Quadrature Signals: Complex, But Not Complicated


 Scarlett Johnson
 2 years ago
 Views:
Transcription
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
DIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN W will assum ha h radr is familiar wih h calculaor s kyboard and h basic opraions. In paricular w hav assumd ha h radr knows h funcions of
More informationEstimating Powers with Base Close to Unity and Large Exponents
Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34 Esimaing Powrs wih Bas Clos o Uniy and Larg Exponns Esimacón d Poncias con Bas Crcana a la Unidad y Grands Exponns Vio Lampr Vio.Lampr@fgg.unilj.si) FGG,
More informationQUALITY OF DYING AND DEATH QUESTIONNAIRE FOR NURSES VERSION 3.2A
UNIVERSITY OF WASHINGTON SCHOOL OF MEDICINE QUALITY OF DYING AND DEATH QUESTIONNAIRE FOR NURSES VERSION 3.2A Plas rurn your compld qusionnair in h nclosd nvlop o: [Rurn Addrss] RNID PID Copyrigh by h Univrsiy
More informationNumerical Algorithm for the Stochastic Present Value of Aggregate Claims in the Renewal Risk Model
Gn. Mah. Nos, Vol. 9, No. 2, Dcmbr, 23, pp. 4 ISSN 229784; Copyrigh ICSRS Publicaion, 23 www.icsrs.org Availabl fr onlin a hp://www.gman.in Numrical Algorihm for h Sochasic Prsn Valu of Aggrga Claims
More informationMTBF: Understanding Its Role in Reliability
Modul MTBF: Undrsanding Is Rol in Rliabiliy By David C. Wilson Foundr / CEO March 4, Wilson Consuling Srvics, LLC dav@wilsonconsulingsrvics.n www.wilsonconsulingsrvics.n Wilson Consuling Srvics, LLC Pag
More informationMany quantities are transduced in a displacement and then in an electric signal (pressure, temperature, acceleration). Prof. B.
Displacmn snsors Many quaniis ar ransducd in a displacmn and hn in an lcric signal (prssur, mpraur, acclraion). Poniomrs Poniomrs i p p i o i p A poniomr is basd on a sliding conac moving on a rsisor.
More informationUnit 2. Unit 2: Rhythms in Mexican Music. Find Our Second Neighborhood (5 minutes) Preparation
Uni 2 Prparaion Uni 2: Rhyhms in Mxican Music Find Our Scond Nighborhood (5 minus) Th Conducor now aks us on a journy from Morningsid Highs, Manhaan, o Eas Harlm, Manhaan, o m our nx singr, Clso. Hav sudns
More informationINTEGRATED SKILLS TEACHER S NOTES
TEACHER S NOTES INTEGRATED SKILLS TEACHER S NOTES LEVEL: Inrmdia + AGE: Tnagrs / Aduls TIME NEEDED: 90 minus + projc LANGUAGE FOCUS: Passiv; undrsand vocabulary in conx; opic words vrbs and nouns LEADIN
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(RsisrCapacir Circuis AP Physics C Circui Iniial Cndiins An circui is n whr yu hav a capacir and rsisr in h sam circui. Supps w hav h fllwing circui: Iniially, h capacir is UNCHARGED ( = 0 and h currn
More informationGENETIC ALGORITHMS IN SEASONAL DEMAND FORECASTING
forcasing, dmand, gnic algorihm Grzgorz Chodak*, Wiold Kwaśnicki* GENETIC ALGORITHMS IN SEASONAL DEMAND FORECASTING Th mhod of forcasing sasonal dmand applying gnic algorihm is prsnd. Spcific form of usd
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) 92.222  Linar Algbra II  Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial
More informationTransient Thermoelastic Behavior of Semiinfinite Cylinder by Using MarchiZgrablich and Fourier Transform Technique
Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : 77698 Volum 1 Issu 5 (May 01) hp://www.ijms.com/ hps://sis.googl.com/si/ijmsjournal/ Transin Thrmolasic Bhavior of Smiinfini Cylindr by Using
More informationKrebs (1972). A group of organisms of the same species occupying a particular space at a particular time
FW 662 Lcur 1  Dnsiyindpndn populaion modls Tx: Golli, 21, A Primr of Ecology Wha is a populaion? Krbs (1972). A group of organisms of h sam spcis occupying a paricular spac a a paricular im Col (1957).
More informationThe example is taken from Sect. 1.2 of Vol. 1 of the CPN book.
Rsourc Allocation Abstract This is a small toy xampl which is wllsuitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of Cnts. Hnc, it can b rad by popl
More informationTerm Structure of Interest Rates: The Theories
Handou 03 Econ 333 Abdul Munasb Trm Srucur of Inrs Ras: Th Thors Trm Srucur Facs Lookng a Fgur, w obsrv wo rm srucur facs Fac : Inrs ras for dffrn maurs nd o mov oghr ovr m Fac : Ylds on shorrm bond mor
More informationOption Pricing with Constant & Time Varying Volatility
Opion Pricing wih Consan & im arying olailiy Willi mmlr Cnr for Empirical Macroconomics Bilfld Grmany; h Brnard chwarz Cnr for Economic Policy Analysis Nw York NY UA and Economics Dparmn h Nw chool for
More informationThe Sensitivity of Beta to the Time Horizon when Log Prices follow an Ornstein Uhlenbeck Process
T Snsiiviy of Ba o Tim Horizon wn Log Prics follow an Ornsin Ulnbck Procss Oc 8, 00) KiHoon Jimmy Hong Dparmn of Economics, Cambridg Univrsiy Ocobr 4, 00 Sv Sacll Triniy Collg, Cambridg Univrsiy Prsnaion
More informationNonHomogeneous Systems, Euler s Method, and Exponential Matrix
NonHomognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous firstordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach
More informationStep Functions; and Laplace Transforms of Piecewise Continuous Functions
Sp Fnion; and Lapla Tranform of Piwi Conino Fnion Th prn objiv i o h Lapla ranform o olv diffrnial qaion wih piwi onino foring fnion ha i, foring fnion ha onain dioninii Bfor ha old b don, w nd o larn
More informationSPECIAL VOWEL SOUNDS
SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)
More informationInterest Parity Conditions. Interest parity conditions are noarbitrage profit conditions for capital. The easiest way to
World Economy  Inrs Ra Pariy (nry wrin for h Princon Encyclopdia of h World Economy) Inrs Pariy Condiions Th pariy condiions Inrs pariy condiions ar noarbirag profi condiions for capial. Th asis way
More informationQUANTITATIVE METHODS CLASSES WEEK SEVEN
QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.
More information5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:
.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This
More information5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power
Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim
More informationFree ACA SOLUTION (IRS 1094&1095 Reporting)
Fr ACA SOLUTION (IRS 1094&1095 Rporting) Th Insuranc Exchang (301) 2791062 ACA Srvics Transmit IRS Form 1094 C for mployrs Print & mail IRS Form 1095C to mploys HR Assist 360 will gnrat th 1095 s for
More informationMathematics. Mathematics 3. hsn.uk.net. Higher HSN23000
hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails
More informationQuestion 3: How do you find the relative extrema of a function?
ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating
More informationYou can recycle all your cans, plastics, paper, cardboard, garden waste and food waste at home.
Your 4 bin srvic You can rcycl all your cans, plasics, papr, cardboard, gardn was and food was a hom. This guid conains imporan informaion abou wha can b rcycld in your bins. Plas ak a momn o rad i. for
More informationCPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions
CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:
More informationSubject: Quality Management System Requirements SOP 4101 
SOP 4101  1.) Scop: This SOP covrs h rquirmns for Pharmco s Qualiy Managmn Sysm. 2.) Normaiv Rfrncs: SOP 442: Qualiy Manual. SOP 423A: Masr Lis of Conrolld Documns SOP 560 Managmn Rsponsibiliy
More informationNew Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ralvalud Fourir sris is xplaind and formula ar givn for convrting
More informationFourier Series and Fourier Transform
Fourier Series and Fourier ransform Complex exponenials Complex version of Fourier Series ime Shifing, Magniude, Phase Fourier ransform Copyrigh 2007 by M.H. Perro All righs reserved. 6.082 Spring 2007
More informationDamon s Newark is hosting an October Charity Fest to help raise money for The Fallen Heroes and Big Brothers and Big Sisters of Licking County.
Rgarding: Ocobr Chariy Fs on Saurday Ocobr 15, 2011. Damon s Nwark is hosing an Ocobr Chariy Fs o hlp rais mony for Th and Big Brohrs and Big Sisrs of Licking Couny. Th purpos of BBBS is o organiz, undr
More informationEstimating Private Equity Returns from Limited Partner Cash Flows
Esimaing Priva Equiy Rurns from Limid Parnr Cash Flows Andrw Ang, Bingxu Chn, William N. Gozmann, and Ludovic Phalippou* Novmbr 25, 2013 W inroduc a mhodology o sima h hisorical im sris of rurns o invsmn
More informationEXTRACTION OF FINANCIAL MARKET EXPECTATIONS ABOUT INFLATION AND INTEREST RATES FROM A LIQUID MARKET. Documentos de Trabajo N.
ETRCTION OF FINNCIL MRKET EPECTTIONS OUT INFLTION ND INTEREST RTES FROM LIQUID MRKET 2009 Ricardo Gimno and José Manul Marqués Documnos d Trabajo N.º 0906 ETRCTION OF FINNCIL MRKET EPECTTIONS OUT INFLTION
More informationRapid Estimation of Water Flooding Performance and Optimization in EOR by Using Capacitance Resistive Model
Iranian Journal of Chmical Enginring Vol. 9, No. 4 (Auumn), 22, IAChE Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl A.R. Basami, M. Dlshad 2, P. Pourafshary
More informationUniplan REIT Portfolio Fiduciary Services Uniplan Investment Counsel, Inc.
Uniplan EIT Porfolio Uniplan Invsmn Counsl, Inc. 22939 W Ovrson d Union Grov, Wisconsin 53182 Syl: SubSyl: Firm AUM: Firm Sragy AUM: EITs EITs Domsic $2.2 billion $1.3 billion Yar Foundd: GIMA Saus: Firm
More informationLecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 1213)
con 37: Answr Ky for Problm St (Chaptr 23) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc
More informationIntroduction to Measurement, Error Analysis, Propagation of Error, and Reporting Experimental Results
Inroducion o Masurmn, Error Analysis, Propagaion of Error, and Rporing Exprimnal Rsuls AJ Pinar, TD Drummr, D Caspary Dparmn of Chmical Enginring Michigan Tchnological Univrsiy Houghon, MI 4993 Spmbr,
More informationISSeG EGEE07 Poster Ideas for Edinburgh Brainstorming
SSG EGEE07 Pos das fo Edinbugh Bainsoming 3xposs, plus hoizonal and vical banns (A0=841mm x 1189mm) Why SSG: anion gabbing: hadlins/shock phoos/damaic ycaching imag Wha is SSG: pojc ovviw: SSG ino, diffnc
More informationCFDCalculation of Fluid Flow in a Pressurized Water Reactor
Journal of Scincs, Islamic Rpublic of Iran 19(3): 7381 (008) Univrsiy of Thran, ISSN 10161104 hp://jscincs.u.ac.ir CFDCalculaion of Fluid Flow in a Prssurizd War Racor H. Farajollahi, * A. Ghasmizad,
More informationBrussels, February 28th, 2013 WHAT IS
Brussls, Fbruary 28h, 2013 WHAT IS 1 OPEN SOURCE 2 CLOUD 3 SERVICES 4 BROKER 5 INTERMEDIATION AGGREGATION ARBITRAGE Cloud Srvics Brokr provids a singl consisn inrfac o mulipl diffring providrs, whhr h
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationLecture 20: Emitter Follower and Differential Amplifiers
Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.
More informationBasis risk. When speaking about forward or futures contracts, basis risk is the market
Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also
More informationCONTINUOUS TIME KALMAN FILTER MODELS FOR THE VALUATION OF COMMODITY FUTURES AND OPTIONS
CONTINUOUS TIME KALMAN FILTER MODELS FOR THE VALUATION OF COMMODITY FUTURES AND OPTIONS ANDRÉS GARCÍA MIRANTES DOCTORAL THESIS PhD IN QUANTITATIVE FINANCE AND BANKING UNIVERSIDAD DE CASTILLALA MANCHA
More informationUniplan REIT Portfolio Select UMA Uniplan Investment Counsel, Inc.
Uniplan EIT Porfolio Uniplan Invsmn Counsl, Inc. 22939 W Ovrson d Union Grov, Wisconsin 53182 Syl: SubSyl: Firm AUM: Firm Sragy AUM: EITs EITs Domsic $2.2 billion $1.3 billion Yar Foundd: GIMA Saus: Firm
More informationVoltage level shifting
rek Applicaion Noe Number 1 r. Maciej A. Noras Absrac A brief descripion of volage shifing circuis. 1 Inroducion In applicaions requiring a unipolar A volage signal, he signal may be delivered from a bipolar
More informationLogo Design/Development 1on1
Logo Dsign/Dvlopmnt 1on1 If your company is looking to mak an imprssion and grow in th marktplac, you ll nd a logo. Fortunatly, a good graphic dsignr can crat on for you. Whil th pric tags for thos famous
More informationMotion Along a Straight Line
Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationLG has introduced the NeON 2, with newly developed Cello Technology which improves performance and reliability. Up to 320W 300W
Cllo Tchnology LG has introducd th NON 2, with nwly dvlopd Cllo Tchnology which improvs prformanc and rliability. Up to 320W 300W Cllo Tchnology Cll Connction Elctrically Low Loss Low Strss Optical Absorption
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationSmall Cap Fiduciary Services
Sragy Saus: Closd Sragy closd o nw accouns and opn o addiional asss Th London Company 1800 Baybrry Cour, Sui 301 ichmond, Virginia 23226 Syl: SubSyl: Firm AUM: Firm Sragy AUM: US Valuorind $10.0 billion
More informationTechnological Entrepreneurship : Modeling and Forecasting the Diffusion of Innovation in LCD Monitor Industry
0 Inrnaional Confrnc on Economics and Financ Rsarch IPEDR vol.4 (0 (0 IACSIT Prss, Singaor Tchnological Enrrnurshi : Modling and Forcasing h Diffusion of Innovaion in LCD Monior Indusry LiMing Chuang,
More informationunion scholars program APPLICATION DEADLINE: FEBRUARY 28 YOU CAN CHANGE THE WORLD... AND EARN MONEY FOR COLLEGE AT THE SAME TIME!
union scholars YOU CAN CHANGE THE WORLD... program AND EARN MONEY FOR COLLEGE AT THE SAME TIME! AFSCME Unitd Ngro Collg Fund Harvard Univrsity Labor and Worklif Program APPLICATION DEADLINE: FEBRUARY 28
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationInvestment Grade Fixed Income Select UMA Cincinnati Asset Management
Invsmn Grad Fixd Incom Cincinnai Ass Managmn 8845 Govrnor's Hill Driv Cincinnai, Ohio 45249 Syl: US Taxabl Cor SubSyl: Taxabl Corpora Firm AUM: $2.7 billion Firm Sragy AUM: $2.0 billion Yar Foundd: GIMA
More informationFactorials! Stirling s formula
Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical
More informationUse a highlevel conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects
Chaptr 3: Entity Rlationship Modl Databas Dsign Procss Us a highlvl concptual data modl (ER Modl). Idntify objcts of intrst (ntitis) and rlationships btwn ths objcts Idntify constraints (conditions) End
More informationThe Land Partnerships Handbook. The Land Partnerships Handbook. Using land to unlock business innovation. Second Edition
Th Land Parnrships Handbook Using land o unlock businss innovaion Scond Ediion Using land o unlock businss innovaion Conns 04 Wha is h Land Parnrships approach? 06 Sp 1: Taking sock 08 Sp 2: Finding h
More informationDept. of Heating, Ventilation and AirConditioning. Zentralschweizerisches Technikum Luzern Ingenieurschule HTL
Znralshwizrishs Thnikum Luzrn Ingniurshul HTL Dp. o Haing, Vnilaion Elkrohnik  Mashinnhnik  Hizungs, Lüungs, Klimahnik  Arhikur  Bauingniurwsn Dvlopd in h proj Low Tmpraur Low Cos Ha Pump Haing Sysm
More informationby John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia
Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs
More informationIT Update  August 2006
IT Nws Saus: No Aciv Til: Da: 7726 Summay (Opional): Body: Wlcom Back! Offic of Infomaion Tchnology Upda: IT Upda  Augus 26 Rob K. Blchman, Ph.D. Associa Dico, Offic of Infomaion Tchnology Whil You W
More informationDensity Forecasting of Intraday Call Center Arrivals. using Models Based on Exponential Smoothing
Dnsiy Forcasing of Inraday Call Cnr Arrivals using Modls Basd on Exponnial Soohing Jas W. Taylor Saïd Businss School Univrsiy of Oxford Managn Scinc, 0, Vol. 58, pp. 534549. Addrss for Corrspondnc: Jas
More informationInvestment Grade Fixed Income Fiduciary Services Cincinnati Asset Management
Invsmn Grad Fixd Incom Cincinnai Ass Managmn 8845 Govrnor's Hill Driv Cincinnai, Ohio 45249 Syl: US Taxabl Cor SubSyl: Taxabl Corpora Firm AUM: $2.5 billion Firm Sragy AUM: $1.7 billion Yar Foundd: GIMA
More informationRemember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D
24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More informationME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
More informationEFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS
25 Vol. 3 () JanuaryMarch, pp.375/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut
More informationEnterprise. Welcome. Enterprise. Highlights. Issue 8  Spring 2014. That s Zen thinking
Enrpris Issu 8  Spring 2014 Wlcom Wlcom o anohr diion of Zn Inrn s Enrpris Nwslr. I s an xciing im for UK businsss, wih nw opions for connciviy, nw applicaions in h cloud, and mos of all nw opporuniis
More informationLateef Investment Management, L.P. 300 Drakes Landing Road, Suite 210 Greenbrae, California 94904
Laf Invsmn Managmn, L.P. 300 Draks Landing oad, Sui 210 Grnbra, California 94904 Sragy Saus: Closd Sragy closd o nw accouns and opn o addiional asss Syl: SubSyl: Firm AUM: Firm Sragy AUM: US Larg Cap
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationEntityRelationship Model
EntityRlationship Modl Kuanghua Chn Dpartmnt of Library and Information Scinc National Taiwan Univrsity A Company Databas Kps track of a company s mploys, dpartmnts and projcts Aftr th rquirmnts collction
More informationCharacterization of semiinsulating GaAs:Cr by means of DCCPM technique
Rvu ds Enrgis Rnouvlabls Vol. 12 N 1 (2009) 125 135 Characrizaion of smiinsulaing GaAs:Cr by mans of DCCPM chniqu. ibrmacin 1* and A. Mrazga 2 1 Laboraoir ds Maériaux Smiconducurs Méalliqus, Univrsié
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationAdverse Selection and Moral Hazard in a Model With 2 States of the World
Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,
More informationShort Term Taxable Fixed Income Fiduciary Services Sage Advisory Services, Ltd. Co.
Shor Trm Taxabl Fixd Incom Sag Advisory Srvics, Ld. Co. 5900 Souhws Parkway Building 1 Ausin, Txas 78735 Syl: Shor Trm Fixd Incom SubSyl: Shor Trm Fixd Incom Firm AUM: $11.2 billion Firm Sragy AUM: $1.9
More informationDeer: Predation or Starvation
: Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,
More informationChapter 28 Magnetic Induction
Chapr 8 Magnic nducion Forad Concpual Probls (a) Th agnic quaor is a lin on h surfac of Earh on which Earh s agnic fild is horizonal. A h agnic quaor, how would you orin a fla sh of papr so as o cra h
More informationFACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data
FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among
More informationSilvercrest Asset Management Group, LLC
Silvrcrs Small Cap Valu Silvrcrs Ass Managmn Group, LLC 1330 Avnu of h Amricas, 38h Floor Nw York, Nw York 10019 Syl: SubSyl: Firm AUM: Firm Sragy AUM: US Small Cap Valu laiv Valu Yar Foundd: GIMA Saus:
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationUS Small Cap Value. $8.2 billion $1.3 billion
Sysmaic Financial Managmn, L.P. 300 Frank W. Burr Blvd.  7h Floor Tanck, Nw Jrsy 07666 PODUCT OVEVIEW Sysmaic Financial Managmn, L.P blivs focusing on companis ha can rir all ousanding db wihin n yars
More informationParallel and Distributed Programming. Performance Metrics
Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationMid Cap Growth Select UMA Congress Asset Management Company
Mid Cap Growh Congrss Ass Managmn Company 2 Sapor Lan, 5h Floor Boson, Massachuss 02210 Syl: SubSyl: Firm AUM: Firm Sragy AUM: US Mid Cap Growh Consrvaiv Growh $5.9 billion $1,035,925.3 billion Yar Foundd:
More informationAnalogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar
Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 073807 Ifeachor
More informationPHYS245 Lab: RC circuits
PHYS245 Lab: C circuis Purpose: Undersand he charging and discharging ransien processes of a capacior Display he charging and discharging process using an oscilloscope Undersand he physical meaning of
More informationa seed career program in the s indus tr career handbook for school counselors and college advisors
a sd carr program No o s Thr a lf : r los o f job opp or uni is in h s d indus r y. ns d u s o k Tal. r u u f h abou g r o f n o D ny a m h u abo s. c r u o s r onlin carr handbook for school counslors
More informationFrequency Modulation. Dr. HweePink Tan http://www.cs.tcd.ie/hweepink.tan
Frequency Modulaion Dr. HweePink Tan hp://www.cs.cd.ie/hweepink.tan Lecure maerial was absraced from "Communicaion Sysems" by Simon Haykin. Ouline Day 1 Day 2 Day 3 Angle Modulaion Frequency Modulaion
More informationImportant Information Call Through... 8 Internet Telephony... 6 two PBX systems... 10 Internet Calls... 3 Internet Telephony... 2
Installation and Opration Intrnt Tlphony Adaptr Aurswald Box Indx C I R 884264 03 02/05 Call Duration, maximum...10 Call Through...7 Call Transportation...7 Calls Call Through...7 Intrnt Tlphony...3 two
More informationExotic Options Pricing under Stochastic Volatility
Exoic Opion Pricing undr Sochaic olailiy Nabil AHANI Prliminary draf April 9h Do no quo Conac informaion: Nabil ahani HEC Monréal Canada Rarch Chair in Rik Managmn 3 Chmin d la CôSainCahrin Monral Qubc
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationNoise Power Ratio (NPR) A 65Year Old Telephone System Specification Finds New Life in Modern Wireless Applications.
TUTORIL ois Powr Ratio (PR) 65Yar Old Tlphon Systm Spcification Finds w Lif in Modrn Wirlss pplications ITRODUTIO by Walt Kstr Th concpt of ois Powr Ratio (PR) has bn around sinc th arly days of frquncy
More information4.2 Trigonometric Functions; The Unit Circle
4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.
More informationLecture 9 Analog and Digital I/Q Modulation
Lecure 9 Analog and Digial I/Q Modulaion Analog I/Q Modulaion Time Domain View Polar View Frequency Domain View Digial I/Q Modulaion Phase Shi Keying Consellaions /4/26 Coheren Deecion Transmier Oupu x()
More informationEXTERNAL DEBT, GROWTH AND SUSTAINABILITY. Abstract. Introduction
ETERNAL EBT, GROWTH AN SUSTAINABILIT Absrac Robro Frnkl * Th papr prsns a odl inndd o dfin and discuss h susainabili of rnal dbs in rgn arks. Th firs susainabili condiion is h isnc of a aiu in h dboupu
More information