Sensors. data, information, signals. Actuators. System Environment. Figure 3.1: A general system immersed in its environment.

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1 Chaper Basics of Sysems. Wha are Sysems? As previously menioned, a signal represens some underlying physical variable of ineres. As an absracion, we consider a signal as simply a real-valued (or someimes complex-valued) funcion defined on some domain. For example, we migh represen a signal as jus a real-valued funcion of ime x( ) where x() represens he value of he signal a ime. Similarly, we will consider an absracion of he general sysem of Figure. from Chaper. Informaion Processing daa, informaion, signals Sensors Acuaors Sysem Environmen Figure.: A general sysem immersed in is environmen. In his figure, we hink of a sysem as some par of he physical world ha ineracs wih is environmen and is designed o carry ou some ask. The box labeled informaion processing receives inpu signals from various sensors and produces oupu signals for various acuaors. Thus, we hink of he sysem as ransforming inpu signals ino oupu signals. A number of subasks wihin c 999- by Sanjeev R. Kulkarni. All righs reserved. Please do no disribue wihou permission of he auhor. Lecure Noes for ELE Inroducion o Elecrical Signals and Sysems. Thanks o Rich Radke and Sean McCormick for producing he figures.

2 CHAPTER. BASICS OF SYSTEMS informaion processing hemselves perform operaions on a signal o produce anoher signal i.e., hey ransform one funcion x() o produce a new funcion y(). The erm sysem is used in his absrac and echnical sense o refer o such mappings ha ake a signal as inpu and produce anoher signal as oupu. As we ll see, by making his absracion and imposing addiional assumpions, we ll be able o sudy special ypes of sysems in a precise way ha leads o useful insighs and resuls. As wih signals, we can have various ypes of sysems such as coninuousime, discree-ime, analog, digial, ec. We will le x() or x[n] denoe he inpu signal for coninuous-ime and discree-ime sysems, respecively, and le y() and y[n] denoe he corresponding he oupu signals. H = H[ ] will denoe he sysem, so ha y() = H[x()] and similarly in he discree-ime case wih () replaced by [n]. I is sandard o represen his relaionship for he sysem H wih a schemaic diagram as shown in Figure.. inpu signal x() H y() oupu signal Figure.: Absracion of a general sysem. The range of possible sysems is so wide ha i is difficul o say much ha applies o compleely general sysems. However, by resricing aenion o special classes of sysems a se of ools can be developed ha helps in he modeling, analysis, and design of sysems. Two paricularly useful properies are lineariy and ime-invariance, which we describe nex. A rich heory has been developed for his class of sysems. Alhough real physical sysems are rarely exacly linear or ime-invarian, resuls for he idealized assumpions sill provide useful models for undersanding more complex sysems.. Lineariy One exremely useful class of sysems is he class of linear sysems. Roughly, a sysem is linear if adding or scaling inpus o he sysem simply resuls in adding or scaling he corresponding oupus. More precisely, a sysem H is linear if he following wo condiions are saisfied:.. for all consans a and inpu signals x(); and H[a x()] = ah[x()] (.) H[x () + x ()] = H[x ()] + H[x ()] (.)

3 .. TIME-INVARIANCE for all inpu signals x (), x (). The firs condiion implies, for example, ha once we know he oupu of he sysem due o x(), hen he oupu due o x() will be wice he original oupu. The second condiion implies ha o find he oupu due o he sum of wo signals x () + x (), we can firs find he oupus due o each signal separaely and hen jus sum hese oupus. The wo condiions (.) and (.) can be combined ino he single condiion H[a x () + a x ()] = a H[x ()] + a H[x ()] (.) for all consans a, a and inpu signals x (), x (). This propery is ofen referred o as superposiion. The definiion of lineariy can also be wrien for discree-ime sysems in he obvious way. The beauy of linear sysems arises from he superposiion propery, which les us analyze oupus due o very complicaed inpus by firs jus considering some sandard se of simple inpus, and hen considering addiive combinaions of hese.. Time-Invariance Time-invariance is anoher useful concep o consider in sudying sysems. Timeinvariance means ha whaever behavior a sysem has, if he inpus were delayed by some amoun of ime, hen he behavior of he sysem would be unchanged oher han jus being delayed by ha same amoun of ime. More precisely, suppose y() denoes he oupu of he sysem due o x(), so ha as before y() = H[x()]. Then H is ime-invarian if H[x( )] = y( ) (.4) for all imes and inpu signals x(). If a sysem is no ime-invarian hen i is said o be ime-varying. In his condiion, denoes he delay, so ha x( ) is he signal x() delayed by ime. Noe ha is allowed o be negaive, so ha he condiion also applies o saring he inpu early as well as delaying i. The lef hand side of (.4) denoes he oupu of he sysem if we apply he delayed inpu x( ). The righ hand side is he oupu y() delayed by ime. Tha is, he lef hand side firs delays he inpu o ge x( ) and hen applies his o he sysem, while he righ hand side firs pus x() ino he sysem and hen delays he resuling oupu. Time-invariance requires hese o give he same final resul. The noion of ime-invariance can be defined in a similar way for discreeime sysems. Of course, in his case, since he ime variable n only akes on ineger values, we only need o consider ineger shifs in he equaion for ime-invariance. Like lineariy, he power of ime-invariance lies in he nice behavioral properies his condiion imposes on he sysem. I allows us o use knowledge of

4 4 CHAPTER. BASICS OF SYSTEMS he sysem behavior on cerain inpus o deermine wha he sysem would do on oher (namely shifed) inpus. The wo condiions ogeher (lineariy and ime-invariance) are paricularly powerful, as we ll see in he nex few secions. In fac, wih hese wo condiions he behavior of a sysem o any inpu can be deermined once we know he behavior due o jus one special inpu namely he dela funcion..4 Impulse Response and Oupu of LTI Sysems We now focus on he class of linear ime-invarian sysems (also known as LTI sysems). Wih hese wo properies (i.e., lineariy and ime-invariance) a rich heory can be developed. Here we will ouch on jus some of he basic ools for describing and analyzing such sysems, bu hese ools form he basis for much of he field of signals and sysems. We will firs consider he case of discree-ime sysems. Suppose we apply he discree-ime dela funcion as he inpu o he sysem, namely we le x[n] = δ[n]. The oupu will be some signal which we will denoe as h[n], so ha h[n] = H[δ[n]] The funcion h[n] is called he impulse response of he sysem. I is he oupu (response) of he sysem when he inpu is a dela funcion (impulse). For he momen, le s no be concerned wih wha paricular values we observe for he impulse response, bu only ha we observe some oupu ha we will record and manipulae in various ways. Our focus is on seeing wheher we can use our knowledge ha he oupu is h[n] when he inpu is δ[n] o predic wha oupu we will ge for some oher inpus. Wihou some resricions or assumpions on he sysem, we have no hope of predicing he oupu on some new inpus. However, recall ha we have assumed ha he sysem is LTI, which are very powerful condiions indeed, as we shall now see. Firs, since he sysem is ime-invarian, we can easily deermine he inpu o he shifed dela funcion δ[n n ]. This inpu is jus a delayed version of he dela funcion δ[n] ha akes he value a ime n = n and is elsewhere. Time-invariance ells us ha he oupu due o his delayed dela funcion is jus h[n n ], namely he oupu due o δ[n] bu jus shifed by he same amoun. In equaion form, H[δ[n n ]] = h[n n ] Now, we can use lineariy o deermine he oupu due o many oher inpus. For example, wha is he oupu due o applying δ[n]? Well, we already know ha he oupu h[n] resuls from he inpu δ[n]. So by he scaling propery of lineariy, we mus ge he oupu h[n] when we apply δ[n]. Wha abou he oupu due o δ[n ]? Time-invariance ells us ha he he oupu mus be he same as he oupu due o δ[n] bu jus shifed by one uni.

5 .4. IMPULSE RESPONSE AND OUTPUT OF LTI SYSTEMS 5 I.e., he oupu is h[n ]. Likewise, if we apply δ[n] + δ[n ], he addiive propery of lineariy ogeher wih ime-invariance ells us ha he oupu mus be h[n] + h[n ]. Now, we can use boh he scaling and addiive properies of lineariy ogeher wih ime-invariance, o deermine ha if we apply he inpu δ[n] + δ[n ], he oupu will be h[n] + h[n ]. Thus, by jus knowing he oupu due o δ[n] and using lineariy and ime-invariance, we can deermine he oupu due o some prey complicaed inpus by jus building up hese inpus in erms of shifed and scaled dela funcions, and hen combining he correspondingly shifed and scaled versions of he impulse response. In fac, by carrying his idea furher, we see ha by appropriae combinaions of scaled and shifed dela funcions, we can in fac synhesize any desired inpu funcion. Suppose we wan o creae an arbirary inpu x[n]. We can sar by making sure we creae he proper value a ime, namely x[]. To do his we include δ[n] scaled by x[], i.e., x[]δ[n]. Nex, o ge he desired value a ime, x[], we simply add x[]δ[n ], which is δ[n ] scaled by x[]. Then o ge he desired value a ime, we add x[ ]δ[n + ]. Noice ha each shifed dela funcion is non-zero only a one ime, so ha adding new erms won ruin he values we ve already creaed. Thus, coninuing in his manner, we can wrie he inpu signal x[n] as x[n] = + x[ ]δ[n + ] + x[ ]δ[n + ] + x[]δ[n] or using summaion noaion +x[]δ[n + ] + x[]δ[n ] + x[n] = i= x[i]δ[n i] (.5) This is acually jus a version of he sifing propery of he dela funcion ha we encounered in Secion XX. Alhough expressing x[n] in his way migh seem degenerae or no paricularly useful, we have acually achieved a grea deal. Iniially, we had no idea wha oupu we would ge if he inpu x[n] was applied o he sysem. Bu now, we can use lineariy and ime-invariance as we did before o see ha he oupu will jus be a combinaion of a bunch (albei, poenially an infinie number) of shifed and scaled versions of he impulse response h[n]. The fac ha he scaling erms are he values of he inpu signal iself shouldn boher or surprise us. On he conrary, ha is exacly wha we did above in finding he oupu due o δ[n] + δ[n ], he only difference being ha we used specific numbers ( and ) for values of he inpu signal. In fac, he oupu of he sysem generally should depend on he inpu signal, oherwise i would be a dull sysem. Formally carrying ou he manipulaions, we ge ha he oupu y[n] due o he inpu x[n] is given by y[n] = H[x[n]]

6 6 CHAPTER. BASICS OF SYSTEMS = H = = [ i= i= i= x[i]δ[n i] x[i]h[δ[n i]] x[i]h[n i] The firs equaliy is he definiion of y[n]. The second uses he expression (.5) for x[n] obained above, which is really he sifing propery of he dela funcion. The hird equaliy follows from lineariy, where we have used boh he addiive propery o bring H inside he sum, and he scaling propery o bring x[i] ou of H[ ]. The fourh equaliy follows from ime-invariance and he definiion of he impulse response h[n]. This final resul is imporan enough o repea as a separae equaion. Namely, if h[n] is he impulse response of a sysem, and we apply an inpu x[n] hen he oupu y[n] is given by y[n] = i= ] x[i]h[n i] (.6) A few commens on his imporan resul are in order. The somewha complicaed/convolued operaion on he righ hand side of Equaion (.6) combining x[n] and h[n] is appropriaely called he convoluion of x[n] and h[n]. I arises so ofen ha i is given he special noaion (x h)[n]. This operaion will be discussed in more deail in he nex secion. Alhough he expression for he convoluion looks complicaed, he remarkable hing is ha his resul implies ha once we know he impulse response of a sysem (i.e., he oupu due o he single inpu δ[n]), hen we can find he oupu due o any inpu! In fac, he oupu is obained by jus convolving he inpu wih he impulse response. We see ha lineariy and ime-invariance impose so much srucure on he sysem ha knowing how he sysem behaves for jus he one special inpu δ[n] deermines he behavior of he sysem for all inpus. Thus, as far as jus he inpu/oupu behavior of a sysem is concerned, he sysem is compleely described by is impulse response. In principle, if we have a black box and know nohing abou is operaion excep ha i happens o be linear and ime-invarian, we can find ou everyhing abou is behavior by doing one simple experimen. We apply he dela funcion δ[n] o he inpu and measure he oupu signal ha resuls. We hen know exacly how he black box will work under oher inpus. Of course, in realiy hings aren quie so simple for a variey of reasons, including he fac ha mos real sysems are neiher exacly linear nor ime-invarian. Neverheless, he resul is exremely useful as we ll see on many occasions. Finally, we should menion ha a resul analogous o he discree-ime case can also be obained for coninuous-ime sysems, as long as he coninuous-

7 .5. CONVOLUTION 7 ime sysem saisfies cerain mild coninuiy condiions ha in pracical cases are always saisfied. In paricular, we le h() denoe he impulse response of he sysem. Tha is, h() is he oupu of he sysem due o he inpu δ(), so ha h() = H[δ()] Then (under suiable coninuiy condiions) he oupu y() due o any inpu x() is given by y() = x(τ) h( τ) dτ (.7) A rough derivaion of his resul follows very similar lines as in he discreeime case. Firs, by he sifing propery of δ(), we can wrie x() = x(τ) δ( τ) dτ Then, using lineariy and ime-invariance we ge y() = H[x()] [ ] = H x(τ)δ( τ) dτ = = x(τ)h[δ( τ)] dτ x(τ)h( τ) dτ The hird equaliy uses lineariy, and his is where he coninuiy condiions are needed in a rigorous derivaion. The las equaliy uses ime-invariance. As in he discree-ime case, he righ-hand-side of Equaion (.7) is called he (coninuous-ime) convoluion of x() and h(), and denoed (x h)(). The resul again implies ha once we know he impulse response, we can deermine he oupu of he sysem due o any inpu via a convoluion. As a pracical maer, finding he impulse response of a coninuous-ime black-box is more challenging since we are we generally unable o produce a coninuous-ime dela funcion. However, approximaions o he dela funcion can be produced and are useful, and in any case he resul is exremely useful as a modeling and analysis ool..5 Convoluion In he previous secion, we inroduced he noion of he convoluion of wo signals. We repea he definiion here for convenience. In discree-ime, he convoluion of x[n] and h[n] is defined by (x h)[n] = x[i]h[n i]. i=

8 8 CHAPTER. BASICS OF SYSTEMS In coninuous-ime, he convoluion x() and h() is defined by (x h)() = x(τ)h( τ) dτ. We have seen ha he convoluion of wo signals arose naurally as a way o undersand he behavior of LTI sysems. Specifically, he oupu of an LTI sysem is simply he inpu convolved wih he impulse response. However, in addiion o his inerpreaion, he convoluion operaion can be moivaed by simply hinking of i as a raher general and useful way o ransform one signal ino anoher signal. If y() = (x h)(), we hink of x() as he original signal, y() as he new signal, and h() as represening he paricular ype of ransformaion we wish o perform. Wih his perspecive, i is immaerial ha he convoluion operaion can be viewed as he oupu of an LTI sysem. In any case, regardless of he paricular view or moivaion of convoluion, i is an operaion ha occurs so frequenly in boh he heory and pracice of signals and sysems ha i well worh spending ime o undersand he convoluion operaion. Le s consider he discree-ime case firs. Le y[n] = (x h)[n] = x[i]h[n i]. i= Firs noice ha he convoluion of wo signals resuls in anoher signal, no jus a single number. Thus o undersand wha he convoluion does, we need o undersand wha i does a each ime. Noice ha for any fixed n, he expression on he righ hand side is he sum of a bunch of (infiniely many) erms. Each erm is he value of x[ ] a some ime and he value of h[ ] a some oher ime. Thus he value of y[n] a some ime n is really jus a weighed sum of he signal x[ ] where he weighs depend on he signal h[ ]. A some differen ime, he convoluion is sill jus a weighed sum where again he weighs depend on h[ ] bu now differen weighs are assigned o values of x[ ]. Noice ha par of he expression for he convoluion sum, namely he x[i] erm, does no depend on n. The summaion index i is jus a dummy variable o carry ou he weighed sum. So we can hink of he convoluion as follows (and as illusraed in Figure.):. Draw he signal x[ ] where we use he dummy variable i, since we will be aking a weighed sum.. Flip he signal h[i] abou he verical (range) axis o ge h[ i] and hen shif by n o ge h[n i].. Muliply he corresponding values of x[i] and h[n i] and add up o find y[n]. In coninuous-ime, he convoluion y() = (x h)() = x(τ)h( τ) dτ.

9 .5. CONVOLUTION 9 Sep x[i] h[i] Sep h[-i] h[-i] Sep x[i] h[-i] Σ y[] Figure.: Schemaic convoluion of x[n] wih h[n]

10 CHAPTER. BASICS OF SYSTEMS has a similar inerpreaion, bu wih he sum replaced by an inegral. Namely, he convoluion a ime is obained as follows:. Consider x(τ) where we now use τ as he dummy variable for he weighed inegral.. Flip he signal h(τ) abou he verical (range) axis o ge h( τ) and hen shif by o ge h( τ).. Muliply he corresponding values of x(τ) and h( τ) and inegrae o find y()..6 Frequency Response of LTI Sysems In his secion, we consider he response of an LTI sysem o a sinusoidal inpu. If he inpu o he sysem is a sinusoid a a paricular frequency, we would like o know wha can be said abou he oupu of he sysem. I urns ou ha he behavior of LTI sysems o sinusoids is simple. All ha an LTI sysem can do o a sinusoidal inpu is o change is ampliude and phase. Tha is, if he inpu o an LTI sysem is a sinusoid a some frequency ω hen he oupu is also a sinusoid a frequency ω, bu possibly wih a differen ampliude and phase. The heory is cleaner if we allow complex sinusoids ha were discussed in Chaper XX. Namely, we consider he inpu x() = e jω = cos ω + j sin ω Now, le s see wha happens if we apply his inpu o an LTI sysem wih impulse response h(). Recall ha he oupu y() is jus he convoluion of x() and h(), i.e., y() = (x h)() Since we have seen ha convoluion is commuaive, we can wrie his as y() = h() x() = h(τ)x( τ) dτ So far, his is compleely general, i.e., we haven ye used he fac ha x() is sinusoidal. We ll now use his fac by subsiuing x() = e jω ino he above expression o ge y() = = h(τ)e jω( τ) dτ h(τ)e jω e jωτ dτ = e jω h(τ)e jωτ dτ

11 .6. FREQUENCY RESPONSE OF LTI SYSTEMS x() x() h(-) h() x()h(-) (a) y()= x()h(-) (b) x() x() h(.5-) h(-) x()h(.5-) 4 Area= x()h(-) 4 y(.5)= x()h(.5-) y()= x()h(-) Heigh a doed line = (c) (d) Figure.4: (a.)the signal, x() op and sysem response h() boom. (b.) x() op. h() flipped abou verical axis middle op. Produc of x(τ) and h(-τ) middle boom. Inegral of produc boom. (c. - h.) Same as (b.) bu furher along in ime.

12 CHAPTER. BASICS OF SYSTEMS x() x() h(-) h(-) x()h(-) 4 x()h(-) y()= x()h(-) y(.5)= x()h(.5-) 4 4 (e) (f ) x() x() h(-) h(.5-) x()h(-) x()h(.5-) y()= x()h(-) 4 y(.5)= x()h(.5-) 4 (g) (h) Figure.5:

13 .6. FREQUENCY RESPONSE OF LTI SYSTEMS Original Signal x() Sysem Response Funcions h() x*h() Convolved Signals h().5 x*h() h().5 x*h() Figure.6: Convolving wih rec funcions of differen lenghs Noice ha on he righ hand side we have he original complex sinusoid e jω imes a erm ha does no depend on. Thus, we have shown he desired resul. Of course, he erm muliplying e jω depends on ω. In fac, his erm is called he frequency response of he sysem, since i ells us wha he sysem does o sinusoidal inpus a each frequency. We will le H(ω) denoe he frequency response, so ha and hen we can wrie H(ω) = y() = H(ω)e jω h(τ)e jωτ dτ (.8) Once we know he impulse response h(), hen in principle we can compue he frequency response H(ω) by simply carrying ou he inegral in Equaion (.8). The fac ha he frequency response can be compued from he impulse response should no longer be surprising o us since we have previously seen ha he impulse response acually deermines he oupu for any inpu. So, in paricular, i deermines he oupu for a sinususoid a frequency ω. Wha may be surprising is ha he behavior of LTI sysems on sinusoids is so simple. Wha may be more surprising is ha no only does he impulse response deermine he frequency response, bu he reverse is also rue. Tha is, if we know he frequency response for all ω hen we can deermine he impulse response, and hence we can deermine he oupu due o any inpu. Thus, once we know wha he sysem does o sinusoids a each frequency, we know how he sysem

14 4 CHAPTER. BASICS OF SYSTEMS responds o all inpus. This resul is closely conneced o exremely useful conceps of frequency domain represenaions of signals and Fourier ransforms, which is he opic we will consider in he nex chaper. In fac, as we ll soon see, he frequency response H(ω) is simply he Fourier ransform (i.e., frequency domain represenaion) of he impulse response. We finish his secion by showing ha similar ideas hold in he case of discree-ime sysems. For he discree-ime case, we consider he discree-ime complex sinusoid x[n] = e jωn As in he coninuous-ime case, he oupu y[n] of an LTI sysem wih impulse response h[n] is given by y[n] = (x h)[n] = (h x)[n] = h[k]x[n k] k= Now, subsiuing he expression for x[n] we ge y[n] = = k= k= = e jωn h[k]e jω(n k) h[k]e jωn e jωk k= h[k]e jωk Thus, in he discree-ime case he frequency response H(ω) is given by H(ω) = The oupu can hen be wrien as k= y[n] = H(ω)e jωn h[k]e jωk (.9) so ha in he case of a discree-ime LTI sysem we also ge a sinusoidal oupu a he same frequency as he inpu.