. Complx rprsntation of continuous-tim priodic signals Eulr s quation jwt cost jsint his is th famous Eulr s quation. Brtrand Russll and Richard Fynman both gav this quation plntiful prais with words such as th most bautiful, profound and subtl xprssion in mathmatics and th most amazing quation in all of mathmatics. Wll ys, it is quit an amazing quation. In signal procssing, this nigmatic quation is quivalnt in importanc to F = ma. his prplxing looking quation was first dvlopd by Eulr (pronouncd Oilr) in th arly800 s. A studnt of Johann Brnoulli, Eulr was th formost scintist of his day. Born in Switzrland, h spnt his latr yars at th Univrsity of St. Ptrsburg in Russia. H prfctd plan and solid gomtry, cratd th first comprhnsiv approach to complx numbrs and is th uncl of modrn calculus. H is not th fathr, that titl is fought ovr by Nwton and Libnitz. Eulr was th first to introduc th concpt of log x x and as functions and it was his fforts that mad th us of iand, th common languag of mathmatics. H drivd th quation and its mor gnral form givn abov. Among his othr contributions wr th consistnt us of th trigonomtric sin, and cosin functions and th us of a symbol for summation. A fathr of 3 childrn, h was a prolific man in all aspcts, in languags, mdicin, botany, gography and all physical scincs. j t h in Eulr s quation is a dciddly confusing concpt. What xactly is th rol of j in j t? W know from algbra that it stands for but what is it doing hr with th sin and cosin? Can w vn visualiz this function? jt h function gos by th nam of complx xponntial (CE). his function is of th gratst importanc in signal procssing and Fourir analysis. W ar going to look at it in this chaptr so both its maning and its application ar clar. jt Lt s ignor th complx xponntial on th lft hand sid and concntrat on th right hand sid of this quation, containing th sin and cosin wavs. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag
jt cos t jsint (.) W can plot this function by stting a valu of, and thn for a rang of t, calculating both cos t and sin t valus. h valu of is a constant. W hav thr valus now, t th indpndnt tim variabl and accosiatd sin and cos valus. With ths thr valus, w can crat th 3-D plot shown in Figur.. im is plottd on th x-axis, and th valus of th two sinusoids on th othr two axs, crating a thr-dimnsional figur. j t Figur. - is a hlix. h xponnt indicats dirction of advanc, (a) positiv xponnt and (b) ngativ xponnt. h xprssion for th ngativ xponntial is writtn as jt cos t jsin t (.) j t h diffrnc btwn j t and can b sn in th Figurs (a) and (b) in that th hlix sms to b moving in th opposit dirction. Now w look at th projction of ach of ths xponntial on th ral and th imaginary plan. In Figur.(a) w plot th projctions of th hlix on Ral and Imaginary plans. In classical trms, ths would b calld th (X, Y) and (Z, Y) plans. h projctions of th complx xponntial ar sin and cosin wavs. h Ral part of th complx xponntial is th cosin wav and Imaginary part is th sin wav. For th ngativ xponnt, or th so calld ngativ complx xponntial, th sin wav is flippd 80 dgrs from th positiv xponntial in Figur.(a). Oftn this xponntial is rfrrd to as having a ngativ frquncy, howvr it is not rally th frquncy that is ngativ. From th xprssion of th ngativ xponnt xponntial in Eq. (.) w s that ngativ sign on th xponntial rsults in th imaginary projction, th sin wav doing a 80 phas chang, or quivalntly bing multiplid by -. h Ral part of th ngativ xponntial is a cosin wav. h imaginary part is a ngativ sin wav. jt jt R cos t Im sint h imaginary part of th positiv xponntial is instad dfind as a positiv sin wav. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag
jt jt R cos t Im sint plot (b) Figur. h projctions of th complx xponntial ar sinusoids. (a) j t 3D plot h ngativ xponntial has as its imaginary part a ngativ sin wav. h positiv xponntial has a positiv sin as its imaginary part. h ral part, which is a cosin, is sam for both. W don t s any ngativ frquncis hr, an ida gnrally associatd with th ngativ complx xponntial. Now that w hav ths two forms of th xponntials, lt s do som math with thm. jwt jwt Adding and subtracting th complx xponntials, and, and thn aftr a littl rarrangmnt, w gt ths nw ways of xprssing a sin and a cosin. j t 3D sin t (cos t jsin t) (cos t j sin t) j j jt jt cos t (cost j sin t) (cost j sin t) jt jt (.3) (.4) Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 3
hs ar vry important xprssions and should b mmorizd if you ar a DSP studnt. If th j in th dnominator of Eq. (.4) bothrs you, thn rmmbr that division by j is sam as multiplication by j. (S appndix A). W can show th quivalnc as follows. sin t j j t j t 0 0 thn jt jt sin t j( ) j( cost jsint cost jsin t) Lt s plot Eq. (.3) and (.4) to s what w gt. Whn w plot th two composit xponntials, w gt just what w show in Eq.(.3) and (.4). h first figur shows that th xponntial has a ral projction of a cosin and th scond only th sin. h hlix is gon, it has bn collapsd into a cosin and a sin. jt jt Figur 3 (a) Plotting / jt jt Imaginary plan (b) Plotting j th ral axis. givs a cosin wav with zro projction on th givs us a sin wav with zro projction on On way to think of this rlationship is to say that both sin and cosin ar composd of complx xponntials. Don t worry about what ths xponntials look lik. hy ar in fact thr-dimnsional functions. It taks two such xponntials to crat on sinusoid. W can crat a cosin by adding th xponntials and crat sin by subtracting. his is a cas of two 3-D functions coming togthr to crat a -D sinusoid. his sounds complicatd but it s actually not an unfamiliar concpt. W can add two -D functions and gt a -D function. An xampl is whn w add a sin and a 80 shiftd sin, w gt a straight lin so a -D function cratd by two 3-D functions should not b a big stumbling block. Unlik th trigonomtric domain whr wavs ar spcifid by asy to undrstand paramtrs (frquncy, amplitud and tim), complx xponntials bing 3-D functions ar spcifid not just by th amplitud but also by th dirction of th phas chang of on Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 4
of its componnts (th sin). In th complx domain, this appars to rsult in frquncy dvloping an additional dgr of frdom, giving it a positiv or ngativ sns. In Cartsian spac, w can s only th sin componnt changing sign. And vn thn, it is not th frquncy that is changing sign but th sign of th function. So how did Eulr s quation com about and why is it so important to signal procssing. W will try to answr that by first loo king at aylor sris rprsntations of th x xponntial, sins and th cosins. h aylor sris xpansion for th two sinusoids is thankfully givn in -D by th infinit sris as 4 6 x x x cos( x)! 4! 6! 3 5 7 x x x sin( x) x 3! 5! 7! (.5) Not that ach on of ths sris is composd of many individual xponntial functions. So vn sin wavs ar actually composd of xponntials. Howvr ths ar ral xponntials that ar non-priodic and not th xact sam thing as th complx xponntials. Complx xponntials ar a spcial class of ral xponntials and ar usd as altrnat fundamntals to th sinusoids, sinc thy ar priodic and offr as of xprssion and calculation which is not obvious at first. Figur -4 - Sin wav as a sum of many xponntials of diffrnt wights. Ral xponntials ar usd in Laplac analysis as th basis st. Ral xponntials ar far mor gnral thn sinusoids and complx xponntials and allow analysis of non-priodic and transint signals, somthing w ar not going to covr in this book. Laplac analysis is a gnral cas of which Fourir is a spcial cas applicabl only to priodic or mostly priodic signals. aylor sris xpansion for th xponntial x givs this sris. (.6) 3 4 x x x x x! 3! 4! Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 5
All thr of ths quations (.5) and (.6) ar straight forward concpts. And indd if w plot ths functions against x, w would gt just what w ar xpcting, th xponntial of and th sinusoids. How clos thy com to th continuous function dpnds on th numbr of trms that ar includd in th summation. h similarity btwn th xponntial and th sinusoids sris in Eq. (.5) and (.6) shows clarly that thr is a rlationship hr. Now lt s chang th xponnt in (.6) from x to j. Not w will us th trm hr instad of t to kp th quation concis. Now w hav by simpl substitution, th xprssion for 3 4 5 ( ) ( ) ( ) ( ) j j j j j j (.7)! 3! 4! 5! j as W know that this sris as j and 4 j, 6 j tc., substituting ths valus, w rwrit 3 4 5 7 j j j j j (.8)! 3! 4! 3! 6! 7! W can sparat out vry othr trm with j as a cofficint to crat a two-part sris, on without th j and th othr with j 4! 4! 6! his is a cosin. 3 5 7 j j j j 3! 3! 7! his is j tims sin. (.9) W s that first part of th sris is a cosin pr Eq. (.5) and th scond part with j as its cofficint is th sris for a sin wav. Hnc w hav just showd that j cos( j ) jsin( j ) (Eq. #) W can now driv som intrsting rsults lik th following. By stting, w can show that By stting 3 j, w can show that j3 / cos( ) j sin( ) 0 j() j cos(3 ) jsin(3 ) 0 j( ) j Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 6
And an anothr intrsting rsult j j cos( ) jsin( ) 0 h purpos of this xrcis is to convinc you that indd th complx xponntial is th sum of only th sins and cosins. h function still rtains a mystrious quality. Howvr, w nd to gt ovr our far of this quation and larn to lov it. h qustion now is why bring up Eulr s quation in contxt of Fourir analysis? Why all this rigmarol about th complx xponntial, why arn t sins and cosins good nough? In Fourir analysis, w computd th cofficints of sins and cosins (th harmonics) sparatly. W also discussd th thr diffrnt formulations of th Fourir sris using sins and cosins, only with cosins and thn with complx xponntials. Fourir analysis using trigonomtric forms is not asy. rig functions ar asy to undrstand but hard to manipulat. Adding and multiplying is a pain. Doing math with xponntials is considrably asir. Now that w hav shown that both sin and cosin ar rlatd to a singl xponntial, w can simplify th math in Fourir sris by working with just on xponntial instad of kping track of both sins and cosin cofficints. his is th main advantag of switching to complx xponntials in formulating a complx form of th Fourir sris. h math looks hard but is actually asir. Howvr complx xponntials bring with thm som concptual difficultis. hy ar hard to visualiz and confusing bcaus w now sm to hav wavs of ngativ frquncis. ak an arbitrary sinusoid. W can dcompos this wav into complx xponntials as follows. ypically whn w dcompos somthing, w do it into a simplr form but hr smingly a mor complx form is bing dployd. A simplr quantity, a cosin wav is now dcomposd into two complx functions. But th nt rsult is that it will mak th procss of th Fourir analysis simplr. W will go from simplicity to complxity and thn to simplicity again. Lt s tak this sinusoid which has a phas trm to complicat things and prsnt it in complx form. x( t) Acos( t ) A A A A j( t ) j( t ) jt j jt j In th last row, w sparatd th xponntial into its powrs. If w xpand this xprssion into trigonomtric domain using Eulr s quation, w s that indd w gt back th trigonomtric cosin wav w startd with. A/ cos( t ) jsin( t ) cos( t ) jsin( t ) Acos( t) Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 7
Using complx xponntials as basis st for th Fourir sris rprsntation Rcall that w usd trigonomtric harmonics as a basis st in chaptr to dvlop th Fourir sris rprsntation. h targt signal was mappd on to a st of sinusoidal harmonics, such as ths basd on fundamntal frquncy of 0. S sin 0t, sin 0t, sin3 0t,, sin k 0t, And of cours w said that this choic of orthogonal signals maks it asy to dtrmin th cofficint of ach bcaus of quation ##. What quation ## says is that th st of sinusoids form an orthogonal basis st. t t sin k t sin( m t) dt 0 whn k m With complx xponntials, w can mak th sam statmnt. A st of complx xponntials givn by st S, can also b usd as a basis st for crating a complx form of th Fourir sris. S,,,, j3 t j t j t jm t 0 0 0 0 hs complx xponntials also form an orthogonal st, making thm asy to sparat from ach othr in an arbitrary signal. his is th main rason why w pick orthogonal signals to rprsnt somthing. Just as our 3-D world is dfind along thr orthogonal axs, X, Y and Z, hr our signals can b similarly projctd on a K-dimnsional orthogonal st. Complx Fourir sris rprsntation Rcall that th Fourir sris is a sum of wightd sinusoids. W assumd that tim is continuous but frquncy rsolution is not. Frquncy on th othr hand taks on discrt harmonic valus. If th fundamntal frquncy is, thn ach is an intgr multipl of or k hnc is discrt no mattr how larg k gts. h distanc btwn ach harmonic rmains th sam, 0. k f ( t) a a cos( t) b sin( t) 0 K k k n k k k K (.0) h cofficints a 0, a k and b k (which w call th trigonomtric cofficints) ar calculatd by (from chaptr ) 0 a0 f () t dt 0 0 Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 8
0 ak f ( t) coskt dt (.) 0 0 0 bk f ( t) sin nt dt 0 0 h prsnc of th intgral tlls us that tim is continuous. Now substitut Eq. (.3), and Eq. (.4) as th dfinition of sin and cosin into Eq. (.9), to gt K K ak jkt jkt bk jkt jkt f ( t) a0 ( ) ( ) j (.) k k Mak th sam substitution into th solution for th cofficints, Eq. (.0). 0 jkt jkt ak f ( t) ( ) dt (.3) 0 0 0 0 0 jkt jkt bk f () t dt (.4) j Rarranging Eq. (.) so that th ach xponntial is sparatd, w gt f ( t) a ( a jb ) ( a jb ) 0 jkt jkt k k k k (.5) k k h cofficints in Eq. (.5) can also b xpandd as follows. 0 0 jkt jkt ak f ( t) dt f ( t) dt (.6) 0 0 0 0 Look at this quation carfully. You s that th trigonomtric cofficint is split into two parts now, on for ach of th xponntials. o mak th nw cofficints concis, lt s rdfin thm with capital lttrs as complx cofficints, A k and B k Ak ( ak jbk ) (.7) Bk ( ak jbk ) (.8) Substituting ths nw dfinitions of th cofficints into Eq. (.5), w gt a much simplr rprsntation. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 9
0 jkt k k k f () t a A B k jkt 0 jkt Ak f () t dt 0 0 0 jkt Bk f () t dt 0 0 (.9) It is clar from this quation that A k can b thought of as th cofficint of th positiv xponntial and B k th cofficint of th ngativ xponntial. hs cofficints ar not th sam as th ons w computd in th trigonomtric form. hy ar complx combinations of th trigonomtric cofficints a k and b k. h trm a 0 stands for th DC componnt. W gnrally do not lik DC trms so w will rmov it by xpanding th rang of th scond trm from 0 to. Rwrit Eq. (.9) as jkt jkt k k (.0) k0 k0 f () t A B h abov quation can b simplifid still furthr by xtnding th rang of cofficints from - to. W can do this by changing th sign of th indx which was on-sidd bcaus w had includd both positiv and ngativ xponntials xplicitly. A positiv indx k allows inclusion of both of ths. Now both trms can b combind into on with a two-sidd indx to writ a much mor compact and lgant quation for th Fourir sris. Now w do not nd th ngativ xponntial in th quation. h indx taks car of that. And hr is a much shortr quation for Fourir sris in th complx domain. jk t f () t C (.) k k h cofficint C k in Eq. (.) is givn by 0 jk0t C f () t dt k 0 (.) 0 C k is also qual to Ck Ak jb (.3) k h Eq. (.) is calld th xponntial or th complx form of th Fourir sris. It is rigorously rlatd to th sinusoidal form but its cofficints C k ar complx. W will us it from now on as it is most usd form of th Fourir sris. It crtainly shortr than th trigonomtric form. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 0
Not that our indx (hnc th frquncy) is always positiv in th trigonomtric form and thrfor th spctrum is on-sidd. h x-axis for th on-sidd spctrum is plottd against frquncy starting at a positiv fundamntal frquncy. All k intgr multipls of th fundamntal frquncy ar positiv as wll. Bcaus th indx is positiv, all frquncis ar positiv. With complx formulation, th indx k can rang from - to +. W start with th ngativ indx, go through calculations of all ngativ xponnt xponntials and thn th positiv ons. Not that at no point is th fundamntal frquncy vr ngativ. Hnc it is not th frquncy of th xponntial that is ngativ but th indx. h xponntial with th ngativ indx k is diffrnt from th positiv xponntial in that th sign of th imaginary part is ngativ. W s nothing hr that says that th frquncy has suddnly bcom ngativ bcaus of th xponntial xponnt bing a ngativ. sin sin jkt cos k t j k t jkt cos k t j k t W now quat this form with th trigonomtric form which smingly had only positiv frquncis. o do that w look at what it taks to rprsnt a sin and a cosin using complx xponntials. cos( k t) jk t sin( k t) jk t j W rquir both a ngativ indx xponntial and a positiv indx xponntial for both th sins and cosins. Whr indx k is always positiv on th lft hand sid of this quation, it is both ngativ and positiv on th right sid. his is th conundrum that traps us. Whr in trigonomtric form a positiv indx is nough to fully and compltly rprsnt th signal, in complx form it taks a doubl-sidd indx. Our spctrum for th complx cas to b complt is plotting th indx tims th frquncy and not just frquncy on th x-axis. But w vry quickly los sight of this fact. W start talking about positiv and ngativ frquncis bcaus w confus th rang of th indx with th frquncy. In a doubl-sidd spctrum w ar using th word frquncy as an altrnat nam for what w ar rally plotting, th indx tims th frquncy. Calling it frquncy allows us to mak som intuitiv sns but thn hav to worry about what a ngativ frquncy mans. W rsort to simpl rprsntation of complx idas for daily us. Bcaus of th plotting convntion, th ngativ indx is oft-forgottn and th axis is rfrrd to as th frquncy axis, spanning in both positiv and ngativ domains. But in this book, I maintain that thr is no such thing as a ngativ frquncy. h ida coms from confusion causd what th x-axis rprsnts. h complx cofficint valus ar on-half of what thy ar calculatd in th trigonomtric domain. (S th factor ½ in Eq. (.).) Now hr w mak up anothr story that this is bcaus th frquncy is bing split into two parts, a ngativ part and positiv part. his is how most books try to xplain th conundrum of positiv and jk t jk t Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag
ngativ frquncy in rlation to Fourir analysis. But thy ar just trying to xplain a plotting convntion. h ral story is that w ar not plotting frquncy on th x-axis but th trm k which is oftn just thought of as plain frquncy. Exampl. Comput complx cofficints of a cosin wav. f( t) Acos t A A 0 j( k ) 0t j( k ) 0t (.4) his is of cours a vry trivial xampl. hr is only on sinusoid hr. From Eq. (.4), w know w can rprsnt a cosin of frquncy by th sum of two complx xponntials of th sam frquncy. his xampl is so simpl that w can dduc th trigonomtric cofficints just by looking at th xprssion. In fact th quation is itslf th prfct rprsntation. In Eq. (.4), w writ this cosin in its xponntial rprsntation. h complx cofficints ar of magnitud A/ locatd at k, and k. W plot th trigonomtric cofficints, c k, in Figur.5(a) as th singl-sidd spctrum and th xponntial cofficints, C k, as th doubl sidd form in.5(b). h x- axis is variabl is k. 0 Not it is not th frquncy that is ngativ but th harmonic indx k. Howvr in a typical plot, th x-axis is labld as frquncy. In ths plots, w hav labld it spcifically as what it is, th trm k. Figur.5 Amplitud spctrum of Acos t Exampl. Comput complx cofficints of a sin wav f ( t) Asin 0 t A A j j j k t j k t 0 0 (.5) Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag
Figur.6 Amplitud spctrum of Asin t his xampl is just th sam as th cosin xampl. h singl-sidd spctrum is asy. It is simply a harmonic of magnitud A and locatd at k = just as it is for th cosin wav in Exampl.. In Eq. (.5), w s th complx form with magnituds of two complx xponntials of A/ and -A/ locatd at k. Howvr thr is a j in th dnominator. What to do with this? h prsnc of j tlls us that th cofficints ar on th Imaginary axis, so thy ar to b plottd on th Imaginary plan, right-angl to th plan on which a cosin lis. Drawn in -D form it has no computational ffct, only that th vrtical axis is calld th Imaginary axis. But whn w hav both cosins and sin wavs prsnt in a signal, th cofficints of ths two hav to b combind not linarly but as a vctor sum as sn in Figur.6b. Whn plotting th magnitud, it no longr falls in purly ral or Imaginary plans so in this cas, w call th vrtical axis, just th magnitud. hr is a bit of trminology sloppinss hr oftn that is what maks signal procssing so confusing. Figur.7-Magnitud of th rsultant vctor Exampl.3 Comput th cofficints of f( t) A(cos t sin t ) W can writ this wav as A A A A j j j t j t j t j t (.6) W can pick out th trigonomtric cofficints from at th first quation. It is simply A for th cosin and A for sin with magnitud qual to squar root of A locatd at =. W gt th complx cofficints by looking at th cofficints of th two xponntials in jt th Eq. (.6). h xponntial has two cofficints, at 90 dgrs to ach othr, ach of magnitud A/. h vctor sum of ths is A. Sam for th ngativ xponntial, Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 3
xcpt th amplitud contribution from th sin is ngativ. Howvr, th vctor sum or th magnitud for both is th sam and always positiv. his is shown in Figur.8(c) drawn in a mor convntional styl showing only th vctor sum. Not that th on-sidd spctrum amplitud is twic that of complx. Figur.8 Amplitud spctrum of Asin t Acos t Exampl.4 Comput cofficints of th complx signal f( t) Acos t jasin t W can writ this as A A A A jt A jt jt jt jt Now hr w s somthing diffrnt. h cofficints from sin and cosin for th ngativ xponntial cancl out. On th positiv sid, th contribution from sin and cosin ar coincidnt and add. So w s a singl valu at positiv indx of k = only. For this signal both th singl and doubl-sidd spctrums ar idntical. his is a surprising and prhaps a countr-intuitiv rsult. Important obsrvation: Only tru ral signals hav symmtrical spctrums about th origin. Complx signals do not. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 4
Figur.9 Doubl-sidd spctrum of Acos t jasint Exampl.5 Comput th cofficints of a constant signal, A. W can writ this function x() t A or as an xponntial of zro frquncy. f ( t) Acos( 0) t A jt A jt A From th first rprsntation, w gt th trigonomtric cofficint = A at = 0. From th complx rprsntation w gt th two complx cofficints, both of amplitud A/ and A/ but both at k= 0 so thir sum is A which is xactly th sam as in th trigonomtric rprsntation. h function f(t), a constant is a non-changing function of tim and w classify it as a DC signal. h DC componnt, if any, always shows up at th origin for this rason. h singl and doubl sidd spctrum hr ar sam as wll. Important Obsrvation: A componnt at zro frquncy mans that th signal is not zroman. Figur -0 Doubl-sidd spctrum of A Exampl.6 Comput cofficints of x t ( ) cos ( t) W can xprss this function in complx form as jt jt jt jt Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 5
Figur - Doubl-sidd amplitud spctrum of cos ( t) Not that th function has a componnt at zro frquncy. h othr componnts ar at k. Important Obsrvation: A squard signal by dfinition always positiv so has a zro-frquncy componnt in th spctrum. Exampl.7 Comput th cofficints of x( t) cos( t )cos( t ) W can xprss this in complx form by making us of a trigonomtric idntity first. cos( t) cos(3 t) jt jt j3t j3t Of cours doing this in trigonomtric form would hav bn just as asy. But that is not always tru. W draw th spctrum as in Figur. Whnvr w multiply a sinusoid with anothr of a diffrnt frquncy w will gt frquncis that ar th sum and diffrnc of th two frquncis in th spctrum. hs ar calld bat frquncis. Figur - Doubl-sidd amplitud spctrum of cos( t)cos( t) Exampl.8 Comput th complx cofficints of this ral signal. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 6
f ( t).8cos 4 t.6sin t.8cos 8 t.3sin 4 t h signal has thr harmonics, at k,, 7. W can writ this quation in complx form as 0.8 0.8 0.6 0.6 j j j k t j k t j k t j k t 0.8 0.8 0.3 0.3 j j j k t j k t j k 7 t j k 7 t Hr w hav contributions from both sin and cosin at k, so ths hav to b vctor summd. h contributions at k coms only from a cosin and at k 7 only from a sin. Not w plot ths on th sam lin at full amplitud as if j dos not xist in th quation. (W will drop mntioning th indx k and call it frquncy to b consistnt with th common usag.) Figur -3- wo-sidd amplitud spctrum Exampl.9 Comput th complx cofficints of this ral signal with phas trms. x( t) 36cos(4 t ) j4sin(4 t 3) j6sin(0 t.5) W convrt this to th complx form as 4t 4 t 4t 3 4 t 3 0t.5 0 t.5 j4t j 3 j j4t j 3 j j0t j.5 j0t j.5 x( t) 3 3 3 3 3 3 3 3 3 3 Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 7
h magnituds of th xponntials com from th phasors in parnthsis. o add thm w nd to convrt thm first to rctangular form as follows. (S Appndix). h CE j 4t has th following cofficints. 3 j4t j 3 j 3cos() cos(3) 3sin() sin(3) 6.6 j4t Similarly, th cofficint of th ngativ xponntial is 3 j4t j 3 j 3cos() cos(3) 3sin() sin(3) 3.46 j4t W draw th spctrum in Figur -4 and not that th spctrum is not symmtric bcaus th signal is complx. Important Obsrvation: Most signals w work with ar complx hnc thir spctrums ar rarly symmtrical. Figur -4 - wo-sidd spctrum of a complx signal Now a difficult but a vry important xampl, a priodic signal of rpating squar pulss. Exampl.0 Comput th Fourir cofficints of th following priodic squar wav. h squar wav is of amplitud v that lasts sconds and rpats in sconds. Figur 5 Squar wav Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 8
First w comput th cofficints for th gnral cas for puls tim qual to rpat tim qual to sc. h trm is calld th duty cycl of th wav. sc. and / jk f0t Ck dt / Not that outsid of t C k, th function is zro. his a prtty asy intgral givn by t j k / jk k / / j k j k / / / / jk k j ( k / 0 ) sin k k ( ) Rplacing th duty cycl trm with r, th quation bcoms fairly asy to undrstand. sin( k r) Ck r r sinc( k r) k r h duty cycl of this signal is qual to r Substituting that in th abov quation, w can writ th xprssion for th cofficints of this signal as Ck sinc( k / ) (.) his is th sinc function. It coms up so oftn in signal procssing that that it is probably th scond most important quation in DSP aftr th Eulr s quation. Now w can plot th cofficints of th rpating continuous tim squar puls cofficints for various duty cycls. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 9
Figur.6 Fourir cofficints of a squar puls Not how th pakdnss changs as th duty cycl of th wav changs. A narrow puls rlativ to th priod in Figur -6(a), has a shallowr rspons than on that taks up mor of th priod. h zro-crossings occur at invrs of th duty cycl. For r = 0.5, th crossing occurs at k =, for r =.5, th crossing is at k = 4 and for r =.75, th crossing occurs at n =.33. At r =.0, th puls would bcom a flat lin and it will hav an impuls rspons. For vry small r, th puls is dlta function-lik and th rspons will go to a flat lin. Although th quation for this function is fairly asy, it taks a whil to dvlop intuitiv fling. I cannot ovr mphasiz th importanc of this signal and you ought to spnd som tim playing around with th paramtrs to undrstand th ffct. W will of cours kp coming back to this in th nxt chaptrs many tims. In this sction, w covrd th complx form of Fourir sris as a prlud to th nxt topic which is Fourir ransform. W s that vn though th tim domain function is a continuous function, th Fourir sris cofficints and hnc th spctrum dvlopd is discrt. In most ral applications th signals ar sampld and hnc ar not continuous. W will s latr how that affcts th Fourir analysis. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 0
Summary of chaptr. W us complx xponntials to rprsnt th sins and cosins in th trigonomtric form of th Fourir sris quation. cos( k t) jk t sin( k t) jk t j jk t jk t. h harmonic indx which was positiv for th trigonomtric, now must xtnd to both sids of th origin, from K k K K 0 k 0 K and this is oftn 0 rad as rprsnting ngativ frquncy whn in fact it is th indx that is ngativ. 3. h x-axis now rprsnt valus from 4. h Fourir sris cofficints instad of bing of thr typs can now b rprsntd by a singl quation. 0 jk0t C f () t dt k 0 0 5. h fundamntal proprtis rmain th sam, th tim in this rprsntation is continuous and frquncy is discrt with indx k, which is an intgr. 6. h amplitud spctrum obtaind from th complx rprsntation looks diffrnt from th on-sidd spctrum. W call this spctrum a two-sidd spctrum. h powr for a particular harmonic is now split in half for th positiv indx (so calld positiv frquncy) and half for th ngativ indx (calld ngativ frquncy). h 0 th componnt howvr rmains th sam. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag
Appndix A: A littl bit about complx numbrs In DSP, w us complx numbrs to dnot quantitis that hav mor than on paramtr associatd with thm. A point in a plan is on xampl. It has a y coordinat and a x coordinat. Anothr xampl is a sin wav, it has a frquncy and a phas. h two parts of a complx numbr ar dnotd by th trms Ral and Imaginary, but th Imaginary part is just as ral as th Ral part. Both ar qually important bcaus thy ar ndd to nail down a physical signal. h signals travling through air ar ral signals and it is only th procssing that is don in th complx domain. hr is a vry ral analogy that will mak this clar. Whn you har a sound, th procssing is don by our brains with two orthogonally placd rcivrs, th ars. h ars har th sound with slightly diffrnt phas and tim dlay. h rcivd signal is diffrnt by th two ars and from this our brains can driv fair amount of information about th dirction, amplitud and frquncy of th sound. So although ys, most signals ar ral, th procssing nds to b don in complx plan if w ar to driv maximum information. Figur A- Multiplication by j shifts th location 0f a point on a plan by 90 o. h concpt of complx numbrs starts with ral numbrs as a point on a lin. Multiplication of a numbr by - rotats that point 80 about th origin on th numbr lin. If a point is 3, thn multiplication by - maks it -3 and it is now locatd 80 from +3 on th numbr lin. Multiplication by - can b sn as 80 shift. Multiplying this rotatd numbr again by -, givs th original numbr back, which is to say by adding anothr 80 shift. So multiplication by rsults in a 360 shift. What do w hav to do to shift a numbr off th lin, say by 90? his is whr j coms in. Multiply 3 by j, so it bcoms 3j. Whr do w plot it now? Hrin lis our answr to what multiplication with j dos. h magnitud of th numbr stays xactly th sam, 3j is th sam as 3, xcpt that multiplication with j shifts th angl of this numbr by +90 o. So instad of an X-axis numbr, it bcoms a Y-axis numbr. It is no longr locatd on th ral numbr lin whr it was. Each subsqunt multiplication by j rotats it furthr by 90 o in anti-clockwis mannr in th X-Y plan as shown in Figur. 3 bcom 3j, thn -3 and thn -3j and back to 3 doing a complt 360 o dgr turn. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag
Qustion: What dos division by j man? Answr: It is sam as multiplying by j. x j jx jx j j his is ssntially th concpt of complx numbrs. Complx numbrs oftn thought of as complicatd numbrs follow all of th common ruls of mathmatics. Prhaps a bttr nam for complx numbrs would hav bn D numbrs. o furthr complicat mattrs, th axs, which wr calld X and Y in Cartsian mathmatics ar now calld rspctivly Ral and Imaginary. Why so? Is th quantity 3j any lss ral than 3? his smantic confusion is th unfortunat rsult of th naming convntion of complx numbrs and hlps to mak thm confusing, complicatd and of cours complx. Now lt s compar how a numbr is rprsntd in th complx plan. Figur A- a. A point is spac on a Cartsian diagram b. Plotting a complx function on a complx plan Plot a complx numbr, 3 + j4. In a Cartsian plot w hav th usual X-Y axs and w writ this numbr as (3, 4) indicating 3 units on th X-axis and 4 units in th Y-axis. W can rprsnt this numbr in a complx plan in two ways. On form is calld th rctangular form and is givn as z x jy h part with th j is calld th imaginary part (although of cours it is a ral numbr) and th on without is calld th ral part. Hr 3 is th Ral part of z and 4 is th Imaginary part. Both ar ral numbrs of cours, I add that just to confus you som mor. Not whn w rfr to th imaginary part, w do not includ j. h symbol j is thr to rmind you that this part (th imaginary part) lis on a diffrnt axis. R z x Im( z) y Altrnat form of a complx numbr is th polar form. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 3
zm Whr M is th magnitud and its angl with th ral axis. h polar form which looks lik a vctor and in ssnc it is, is calld a phasor in signal procssing. his form coms from circuit analysis and is vry usful is that ralm. W also us it in signal procssing but it sms to caus som concptual difficulty hr. Mainly bcaus, unlik in circuit analysis, in signal procssing tim is important. W ar intrstd in signals in tim domain and th phasor which a tim-lss concpt is confusing. h phas as th trm is usd in signal procssing is kind of th initial valu of phas, whr it is an angl in vctor trminology. j t Qustion: If z A thn what is its rctangular form? Answr: z Acost Aj sint. W just substitutd th Eulr s quation for th complx xponntial cost jsin t R Im j t j t. hink of as a shorthand functional notation for th xprssion.h ral and imaginary parts of z ar givn by z Acost z Asint Convrting forms Rul:. Givn a rctangular form z x jy thn its polar form is qual to M x y tan y / x x y y x tan / if x <0. Givn a polar form zm thn its rctangular form is givn by xjy Mcos jmsin Exampl: Convrt z 5.97 to rctangular form Ral part = 5cos(.97) 3 Imaginary part = 5sin(.97) 4 Z = 3 + j4 Exampl: Convrt z j to rctangular form j M Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 4
y arctan x y arctan If x < 0 x =arctan 3 / 4 z 3 / 4 Exampl: convrt z = +j to polar form j and arctan() /4.785 Z = Adding and multiplying Add in rctangular form, multiply in polar. Its asir this way. Rul: Givn z a jb and z c jd thn z z ( a c) j( b d). Rul: Givn z M and z M thn z z z M M ( ) 3 Exampl: Add z.785 and z 5.97 Convrt both to rctangular form. z j and z 3 4j z3 ( 3) j( 4) 4 j5 Exampl: Multiply z j and z 3 4j First convrt to polar form and thn multiply. Although multiplying ths two complx numbrs in rctangular format looks asy in gnral that is not th cas. Polar form is bttr for multiplication and division. z.785 z 5.97 =5.7 Exampl: Divid z j and z 3 4j z 5.97 z.785 z 5 z3.97.785 z thn Conjugation h conjugat for a complx numbr z, is givn by * z x jy. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 5
j t For a complx xponntial j t is th complx conjugat of. In polar format th complx conjugat is sam phasor but rotating in th opposit dirction. z M * z M Usful proprtis of complx conjugats z * zz his rlationship is usd to comput th powr of th signal. h magnitud of th signal can b computd by half th sum of th signal and its complx conjugat. Not th imaginary part cancls out in this sum. * z z z rms usd in Chaptr j t Continuous-tim Complx xponntial of frquncy jwt R cos( t ) jwt Im g sin( t ) jwt R cos( t ) jwt Im g sin( t ) C Complx Cofficints of Fourir sris k Squar puls tim Copyright 05 Charan Langton All Rights rsrvd. Chap - Complx rprsntation of continuous-tim priodic signals Charan Langton Pag 6