquncy point. That is, th in-phas part's positiv and ngativ frquncy componnts ar mirror imags of ach othr. Convrsly, th positiv and ngativ frquncy componnts of its quadratur (imaginary) spctrum ar always ngativs of ach othr. This mans that th phas angl of any givn positiv quadratur frquncy componnt is th ngativ of th phas angl of th corrsponding quadratur ngativ frquncy componnt as shown by th thin solid arrows in Figur 11(a). This 'conjugat symmtry' is th invariant natur of ral signals whn thir spctra ar rprsntd using complx notation. Quadratur phas (Imag.) cos( fot+ ) In-phas () (a) Complx xponntial In-phas (ral) part Quadratur phas (imaginary) part 1/ Quadratur phas (Imag.) In-phas () (b) B Quadratur phas (Imag.) In-phas () Quadratur phas (Imag.) In-phas () B B (c) (d) Figur 11. Quadratur rprsntation of signals: (a) sinusoid cos( t + ), (b) bandpass signal containing six sinusoids ovr bandwidth B; (c) bandpass signal containing an infinit numbr of sinusoids ovr bandwidth B Hz; (d) Complx bandpass signal of bandwidth B Hz. Lt's rmind ourslvs again, thos bold arrows in Figur 11(a) and (b) ar not rotating phasors. Thy'r frquncy-domain impuls symbols indicating a singl complx xponntial j ft. Th dirctions in which th impulss ar pointing show th rlativ phass of th spctral componnts. Thr's an important principl to kp in mind bfor w continu. Copyright April 13, Richard Lyons, All Rights Rsrvd

11 Multiplying a tim signal by th complx xponntial j t, what w can call quadratur mixing (also calld complx mixing), shifts that signal's spctrum upward in frquncy by Hz as shown in Figur 1 (a) and (b). Likwis, multiplying a tim signal by -j t shifts that signal's spctrum down in frquncy by Hz. Quad. phas In-phas Quad. phas In-phas Quad. phas In-phas (a) (b) (c) Figur 1. Quadratur mixing of a signal: (a) Spctrum of a complx signal x(t), (b) Spctrum of x(t) j fot, (c) Spctrum of x(t) -j fot. A Quadratur-Sampling Exampl W can us all that w'v larnd so far about quadratur signals by xploring th procss of quadratur-sampling. Quadratur-sampling is th procss of digitizing a continuous (analog) bandpass signal and translating its spctrum to b cntrd at zro Hz. Lt's s how this popular procss works by thinking of a continuous bandpass signal, of bandwidth B, cntrd about a carrir frquncy of f c Hz. Original Continuous Spctrum f c X(f) B f c Dsird Digitizd "Basband" Spctrum f s X(m) f s (m) Figur 13. Th 'bfor and aftr' spctra of a quadratur-sampld signal. Our goal in quadratur-sampling is to obtain a digitizd vrsion of th analog bandpass signal, but w want that digitizd signal's discrt spctrum cntrd about zro Hz, not f c Hz. That is, w want to mix a tim signal with -j f ct to prform complx down-convrsion. Th frquncy fs is th digitizr's sampling rat in sampls/scond. W show rplicatd spctra at th bottom of Figur 13 just to rmind ourslvs of this ffct whn A/D convrsion taks plac. OK,... tak a look at th following quadratur-sampling block diagram known as I/Q dmodulation (or 'Wavr dmodulation' for thos folk with xprinc in communications thory) shown at th top of Figur 14. That arrangmnt of two sinusoidal oscillators, with thir rlativ 9 o phas diffrnc, is oftn calld a quadratur-oscillator. Thos j f ct and -j f c t trms in that busy Figur 14 rmind us that th constitunt complx xponntials comprising a ral cosin duplicats ach part of th X bp (f) spctrum to produc th X i (f) spctrum. Th Figur shows how w gt th filtrd continuous in-phas portion our dsird complx quadratur signal. By dfinition, thos X i (f) and I(f) spctra ar tratd as 'ral Copyright April 13, Richard Lyons, All Rights Rsrvd

12 only'. Continuous Discrt x (t) bp Continuous spctrum In-phas continuous spctrum LP filtrd in-phas continuous spctrum cos( f t) c sin( f c t) o 9 x (t) i x q (t) f c LPF LPF X (f) i i(t) q(t) X bp (f) f s A/D A/D i(n) q(n) i(n) jq(n) f B/ +B/ c f c f c f c I(f) B/ B/ B f c j fct j fct part Filtrd ral part complx squnc is: Figur 14. Quadratur-sampling block diagram and spctra within th in-phas (uppr) signal path. Likwis, Figur 15 shows how w gt th filtrd continuous quadratur phas portion our complx quadratur signal by mixing x bp (t) with sin( f c t). Continuous spctrum f c X bp (f) B f c Quadratur continuous spctrum part X (f) q f c f c f c Ngativ du to th minus sign of th sin's f c j j ft Q(f) LP filtrd quadratur continuous spctrum B/ B/ Filtrd imaginary part Figur 15. Spctra within th quadratur phas (lowr) signal path of th block diagram. Hr's whr w'r going: I(f) - jq(f) is th spctrum of a complx rplica of our original bandpass signal x bp (t). W show th addition of thos two spctra in Figur 16. Copyright April 13, Richard Lyons, All Rights Rsrvd

13 I(f) Filtrd continuous in-phas (ral only) B/ B/ B/ Q(f) Filtrd continuous quadratur (imaginary only) B/ I(f) jq(f) Spctrum of continuous complx signal: i(t) jq(t) B/ B/ Figur 16. Combining th I(f) and Q(f) spctra to obtain th dsird 'I(f) - jq(f)' spctra. This typical dpiction of quadratur-sampling sms lik mumbo jumbo until you look at this situation from a thr-dimnsional standpoint, as in Figur 17, whr th -j factor rotats th 'imaginary-only' Q(f) by -9 o, making it 'ral-only'. This -jq(f) is thn addd to I(f). ( j ) A 3-dimnsional Viw ( j ) I(f) Q(f) ( j ) ( j ) jq(f) I(f) jq(f) Figur D viw of combining th I(f) and Q(f) spctra to obtain th I(f) - jq(f) spctra. Th complx spctrum at th bottom Figur 18 shows what w wantd, a digitizd vrsion of th complx bandpass signal cntrd about zro Hz. Copyright April 13, Richard Lyons, All Rights Rsrvd

14 Spctrum of continuous complx signal: i(t) jq(t) -B/ B/ This is what w wantd. A digitizd complx vrsion of th original x bp (t), but cntrd about zro Hz. A/D convrsion Spctrum of discrt complx squnc: i(n) jq(n) f f B/ s f s B/ s f s Figur 18. Th continuous complx signal i(t) - q(t) is digitizd to obtain discrt i(n) - jq(n). Som advantags of this quadratur-sampling schm ar: - Each A/D convrtr oprats at half th sampling rat of standard ral-signal sampling, - In many hardwar implmntations oprating at lowr clock rats sav powr. - For a givn f s sampling rat, w can captur widr-band analog signals. - Quadratur squncs mak FFT procssing mor fficint du to covring a widr frquncy rang than whn an FFT s input is a ral-valud squnc. - Bcaus quadratur squncs ar ffctivly ovrsampld by a factor of two, signal squaring oprations ar possibl without th nd for upsampling. - Knowing th phas of signals nabls cohrnt procssing. - Quadratur-sampling maks it asir to masur th instantanous magnitud and phas of a signal during dmodulation. Rturning to th Figur 14 block diagram rminds us of an important charactristic of quadratur signals. W can snd an analog quadratur signal to a rmot location. To do so w us two coax cabls on which th two ral i(t) and q(t) signals travl. (To transmit a discrt tim-domain quadratur squnc, w'd nd two multi-conductor ribbon cabls as indicatd by Figur 19.) Continuous Discrt x (t) i LPF i(t) A/D i(n) f s x q (t) LPF q(t) A/D q(n) Rquirs two coax cabls to transmit quadratur analog signals i(t) and q(t) Rquirs two ribbon cabls to transmit quadratur discrt squncs i(n) and q(n) Figur 19. Ritration of how quadratur signals compris two ral parts. To apprciat th physical maning our discussion hr, lt's rmmbr that a continuous quadratur signal xc(t) = i(t) + jq(t) is not just a mathmatical Copyright April 13, Richard Lyons, All Rights Rsrvd

15 abstraction. W can gnrat xc(t) in our laboratory and transmit it to th lab down th hall. All w nd is two sinusoidal signal gnrators, st to th sam frquncy. (Howvr, somhow w hav to synchroniz thos two hardwar gnrators so that thir rlativ phas shift is fixd at 9 o.) Nxt w connct coax cabls to th gnrators' output connctors and run thos two cabls, labld 'i(t)' for our cosin signal and 'q(t)' for our sin wav signal, down th hall to thir dstination Now for a two-qustion pop quiz. In th othr lab, what would w s on th scrn of an oscilloscop if th continuous i(t) and q(t) signals wr connctd to th horizontal and vrtical input channls, rspctivly, of th scop? (Rmmbring, of cours, to st th scop's Horizontal Swp control to th 'Extrnal' position.) q(t) = sin( t) i(t) = cos( t) O-scop Vrt. In Horiz. In Figur. Displaying a quadratur signal using an oscilloscop. Nxt, what would b sn on th scop's display if th cabls wr mislabld and th two signals wr inadvrtntly swappd? Th answr to th first qustion is that w d s a bright 'spot' rotating countrclockwis in a circl on th scop's display. If th cabls wr swappd, w'd s anothr circl, but this tim it would b orbiting in a clockwis dirction. This would b a nat littl dmonstration if w st th signal gnrators' frquncis to, say, 1 Hz. This oscilloscop xampl hlps us answr th important qustion, "Whn w work with quadratur signals, how is th j-oprator implmntd in hardwar? Th answr is w can t go to Radio Shack and buy a j-oprator and soldr it to a circuit board. Th j-oprator is implmntd by how w trat th two signals rlativ to ach othr. W hav to trat thm orthogonally such that th in-phas i(t) signal rprsnts an East-Wst valu, and th quadratur phas q(t) signal rprsnts an orthogonal North-South valu. (By orthogonal, I man that th North-South dirction is orintd xactly 9 o rlativ to th East-Wst dirction.) So in our oscilloscop xampl th j-oprator is implmntd mrly by how th connctions ar mad to th scop. Th in-phas i(t) signal controls horizontal dflction and th quadratur phas q(t) signal controls vrtical dflction. Th rsult is a two-dimnsional quadratur signal rprsntd by th instantanous position of th dot on th scop's display. A prson in th lab down th hall who's rciving, say, th discrt squncs i(n) and q(n) has th ability to control th orintation of th final complx spctra by adding or subtracting th 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 th sign of q(n) to control spctral orintation. Th top path in Figur 1 is quivalnt to multiplying th original x bp (t) by -j f ct, and th bottom path is quivalnt to multiplying th xbp (t) by j f ct. Thrfor, had th quadratur portion our quadratur-oscillator at th top Copyright April 13, Richard Lyons, All Rights Rsrvd

16 of Figur 14 bn ngativ, -sin( f c t), th rsultant complx spctra would b flippd (about Hz) from thos spctra shown in Figur 1. Whil w r thinking about flipping complx spctra, lt s rmind ourslvs that thr ar two simpl ways to rvrs (invrt) an x(n) = i(n) + jq(n) squnc s spctral magnitud. As shown in Figur 1, w can prform conjugation to obtain an x'(n) = i(n) - jq(n) with an invrtd magnitud spctrum. Th scond mthod is to swap x(n) s individual i(n) and q(n) sampl valus to crat a nw squnc y(n) = q(n) + ji(n) whos spctral magnitud is invrtd from x(n) s spctral magnitud. (Not, whil x'(n) s and y(n) s spctral magnituds ar qual, thir spctral phass ar not qual.) Conclusions This nds our littl quadratur signals tutorial. W larnd that using th complx plan to visualiz th mathmatical dscriptions of complx numbrs nabld us to s how quadratur and ral signals ar rlatd. W saw how thr-dimnsional frquncy-domain dpictions hlp us undrstand how quadratur signals ar gnratd, translatd in frquncy, combind, and sparatd. Finally w rviwd an xampl of quadratur-sampling and two schms for invrting th spctrum of a quadratur squnc. Rfrncs [1] D. Struik, A Concis History of Mathmatics, Dovr Publications, NY, [] D. Brgamini, Mathmatics, Lif Scinc Library, Tim Inc., Nw York, [3] N. Boutin, "Complx Signals," RF Dsign, Dcmbr Answr to trivia qustion just following Eq. (5) is: Th scarcrow in Th Wizard of Oz. Hav you hard this littl story? Whil in Brlin, Lonhard Eulr was oftn involvd in philosophical dbats, spcially with Voltair. Unfortunatly, Eulr's philosophical ability was limitd and h oftn blundrd to th amusmnt of all involvd. Howvr, whn h rturnd to Russia, h got his rvng. Cathrin th Grat had invitd to hr court th famous Frnch philosophr Didrot, who to th chagrin of th czarina, attmptd to convrt hr subjcts to athism. Sh askd Eulr to quit him. On day in th court, th Frnch philosophr, who had no mathmatical knowldg, was informd that somon had a mathmatical proof th xistnc of God. H askd to har it. Eulr thn stppd forward and statd: "Sir, a + bn n = x, hnc God xists; rply!" Didrot had no ida what Eulr was talking about. Howvr, h did undrstand th chorus of laughtr that followd and soon aftr rturnd to Franc. Although it's a cut story, srious math historians don't bliv it. Thy know that Didrot did hav som mathmatical knowldg and thy just can t imagin Eulr clowning around in that way. Copyright April 13, Richard Lyons, All Rights Rsrvd