AND SYSTEMS 2.0 INTRODUCTION

Transcription

1 2 DSCRETE-TME SGNALS AND SYSTEMS 2.0 NTRODUCTON The term sigal is geerally applied to somethig that coveys iformatio. Sigals geerally covey iformatio about the state or behavior of a physical system, ad ofte, sigals are sythesized for the purpose of commuicatig iformatio betwee humas or betwee humas ad machies. Although sigals ca be represeted i may ways, i all cases the iformatio is cotaied i some patter of variatios. Sigals are represeted mathematically as fuctios of oe or more idepedet variables. For example, a speech sigal is represeted mathematically as a fuctio of time, ad a photographic image is represeted as a brightess fuctio of two spatial variables. A commo covetio-ad oe that usually will be followed i this book-is to refer to the idepedet variable of the mathematical represetatio of a sigal as time, although i specific examples the idepedet variable may i fact ot represet time. The idepedet variable i the mathematical represetatio of a sigal may be either cotiuous or discrete. Cotiuous-time sigals are defied alog a cotiuum of times ad thus are represeted by a cotiuous idepedet variable. Cotiuoustime sigals are ofte referred to as aalog sigals. Discrete-time sigals are defied at discrete times, ad thus, the idepedet variable has discrete values; i.e., discrete-time sigals are represeted as sequeces of umbers. Sigals such as speech or images may have either a cotiuous- or a discrete-variable represetatio, ad if certai coditios hold, these represetatios are etirely equivalet. Besides the idepedet variables beig either cotiuous or discrete, the sigal amplitude may be either cotiuous or discrete. Digital sigals are those for which both time ad amplitude are discrete. Sigal-processig systems may be classified alog the same lies as sigals. That is, cotiuous-time systems are systems for which both the iput ad the output are 8

2 Sec. 2.1 Discrete-Time Sigals: Sequeces 9 cotiuous-time sigals, ad discrete-time systems are those for which both the iput ad the output are discrete-time sigals. Similarly, a digital system is a system for which both the iput ad the output are digital sigals. Digital sigal processig, the, deals with the trasformatio of sigals that are discrete i both amplitude ad time. The pricipal focus i this book is o discrete-time (rather tha digital) sigals ad systems. However, the theory of discrete-time sigals ad systems is also exceedigly useful for digital sigals ad systems, particularly if the sigal amplitudes are fiely quatized. The effects of sigal amplitude quatizatio are cosidered i Sectios 4.8, , ad 9.7. Discrete-time sigals may arise by samplig a cotiuous-time sigal, or they may be geerated directly by some discrete-time process. Whatever the origi of the discretetime sigals, discrete-time sigal-processig systems have may attractive features. They ca be realized with great flexibility with a variety of techologies, such as charge trasport devices, surface acoustic wave devices, geeral-purpose digital computers, or high-speed microprocessors. Complete sigal-processig systems ca be implemeted usig VLS techiques. Discrete-time systems ca be used to simulate aalog systems or, more importatly, to realize sigal trasformatios that caot be implemeted with cotiuous-time hardware. Thus, discrete-time represetatios of sigals are ofte desirable whe sophisticated ad flexible sigal processig is required. this chapter, we cosider the fudametal cocepts of discrete-time sigals ad sigal-processig systems for oe-dimesioal sigals. We emphasize the class of liear time-ivariat discrete-time systems. May of the properties ad results that we derive i this ad subsequet chapters will be similar to properties ad results for liear timeivariat cotiuous-time systems, as preseted i a variety of texts. (See, for example, Oppeheim ad Willsky, 1997.) fact, it is possible to approach the discussio of discrete-time systems by treatig sequeces as aalog sigals that are impulse trais. This approach, if implemeted carefully, ca lead to correct results ad has formed the basis for much of the classical discussio of sampled data systems. (See, for example, Phillips ad Nagle, 1995.) However, ot all sequeces arise from samplig a cotiuous-time sigal, ad may discrete-time systems are ot simply approximatios to correspodig aalog systems. Furthermore, there are importat ad fudametal differeces betwee discrete- ad cotiuous-time systems. Therefore, rather tha attempt to force results from cotiuous-time system theory ito a discrete-time framework, we will derive parallel results startig withi a framework ad with otatio that is suitable to discretetime systems. Discrete-time sigals will be related to cotiuous-time sigals oly whe it is ecessary ad useful to do so. 2.1 DSCRETE-TME SGNALS: SEQUENCES Discrete-time sigals are represeted mathematically as sequeces of umbers. A sequece of umbers x, i which the th umber i the sequece is deoted x[],l is formally writte as. x = {x(j}, -co < < co, (2.1) where is a iteger. a practical settig, such sequeces ca ofte arise from periodic A sequece is simply a fuctio whose domai is the set of itegers. :'\ote that we use [ ] to eclose the idepedet variable of such fuctios. ad we use ( ) to eclose the imkpedel variable of cotiuousvariable fuctios.

3 10 Discrete-Time Sigals ad Systems Chap. 2 samplig of a aalog sigal. this case, the umeric value of the th umber sequece is equal to the value of the aalog sigal, xa(t), at time T; i.e., i the x[] = xa(t), -(X) < < 00. (2.2) The quatity T is called the samplig period, ad its reciprocal is the samplig frequecy. Although sequeces do ot always arise from samplig aalog waveforms, it is coveiet to refer to x[] as the "th sample" of the sequece. Also, although, strictly speakig,x[] deotes the th umber i the sequece, the otatio ofeq. (2.1) is ofte uecessarily cumbersome, ad it is coveiet ad uambiguous to refer to "the sequece x[]" whe we mea the etire sequece, just as we referred to the "aalog sigal xa(t).".discrete-time sigals (i.e., sequeces) are ofte depicted graphically as show i Figure 2.1. Although the abscissa is draw as a cotiuous lie, it is importat to recogize that x[ lis defied oly for iteger values of. t is ot correct to thik of x[ ] as beig zero for is ot a iteger; x[] is simply udefied for oiteger values of. X [-2] T i X [-] x[o] tx[l] X [2] x[] Figure 2.1 Graphical represetatio of a discrete-time sigal. As a example, Figure 2.2( a) shows a segmet of a speech sigal correspodig to acoustic pressure variatio as a fuctio of time, ad Figure 2.2(b) presets a sequece ~ 32ms (a) 1 i ~ 256 samples (b) Figure 2.2 (a) Segmet of a cotiuous-time speech sigal. (b) Sequece of samples obtaied from part (a) with T = 125 MS.

4 Sec. 2.1 Discrete-Time Sigals: Sequeces 11 of samples of the speech sigal. Although the origial speech sigal is defied at all values of time t, the sequece cotais iformatio about the sigal oly at discrete istats. From the samplig theorem, discussed i Chapter 4, the origial sigal ca be recostructed as accurately as desired from a correspodig sequece of samples if the samples are take frequetly eough Basic Sequeces ad Sequece Operatios the aalysis of discrete-time sigal-processig systems, sequeces are maipulated i several basic ways. The product ad sum of two sequeces x[] ad y[] are defied as the sample-by-sample product ad sum, respectively. Multiplicatio of a sequece x[] by a umber a is defied as multiplicatio of each sample value by a. A sequece y[] is said to be a delayed or shifted versio of a sequece x[] if y[] = x[ - o], (2.3) where o is a iteger. discussig the theory of discrete-time sigals ad systems, several basic sequeces are of particular importace. These sequeces are show i Figure 2.3 ad are discussed ext. The uit sample sequece (Figure 2.3a) is defied as the sequece 8[] = {O, 1, i= 0, =0. (2.4) As we will see, the uit sample sequece plays the same role for discrete-time sigals ad systems that the uit impulse fuctio (Dirac delta fuctio) does for cotiuous-time sigals ad systems. For coveiece, the uit sample sequece is ofte referred to as a discrete-time impulse or simply as a impulse. t is importat to ote that a discrete-time impulse does ot suffer from the mathematical complicatios of the cotiuous-time impulse; its defiitio is simple ad precise. As we will see i the discussio of liear systems, oe of the importat aspects of the impulse sequece is that a arbitrary sequece ca be represeted as a sum of scaled, delayed impulses. For example, the sequece as p[] i Figure 2.4 ca be expressed p[] = a_38[ + 3] + a18[ - 1] + a28[ - 2] + a78[ - 7]. (2.5) More geerally, ay sequece ca be expressed as 00 x[] = ~ x[k]8[ - k]. We will make specific use of Eg. (2.6) i discussig the represetatio liear systems. Te uit sfep sequece (Figure 2.3b) is give by ::: O. < 0. (2.6) of discrete-time (2.7)

5 12 Discrete-Time Sigals ad Systems Chap '1~ o (a) Uit sample 1 Uit step o (b) Real expoetial o (c) Siusoidal (d) Figure 2.3 Some basic sequeces. The sequeces show play importat roles i the aalysis ad represetatio of discrete-time sigals ad systems. p[] 8 Figure 2.4 Example of a sequece to be represeted as a sum of scaled, delayed impulses. The uit step is related to the impulse by l[] = :L o[k]: k=-x that is. the value of the uit step sequece at (time) idex is equal to the accumulated sum ()f the value at idex ad all previous values of the impulse sequece. A alterative represetatio of the uit step i terms of the impulse is obtaied by iterpretig (2.8)

6 Sec.2.1 Discrete-Time Sigals: Sequeces 13 the uit step i Figure 2.3(b) i terms of a sum of delayed impulses as i Eq. (2.6). this case, the ozero values are all uity, so or u[] = 8[] + 8[ -1] + 8[ - 2] ur] = L 8[ - k=o k]. (2.9a) (2.9b) Coversely, the impulse sequece ca be expressed as the first backward differece of the uit step sequece, i.e., 8[] = u[] - u[ -1]. (2.10) Expoetial sequeces are extremely importat i represetig ad aalyzig liear time-ivariat discrete-time systems. The geeral form of a expoetial sequece S (2.11) f A ad a are real umbers, the the sequece is real. f 0 < a < 1 ad A is positive, the the sequece values are positive ad decrease with icreasig, as i Figure 2.3( c). For -1 < a < 0, the sequece values alterate i sig, but agai decrease i magitude with icreasig. f lal > 1, the the sequece grows i magitude as icreases. Example 2.1 Combiig Basic Sequeces We ofte combie basic sequeces to form simple represetatios of other sequeces. f we wat a expoetial sequece that is zero for < 0, we ca write this as the somewhat cumbersome expressio x[] = {Aa, 2: 0, \. 0, < O. (2.12) A much simpler expressio is x[] = Aau[J. Siusoidal sequeces are also very importat. A siusoidal sequece has the geeralform x[] = Acos(wo+ ), for all, (2.13) with A ad real costats, ad is illustrated i Figure 2.3( d). The expoetial sequece Aa with complex a has real ad imagiary parts that are expoetially weighted siusoids. Specifically, if a = lalejwq ad A = AleN, the sequece Aa ca be expressed i ay of the followig ways: x[] = Aa = AleNlalejwoll = Aai cos(wo + ) + jlallal" si(wo + ). (2.14 )

7 14 Discrete-Time Sigals ad Systems Chap. 2 The sequece oscillates with a expoetially growig evelope if Ja > 1 or with a expoetially decayig evelope if lal < 1. (As a simple example, cosider the case Wo = Jr.) Whe lal = 1, the sequece is referred to as a complex expoetial sequece ad has the form x[j = Alej(woH) = A cos(wo + 4»+ jalsi(wo+4»; (2.15) that is, the real ad imagiary parts of ejwo vary siusoidally with. By aalogy with the cotiuous-time case, the quatity Wo is called the frequecy of the complex siusoid or complex expoetial, ad 4> is called the phase. However, ote that is a dimesioless iteger. Thus, the dimesio of Wo must be radias. f we wish to maitai a closer aalogy with the cotiuous-time case, we ca specify the uits of Wo to be radias per sample ad the uits of to be samples. Thefactthatis always aitegerieq. (2.15) leads to some importat differeces betwee the properties of discrete-time ad cotiuous-time complex expoetial sequeces ad siusoidal sequeces. A importat differece betwee cotiuous-time ad discrete-time complex siusoids is see whe we cosider a frequecy (wo + 2Jr). this case, x[j = Aej(wo+2Jr) = Aejwoej2Jr = Aejwo. (2.16) More geerally, we ca easily see that complex expoetial sequeces with frequecies (wo + 2Jrr), where r is a iteger, are idistiguishable from oe aother. A idetical statemet holds for siusoidal sequeces. Specifically, it is easily verified that x[j = Acos[(wo+2Jrr)+4>J = A cos(wo + 4». (2.17) The implicatios of this property for sequeces obtaied by samplig siusoids ad other sigals will be discussed i Chapter 4. For ow, we simply coclude that, whe discussig complex expoetial sigals of the form x[ ] = Aejwo or real siusoidal sigals of the form x[] = Acos(wo + 4», we eed oly cosider frequecies i a iterval of legth 2Jr, such as -Jr < Wo.:::Jr or 0.:::Wo < 2Jr. Aother importat differece betwee cotiuous-time ad discrete-time complex expoetials ad siusoids cocers their periodicity. the cotiuous-time case, a siusoidal sigal ad a complex expoetial sigal are both periodic, with the period equal to 2Jr divided by the frequecy. the discrete-time case, a periodic sequece is a sequece for which x[j = x[ + NJ, for all, (2.18) where the period N is ecessarily a iteger. f this coditio for periodicity is tested for the discrete-time siusoid, the which requires that A cos(wo + 4» = A cos(wo + (rjon +4». won = 2;[ k, (2.19) (2.20) \\'here k is a iteger. A similar statemet holds for the complex expoetial sequece

8 Sec. 2.1 Discrete-Time Sigals: Sequeces 15 Cejwo; that is, periodicity with period N requires that (2.21) which is true oly for wan = 2 k, as i Eq. (2.20). Cosequetly, complex expoetial ad siusoidal sequeces are ot ecessarily periodic i with period (2jwa) ad, depedig o the value of Wa, may ot be periodic at all. Example 2.2 Siusoids Periodic ad Aperiodic Discrete-Time = cos(rr/4). This sigal has a period of N = 8. To show Cosider the sigal xl[] this, ote that x[ + 8] = cos(rr( + 8)/4) = cos(rr/4 + 2rr) = cos(rr/4) = x[], satisfyig the defiitio of a discrete-time periodic sigal. Cotrary to our ituitio from cotiuous-time siusoids, icreasig the frequecy of a discrete-time siusoid' does ot ecessarily decrease the period of the sigal. Cosider the discrete-tim~ siusoid x2[] = cos(3rr/8), which has a higher frequecy tha Xl []. However, x2[] is ot periodic with period 8, sice x2[ +8] = cos(3rr( + 8)/8) = cos(3rr/8 +3rr) = -x2[]. Usig a argumet aalogous to the oe for Xl [], we ca show that x2[] has a period of N = 16. Thus, icreasig the frequecy from (,Va = 2rr/8 to (,Va = 3rr/8 also icreases the period of the sigal. This occurs because discrete-time sigals are defied oly for iteger idices. The iteger restrictio o causes some siusoidal sigals ot to be periodic at all. For example, there is o iteger N such that the sigal x3[] = cos() satisfies the coditio x3[ + N] = x3[] for all. These ad other properties of discrete-time siusoids that ru couter to their cotiuous-time cquterparts are caused by the limitatio of the time idex to itegers for discrete-time sigals ad systems. Whe we combie the coditio of Eq. (2.20) with our previous observatio that Wa ad (wo + 2r) are idistiguishable frequecies, it becomes clear that there are N distiguishable frequecies for which the correspodig sequeces are periodic with period N. Oe set of frequecies is Wi: = 27T kj N, k = 0, 1,..., N - 1. These properties of complex expoetial ad siusoidal sequeces are basic to both the theory ad the desig of computatioal algorithms for discrete-time Fourier aalysis, ad they will be discussed i more detail i Chapters 8 ad 9. Related to the precedig discussio is the fact that the iterpretatio of high ad low frequecies is somewhat differet for cotiuous-time ad discrete-time siusoidal ad complex expoetial sigals. For a cotiuous-time siusoidal sigal x(t) = A cos( Qat + ), as Q 0 icreases, x(t) oscillates more ad more rapidly. For the discretetime siusoidal sigal x(j = A cos(wo + ), as Wo icreases from Wo = 0 toward Wo =, x(j oscillates more ad more rapidly. However, as Wo icreases from Wa = to Wo = 2, the oscillatios become slower. This is illustrated i Figure 2.5. fact, because of the periodicity i Wo of siusoidal ad complex expoetial sequeces, Wa = 2 is idistiguishable from Wo = 0, ad, more geerally, frequecies aroud Wo = 2 are idistiguishable from frequecies aroud (Va = O. As a cosequece, for siusoidal ad complex expoetial sigals, values of Wa i the viciity of Wa = 2rr k for ay iteger value of k are typically referred to as low frequecies (relatively slow oscillatios), while values of Wa i the viciity of (Ull = ( + 2rrk) for ay iteger value of k are typically referred to as high frequecies (relatively rapid oscillatios).._ _.~,.>::"... '...

9 16 Discrete-Time Sigals ad Systems Chap. 2 Wo = 0 or Wo = 2'TT u, 1 o (a) (b)... W 0= 'TT/4 or Wo = 7'TT/4 o W (c) i... j... 0= 'TT 1 (d) 2.2 DSCRETE-TME SYSTEMS Figure 2.5 cos wo for several differet values of woo As Wo icreases from zero toward (parts a-d), the sequece oscillates more rapidly. As Wo icreases from to 2 (parts d-a), the oscillatios become slower. A discrete-time system is defied mathematically as a trasformatio or operator that maps a iput sequece with values x[] ito a output sequece with values y[]. This ca be deoted as y[j = T{xl]} (2.22)

10 Sec. 2.2 Discrete-Time Systems 17 ~~ ~T_{'_} ~~ Figure 2.6 Represetatio of a discrete-time system, Le., a trasformatio that maps a iput sequece x[] ito a uique output sequece y[]. ad is idicated pictorially i Figure 2.6. Equatio (2.22) represets a rule or formula for computig the output sequece values from the iput sequece values. t should be emphasized that the value of the output sequece at each value of the idex may deped o x[] for all values of. The followig examples illustrate some simple ad useful systems. Example 2.3 The deal Delay System The ideal delay system is defied by the equatio y[] = x[ - d], -CXJ < < CXJ, (2.23) where d is a fixed positive iteger called the delay of the system. words, the ideal delay system simply shifts the iput sequece to the right by d samples to form the output. f, i Eq. (2.23), d is a fixed egative iteger, the the system would shift the iput to the left by dl samples, correspodig to a time advace. Example 2.3, oly oe sample of the iput sequece is ivolved i determiig a certai output sample. the followig example, this is ot the case. Example 2.4 Movig Average The geeral movig-average system is defied by the equatio 1 Mz y[] = -'-1 -+-M--+-1 L x[ - k] 1 2 k=-mj {x[ M] +M2+1 + x[ - 1] x[ - M2]}. + Md + x[ + M] -1] x[] -5 (2.24) x[k] k Figure 2.7 Sequece values ivolved i computig a causal movig average. This system computes the th sample of the output sequece as the average of (11,1 + k/2 + ) samples of the iput sc:quece aroud the 11thsample. Figure 2.7 shows a

11 18 Discrete-Time Sigals ad Systems Chap. 2 iput sequece plotted as a fuctio of a dummy idex k ad the samples ivolved i the computatio ofthe output sample y[] for = 7, M1 = 0, ad M2 = 5.The output sample y[7] is equal to oe-sixth of the sum of all the samples betwee the vertical dotted lies. To compute y[8], both dotted lies would move oe sample to the right. Classes of systems are defied by placig costraits o the properties of the trasformatio T {.}. Doig so ofte leads to very geeral mathematical represetatios, as we will see. Of particular importace are the system costraits ad properties, discussed i Sectios Memoryless Systems A system is referred to as memoryless'if the output y[] at every value of depeds oly o the iput x[] at the same value of.. Example 2.5 A Memoryless System A example of a memoryless system is a system for which x[] ad y[] are related by y[] = (x[]f, for each value of. (2.25) The system i Example 2.3 is ot memoryless uless d = 0; i particular, this system is referred to as havig "memory" whether d is positive (a time delay) or egative (a time advace). The system i Example 2.4 is ot memoryless uless Ml = M2 = O Liear Systems The class of liear systems is defied by the priciple of superpositio. f Yl [] ad Y2[] are the resposes of a system whe xl[] ad x2[] are the respective iputs, the the system is liear if ad oly if ad T{xf] + x2[]} = T{x2[]} + T{x2[]} = Yl[] + Y2[] T{ax[]} = a T{x[]} = ay[]. (2.26a) (2.26b) where a is a arbitrary costat. The first property is called the additivity property, ad the secod is called the homogeeity or scalig property. These two properties ca be combied ito the priciple of superpositio, stated as for arbitrary costats a ad b. This equatio ca be geeralized to the superpositio of may iputs. Specifically, if (2.27) the the output x[] = Lakxk[], k of a liear system will be (2.28a) y[] ~ LakYd]. (2.28b) k where yd ] is the system respose to the iput xt;[]. By usig the defiitio of the priciple of superpositio. we ca easily show that the systems of Examples 2.3 ad 2.4 are liear systems. (See Problem 2.23.) A example of a ol!ear system is the system i Example 2.:'.

12 Sec.2.2 Discrete-Time Systems 19 Example 2.6 The Accumulator System The system defied by the iput-output y[] equatio = L x[k] (2.29) is called the accumulator system, sice the output at time is just the sum of the preset ad all previous iput samples. The accumulator system is a liear system. order to prove this, we must show that it satisfies the superpositio priciple for all iputs, ot just ay specific set of iputs. We begi by defiig two arbitrary iputs xl[] ad x2[] ad their correspodig outputs Yl[] = L Xl [k], (2.30) Y2[] = L x2[k]. (2.31) Whe the iput is x3[] = axl[] + bx2[], the superpositio priciple requires the output Y3 [] = ayl [] + by2 [] for all possible choices of a ad b. We ca show this by startig from Eq. (2.29): - Y3[] = L X3 [k], (2.32) = L (axl[k] + bx2[k]), = a L xdk] + b L x2[k], (2.33) (2.34) (2.35' Thus, the accumulator system of Eq. (2.29) satisfies the superpositio priciple for a' iputs ad is therefore liear. geeral, it may be simpler to prove that a system is ot liear (if it is ot) thd' to prove that it is liear (if it is). We simply must fid a iput or set of iputs for whic the system does ot satisfy the coditios of liearity. Example 2.7 A Noliear System Cosider the system defied by w[] = loglo (lx[]). (2.3(. This system is ot liear. order to prove this, we oly eed to fid o couterexample-that is. oe set of iputs ad outputs which demostrates that the sy' ter violates the superpositio priciple, Eg. (2.27). The iputs x, [] = 1 ad X::, (] = 1\ are a couterexample. The output for the first sigal is w, [] = O. while for the seeo, a'::,[] =. The scalig property of liear systems requires that. sice X::, (] = lox, (r: if the system is liear. it must be true that w::,(tz] = lchvdll]. Sice this is ot so fl, Eq. (2.~'h! for this set,f iputs ~llldoutpus. the ~\stcr is l,'! lie:l!"

13 20 Discrete-Time Sigals ad Systems Chap. 2 l Time-variat Systems A time-ivariat system (ofte referred to equivaletly as a shift-ivariat system) is a system for which a time shift or delay of the iput sequece causes a correspodig shift i the output sequece. Specifically, suppose that a system trasforms the iput sequece with values x[] ito the output sequece with values y[]. The the system is said to be time ivariat if, for all o, the iput sequece with values xl[] = x[ - o] produces the output sequece with values Yl[] = y[ - o]. As i the case of liearity, provig that a system is time ivariat requires a geeral proof makig o specific assumptios about the iput sigals. All of the systems i Examples are time ivariat. The style of proof for time ivariace is illustrated i Examples 2.8 ad 2.9.! Example 2.8 The Accumulator as a Time-variat System Cosider the accumulator from Example 2.6. We defie x1[] = x[ - o]. To show time ivariace, we solve for both y[ - o] ad Y1[] ad compare them to see whether they are equal. First, Next, we fid -o y[ - o] = L x[k]. Y1[] = L x1[k] = L x[k-o]. (2.37) (2.38) (2.39) Substitutig the chage of variables k1 = k - o ito the summatio gives -o Yl[] = L x[k1] = y[ - o]. kj=-oo Thus, the accumulator is a time-ivariat system. (2.40) The followig example illustrates a system that is ot time ivariat. Example 2.9 The Compressor System The system defied by the relatio y[] = x[m], -00 < < 00, (2.41) with M a positive iteger, is called a compressor. Specifically, it discards (M - 1) samples out of M; i.e., it creates the output sequece by selectig every Mth sample. This system is ot time ivariat. We ca show that it is ot by cosiderig the respose Y [] to the iput Xl [] = x[ - o]. order for the system to be time ivariat. the output of the system whe the iput is X, [] must be equal to y[ - o]. The output )'1 [] that results from the iput x, [ J ca be directly computed from Eq. (2.-:11) to be yd] = x] [M] = x[m - o]. (2.42)

14 Sec. 2.2 Discrete-Time Systems 21, Delayig the output y[] by o samples yields y[ - o] = x[m( - o)]. (2.43) Comparig these two outputs, we see that y[ - o] is ot equal to Yl[] for all M ad o, ad therefore, the system is ot time ivariat. t is also possible to prove that a system is ot time ivariat by fidig a sigle couterexample that violates the time-ivariace property. For istace, a couterexample for the compressor is the case whe M = 2, x[] = 8[], ad xl[] = 8[ -1]. For this choice of iputs ad M, y[] = 8[], but Yl[] = 0; thus, it is clear that Yl[] =- y[ -1] for this system Causality A system is causal if, for every choice of o, the output sequece value at the idex = o depeds oly o the iput sequece values for.::::o. This implies that if xl[] = x2[] for.:::: o, the yd] = Y2[] for.::::o. That is, the system is oaticipative. The system of Example 2.3 is causal for d 2: 0 ad is ocausal for d < O. The system of Example 2.4 is causal if -M1 2: 0 ad M2 2: 0; otherwise it is ocausal. The system of Example 2.5 is causal, as is the accumulator of Example 2.6 ad the oliear system i Example 2.7. However, the system of Example 2.9 is ocausal if M > 1, sice y[l] = x[ M]. Aother ocausal system is give i the followig example. Example 2.10 Systems The Forward ad Backward Differece Cosider the forward differece system defied by the relatioship y[] = x[ + 1] - x[]. (2.44) This system is ot causal, sice the curret value of the output depeds o a future value of the iput. The violatio of causality ca be demostrated by cosiderig the two iputs xl[] = 8[ - 1] ad x2[] = 0 ad their correspodig outputs yd] = 8[] - 8[ - 1] ad Y2[] = O. Note that xdll] = X2[1l] for 11:s 0, so the defiitio of causality requires that Yl [] = Y2 [] for 17 :s 0, which is clearly ot the case for 11 = O. Thus, by this couterexample, we have show that the system is ot causal. The backward differece system, defied as y[] = x[] - x[ - 1]. (2.45) has a output that depeds oly o the preset ad past values of the iput. Because there is o way for the output at a specific time y[ o] to icorporate values of the iput for > o, the system is causal Stability A system is stable i the bouded-iput. bouded-output (BlBO) sese if ad oly if every bouded iput sequece produces a bouded output sequece. The iput x[] is bouded if there exists a fixed positive fiite value Br such that x[]1.::::b, < 00. for all t1. (2.46)

15 22 Discrete-Time Sigals ad Systems Chap. 2 Stability requires that, for every bouded iput, there exist a fixed positive fiite value By such that ly[]1 ::::By < co, for all. (2.47) t is importat to emphasize that the properties we have defied i this sectio are properties of systems, ot of the iputs to a system. That is, we may be able to fid iputs for which the properties hold, but the existece of the property for some iputs does ot mea that the system has the property. For the system to have the property, it must hold for all iputs. For example, a ustable system may have some bouded iputs for which the output is bouded, but for the system to have the property of stability, it must be true that for all bouded iputs, the output is bouded. f we ca fid just oe iput for which the system property does ot hold, the we have show that the system does ot have that property. The followig example illustrates the testig of stability for several of the systems that we have defied. "* Example 2.11 Testig for Stability or stability The system of Example 2.5 is stable. To see this, a~sume that the iputx[] is bouded such that x [] ::s Bx for all. The ly[]1 = x[]12 ::s B;. Thus, we ca choose By = B; ad prove that y[] is bouded. Likewise, we ca see that the system defied i Example 2.7 is ustable, sice y[] = 10glO(!x[]l) = -00 for ay values of the time idex at which x[] = 0, eve though the output will be bouded for ay iput samples that are ot equal to zero. The accumulator, as defied i Example 2.6 by Eq. (2.29), is also ot stable. For example, cosider the case whe x[] = ur], which is clearly bouded by Bx = 1. For this iput, the output of the accumulator is y[] =.L u[k] = { O. ( + 1), < 0, :o:: O. (2.48) (2.49) There is o fiite choice for By such that ( + 1) ::s By < 00 for all ; thus, the system is ustable. Usig similar argumets, it ca be show that the systems i Examples 2.3, 2.4, 2.9 ad 2.10 are all stable. 2.3 LNEAR TME-NVARANT SYSTEMS A particularly importat class of systems cosists of those that are liear ad time ivariat. These two properties i combiatio lead to especially coveiet represetatios for such systems. Most importat, this class of systems has sigificat sigal-processig applicatios. The class of liear systems is defied by the priciple of superpositio i Eq. (2.27). f the liearity property is combied with the represetatio of a geeral sequece as a liear combiatio of delayed impulses as i Eq. (2.6), it follows that a liear system ca be completely characterized by its impulse respose. Specifically. let hk[] be the respose of the system to b[ - kj. a impulse occurrig at = k. The,

16 Sec. 2.3 Liear Time-variat Systems 23 from Eq. (2.6), y[] ~ T Ltw X [k]o[ - k]}. From the priciple of superpositio i Eq. (2.27), we ca write y[] = L x[k]t{8[ - k]} = L x[k]hk[]. (2.50) (2.51) Accordig to Eq. (2.51), the system respose to ay iput ca be expressed i terms of the resposes of the system to the sequeces 8[ - k]. f oly liearity is imposed, hk[] will deped o both ad k, i which case the computatioal usefuless of Eq. (2.51) is limited. We obtai a more useful result if we impose the additioal costrait of time mvaace. The property of time ivariace implies that if h[] is the respose to 8[], the the respose to 8[ - k] is h[ - k]. With this additioal costrait, Eq. (2.51) becomes 00 y[] = L x[k]h[ - k]. (2.52) As a cosequece ofeq. (2.52), a liear time-ivariat system (which we will sometimes abbreviate as LT) is completely characterized by its impulse respose h[] i the sese that, give h[], it is possible to use Eq. (2.52) to compute the output y[] due to ay iput x[]. Equatio (2.52) is commoly called the covolutio Slim. f y[] is a sequece whose values are related to the values of two sequeces h[] ad x[] as i Eq. (2.52), we say that y[] is the covolutio of x[] with h[] ad represet this by the otatio y[] = x[] * h[]. (2.53) The operatio of discrete-time covolutio takes two sequeces x[] ad h[] ad produces a third sequece y[]. Equatio (2.52) expresses each sample of the output sequece i terms all of the samples of the iput ad impulse respose sequeces. The derivatio of Eq. (2.52) suggests the iterpretatio that the iput sample at = k, represeted as x[k]<5[ - k], is trasformed by the system ito a output sequece x[k]h[ - k], for -00 <.: < 00, ad that, for ooch k, these sequeces are superimposed to form the overall output sequellce. This iterpretatio is illustrated i Figure 2.8, which shows a impulse respose, a simple iput sequece havig three ozero samples, the idividual outputs due to each sample, ad the composite output due to all the samples i the iput sequece. Specifically, x[] ca be decomposed as the sum of the three sequeces x[-2]8[ + 2], x[o]8[], ad x[3]8[ - 3] represetig the three ozero values i the sequece x[]. The sequeces x[-2]h[ + 2], x[o]h[], ad x[3]h[ - 3] are the system resposes to x[ -2]8[ + 2], x[o]<5[], ad x[3]8[ - 3J, respectively. The respose to x [] is the the sum of these three idividual resposes.. Although the covolutio-sum expressio is aalogous to the covolutio itegral of cotiuous-time liear system theory, the covolutio sum should ot be thought of as a approximatio to the covolutio itegral. The covolutio itegral plays maily a theoretical role i cotiuous-time liear system theory: we will see that the covolutio sum, i additio to its theoretical importace. ofte serves as a explicit realizatio of,j discrete-time liear system. Thus. it is importam to gai some isight ito the propertie~ of the covolutio sum i actual calculatios.

17 24 Discrete-Time Sigals ad Systems Chap h[] x[].1l 0 x_2[] Y_2[] == ==x[-2]h[ x[-2]s[ + 02] 2] ] ll 0 xo[] == x[o]s[].11.yo:j ~X[OJh[J o Y3[] o -- 3 ==x[3]h[-3] ~ y[] == Y_2[J + Yo[] + Y3[] -2 o 3 T 1 Figure 2.8 Represetatio of the output of a liear time-ivariat system as the superpositio of resposes to idividual samples of the iput.

18 .. Sec. 2.3 Liear Time-variat Systems 25 The precedig iterpretatio of Eq. (2.52) emphasizes that the covolutio sum is a direct result of liearity ad time ivariace. However, a slightly differet way of lookig at Eq. (2.52) leads to a particularly useful computatioal iterpretatio. Whe viewed as a formula for computig a sigle value of the output sequece, Eq. (2.52) dictates that y[] (i.e., the th value of the output) is obtaied by multiplyig the iput sequece (expressed as a fuctio of k) by the sequece whose values are h[ - k], -00 < k < 00,ad the, for ay fixed value of, summig all the values of the products x[k]h[ - k], with k a coutig idex i the summatio process. Therefore, the operatio of covolvig two sequeces ivolves doig the computatio for all values of, thus geeratig the complete output sequece y[], -00 < < 00.The key to carryig out the computatios ofeq. (2.52) to obtai y[] is uderstadig how to form the sequece h[ - k], -00 < k < 00,for all values of that are of iterest. To this ed, it is useful to ote that h[ - k] = h[-(k- )]. (2.54) The iterpretatio Example2.12 of Eq. (2.54) is best doe with a example. Computatio of the Covolutio Sum Suppose h[k] is the sequece show i Figure 2.9(a) ad we wish to fid h[ - k) = h[-(k -»). Defie h1[k) to be h[-k], which is show i Figure 2.9(b). Next, defie h[kj -3 o 6 k (a) h[-k]=h[o-k] o (b) 3 k h[-kj=h[-(k-)] -6 o +3 k (c) Figure 2.9 Formig the sequece h[ - k]. (a) The sequece h[k] as a fuctio of k. (b) The sequece h[ -k] as a fuctio of k. (c) The sequece h[ - k] = h[-(k - }] as a fuctio of k for = 4.

19 26 Discrete-Time Sigals ad Systems Chap. 2 h2[k] to be h1[k], delayed, by samples o the kaxis,i.e.,h2[k] = hdk-]. Figure2.9(c) shows the sequece that results from delayig the sequece i Figure 2.9(b) by samples. Usig the relatioship betwee hl[k] ad h[k], we ca show that h2[k] = hdk - ] = h[ -(k - )] = h[ - k], ad thus, the bottom figure is the desired sigal. To summarize, to compute h[ - k] from h[k], we first reverse h[k] i time about k = 0 ad the delay the time-reversed sigal by samples. From Example 2.3, it should be clear that, i geeral, the sequece h[ - k], -00 < k < 00,is obtaied by 1. reflectig h[k] about the origi to obtai h[ -k]; 2. shiftig the origi of the reflected sequece to k =. To implemet discrete-time covolutio, the two sequeces x[k] ad h[ - k] are multiplied together for -00 < k < 00, ad the products are summed to compute the output sample y[]. To obtai aother output sample, the origi of the,sequece h[-k] is shifted to the ew sample positio, ad the process is repeated. This computatioal procedure applies whether the computatios are carried out umerically o sampled data or aalytically with sequeces for which the sample values have simple formulas. The followig example illustrates discrete-time covolutio for the latter case. Example 2.13 Aalytical Evaluatio of the Covolutio Sum Cosider a system with impulse respose h[] = ur] - ur - N] The iput is - {0: 0<<N-1 othe~se. ' To fid the output at a particular idex, we must form the sums over all k of the product x[kjh[ - k]. this case, we ca fid formulas for y[] for differet sets of values of. For example, Figure 2.1O(a) shows the sequeces x[k] ad h[ - k], plotted for a egative iteger. Clearly, all egative values of give a similar picture; i.e., the ozero portios of the sequeces x[k] ad h[ - k] do ot overlap, so y[] = 0, < O. Figure 2.10(b) illustrates the two sequeces whe 0 ::s ad - N + 1 ::s O.These two coditios ca be combied ito the sigle coditio 0 ::s ::s N - 1. By cosiderig Figure 2.1O(b), we see that, sice x[kjh[ - kj = ak, it follows that y[l = 2.::>k, k=o for lj ::s ::s N - 1. (2.55)

20 Sec. 2.3 Liear Time-variat Systems 27 x x x x o h[-k] x x[k] t 0 x x ~ r k 0 (b) (a) k k f X X X X X y[j,,,,-j! 1 k (d) Figure 2.10 Sequece ivolved i computig a discrete covolutio. (a)-(c) The sequeces x[k] ad h[ - k] as a fuctio of k for differet values of. (Oly ozero samples are show.) (d) Correspodig output sequece as a fuctio of. The limits o the sum are determied directly from Figure 2.10(b). Equatio (2.55) shows that y[j is the sum of il + 1 terms of a geometric series i which the ratio of terms is a. This sum ca be expressed i closed form usig the geeral formula (2.56)

21 28 Discrete-Time Sigals ad Systems Chap. 2 Applyig this formula to Eq. (2.55), we obtai 1- a+1 y[] = " ' O:::c:::cN-l. -a Fially, Figure 2.10(c) shows the two sequeces whe 0 < - N + 1 or N -1 before, but ow the lower limit o the sum is - N + 1, as see i Figure 2.10(c). Thus, Usig Eq. (2.56), we obtai x[k]h[ - k] = ak, y[] = :L ak, k=-n+l -N+l < k:::c, (2.57) <. As for N -1 <. (2.58) or y[] = a-n+1 - a+1 (2.59) y[] = a-n+1 (1- -a an). Thus, because of the piecewise-expoetial ature of both the iput ad the uit sample respose, we have bee able to obtai the followig closed-form expressio for y[ ] as a fuctio of the idex : y[] = 0, 1- a+1 -a a_n+1(1-an) 1- a ' This sequece is show i Figure 2.10(d). < 0, O:::c :::c N -1, N -1 <. (2.60) Example 2.13 illustrates how the covolutio sum ca be computed aalytically whe the iput ad the impulse respose are give by simple formulas. such cases, the sums may have a compact form that may be derived usig the formula for the sum of a geometric series or other "closed-form" formulas? Whe o simple form is available, the covolutio sum ca still be evaluated umerically usig the techique illustrated i Example 2.13 wheever the sums are fiite, which will be the case if either the iput sequece or the impulse respose is of fiite legth, i.e., has a fiite umber of ozero samples. 2.4 PROPERTES OF LNEAR TME-NVARANT SYSTEMS Sice all liear time-ivariat systems are described by the covolutio sum of Eq. (2.52), the properties of this class of systems are defied by the properties of discretetime covolutio. Therefore, the impulse respose is a complete characterizatio of the properties of a specific liear time-ivariat system. :Such results are discussed. for example. i Grossma (19':12).

22 Sec.2.4 Properties of Liear Time-variat Systems 29 Some geeral properties of the class of liear time-ivariat systems ca be foud by cosiderig properties of the covolutio operatio. For example, the covolutio operatio is commutative: x[] * h[] = h[] * x[]. (2.61) This ca be show by applyig a substitutio of variables to Eq. (2.52). Specifically, with m=-k, y[] = L x[ - m]h[m] = L h[m]x[ - m] = h[] * x[], (2.62) m=oo m=-oo so the roles of x[] ad h[] i the summatio are iterchaged. That is, the order of the sequeces i a covolutio is uimportat, ad hece, the system output is the same if the roles of the iput ad impulse respose are reversed. Accordigly, a liear time-ivariat system with iput x[] ad impulse respose h[] will have the same output as a liear time-ivariat system with iput h[] ad impulse respose x[]. The covolutio operatio also distributes over additio; i.e., x[] * (h1[] + hz[]) = x[] * h1[] + x[] * hz[]. This follows i a straightforward way from Eq. (2.52) ad is a direct result of the liearity ad commutativity of covolutio. a cascade coectio of systems, the output of th~ first system is the iput to the secod, the output of the secod is the iput to the third, etc. The output of the last system is the overall output. Two liear time-ivariat systems i cascade correspod to a liear time-ivariat system with a impulse respose that is the covolutio of the impulse resposes of the two systems. This is illustrated i Figure the upper block diagram, the output of the first system will be h1[ ] if x[] = 8[]. Thus, the output of the secod system (ad, by defiitio, the impulse respose of the overall system) will be (2.63) As a cosequece of the commutative property of covolutio, the impulse respose of a cascade combiatio of liear time-ivariat systems is idepedet of the order i which they are cascaded. This result issummarized i Figure 2.11, where the three systems all have the same impulse respose. Figure 2.11 Three liear time-ivariat systems with idetical impulse resposes.

23 30 Discrete-TimeSigalsad Systems Chap. 2 x[] (a) x[] (b) y[] Figure 2.12 (a) Parallelcombiatio of lieartime-ivariatsystems. (b) A equivaletsystem. a parallel coectio, the systems have the same iput, ad their outputs are summed to produce a overall output. t follows from the distributive property of covolutio that the coectio of two liear time-ivariat systems i parallel is equivalet to a sigle system whose impulse respose is the sum of the idividual impulse resposes; 1.e., h[] = h1[] + h2[]. (2.64) This is depicted i Figure The costraits of liearity ad time ivariace defie a class of systems with very special properties. Stability ad causality represet additioal properties, ad it is ofte importat to kow whether a liear time-ivariat system is stable ad whether it is causal. Recall from Sectio that a stable system is a system for which every bouded iput produces a bouded output. Liear time-ivariat systems are stable if ad oly if the impulse respose is absolutely summable, i.e., if co S = L h[k]1 < 00. k=-co This ca be show as follows. From Eq. (2.62), f x[] is bouded, so that co ly[]1 = k~co h[k]x[ - k]! ::: k~co Mk]llx[ - k]l x[]1 ::: Ex, the substitutig Ex for x[ - k]1 ca oly stregthe the iequality. Hece, co (Xl y[]j ::: Ex L h[k]l k=-co (2.65) (2.66) (2.67) Thus, y[j is bouded if Eq. (2.65) holds; i other words, Eq. (2.65) is a sufficiet coditio for stability. To show that it is also a ecessary coditio, we must show that if S = 00, the a bouded iput ca be foud that will cause a ubouded output. Such a iput is the sequece with values x[]= ---. h[] i O. h[--]i {/ O. h~[-/] h[j = 0, (2.68)

24 Sec.2.4 Properties of Liear Time-variat Systems 31 where h*[] is the complex cojugate of h[]. The sequece x[] is clearly bouded by uity. However, the value of the output at = 0 is y[o] = tx[-k]h[k] = f h[[k]]12= s. k=-00 k~oo h k (2.69) Therefore, if S = 00, it is possible for a bouded iput sequece to produce a ubouded output sequece. The class of causal systems was defied i Sectio as those systems for which the output y[o] depeds oly o the iput samples x[], for ::: o. t follows from Eq. (2.52) or Eq. (2.62) that this defiitio implies the coditio h[] = 0, < 0, (2.70) for causality of liear time-ivariat systems. (See Problem 2.62.) For this reaso, it is. sometimes coveiet to refer to a sequece that is zero for < 0 as a causal sequece, meaig that it could be the impulse respose of a causal system. To illustrate how the properties of liear time-ivariat systems are reflected i the impulse respose, let us cosider agai some of the systems defied i Examples First ote that oly the systems of Examples 2.3, 2.4, 2.6, ad 2.10 are liear ad time ivariat. Although the impulse respose of oliear or time-varyig systems ca be foud, it is geerally of limited iterest, sice the covolutio-sum formula ad Eqs. (2.65) ad (2.70), expressig stability ad causality, do ot apply to such systems. First, let us fid the impulse resposes ofthe systems i Examples 2.3, 2.4, 2.6, ad We ca do this by simply computig the respose of each system to 8[], usig the defiig relatioship for the system. The resultig impulse resposes are as follows: deal Delay (Example 2.3) h[] = 8[ - d], d a positive fixed iteger. (2.71) Movig Average (Example 2.4) (2.72) Accumulator (Example 2.6) h[] = L 8[k] _- {1, 0,?: 0, < 0, (2.73) = u[]. Forward Differece (Example 2.10) h[] = 8[ + 1] - 8[]. (2.74)

25 32 Discrete-Time Sigals ad Systems Chap. 2 Backward Differece (Example 2.10) h[] = 8[] - 8[ -1]. (2.75) Give the impulse resposes of these basic systems [Eqs. (2.71)-(2.75)], we ca test the stability of each oe by computig the sum 00 s = : h[]l =-oo For the ideal delay, movig-average, forward differece, ad backward differece examples, it is clear that S < 00, sice the impulse respose has oly a fiite umber of ozero samples. Such systems are called fiite-duratio impulse respose (FR) systems. Clearly, FR systems will always be stable, as log as each of the impulse respose values is fiite i magitude. The accumulator, however, is ustable because 00 S = :u[] = 00. =O Sectio 2.2.5, we also demostrated the istability of the accumulator by givig a example of a bouded iput (the uit step) for which the output is ubouded. The impulse respose of the accumulator is ifiite i duratio. This is a example of the class of systems referred to as ifiite-duratio impulse respose (R) systems. A example of a R system that is stable is a system whose impulse respose is h[] = au[] with lal < 1. this case, 00 (2.76) f lal < 1, the formula for the sum of the terms of a ifiite geometric series gives 1 S= -- <00. 1-lal (2.77) f, o the other had, la 2:: 1, the sum is ifiite ad the system is ustable. To test causality of the liear time-ivariat systems i Examples 2.3, 2.4, 2.6, ad 2.10, we ca check to see whether h[] = 0 for < O.As discussed i Sectio 2.2.4, the ideal delay [d ::: 0 i Eq. (2.23)] is causal. f d < 0, the system is ocausal. For the movig average, causality requires that - M1 ::: 0 ad M2 ::: O. The accumulator ad backward differece systems are causal, ad the forward differece system is ocausal. The cocept of covolutio as a operatio betwee two sequeces leads to the simplificatio of may problems ivolvig systems. A particularly useful result ca be stated for the ideal delay system. Sice the output of the delay system is y[ ] = x[ - d], ad sice the delay system has impulse respose h[] = 8[ - d], it follows that x[] * 8[ - d = 8[ - d] * x[] = x[ - d]' (2.78) That is, the covolutio of a shifted impulse sequece with ay sigal x[ ] is easily evaluated by simply shiftig x[] by the displacemet of the impulse. Sice delay is a fudametal operatio i the implemetatio of liear systems. the precedig result is ofte useful i the aalysis ad simplificatio of itercoectios of liear time-ivariat systems. As a example. cosider the systerr. of Figure 2.13( a).

26 Sec.2.4 Properties of Liear Time-variat Systems 33 (a) (b) (c) Figure 2.13 Equivalet systems foud by usig the commutative property of covolutio. which cosists of a forward differece system cascaded with a ideal delay of oe sample. Accordig to the commutative property of covolutio, the order i which systems are cascaded does ot matter, as log as they are liear ad time ivariat. Therefore, we obtai the same result whe we compute the forward differece of a sequece ad delay the result (Figure 2.13a) as whe we delay the sequece first ad the compute the forward differece (Figure 2.13b). Also, it follows from Eq. (2.63) that the overall impulse respose of each cascade system is the covolutjo of the idividual impulse resposes. Cosequetly, h[] = (o[ + 1] - o[j) * o[ - 1] = o[ -1] * (o[ + 1]- 8[J) = o[] - o[ - 1]. (2.79) Thus, h[ ] is idetical to the impulse respose of the backward differece system; that is, the cascaded systems of Figures 2.13(a) ad 2.13(b) ca be replaced by a backward differece system, as show i Figure 2.13(c). Note that the ocausal forward differece systems i Figures 2.13(a) ad (b) have bee coverted to causal systems by cascadig them with a delay. geeral, ay ocausal FR system ca be made causal by cascadig it with a sufficietly log delay. Aother example of cascaded systems itroduces the cocept of a iverse system. Cosider the cascade of systems i Figure The impulse respose of the cascade system is h[] = u[] * (o[] - o[ -1]) =u[]-u[-l] (2.80) = 8[]. That is, the cascade combiatio of a accumulator followed by a backward differece (or vice versa) yields a system whose overall impulse respose is the impulse. Thus, the output of the cascade combiatio will always be equal to the iput, sice x[] * t5[] = x[]. this case. the backward differece system compesates exactly for (or iverts) the effect of the accumulator; that is. the backward differece system is the iverse

27 34 Discrete-Time Sigals ad Systems Chap. 2 x[] Accumulator system y[] Backwarddifferece system x[] Figure 2.14 A accumulator i cascade with a backward differece. Sice the backward differece is the iverse system for the accumulator, the cascade combiatio is equivalet to the idetity system. system for the accumulator. From the commutative property of covolutio, the accumulator is likewise the iverse system for the backward differece system. Note that this example provides a system iterpretatio of Eqs. (2.8) ad (2.10). geeral, if a liear time-ivariat system has impulse respose h[), the its iverse system, if it exists, has impulse respose hi [) defied by the relatio h[) * hi[) = hi[) * h[) = 8[). (2.81) verse systems are useful i may situatios i which it is ecessary to compesate for the effects of a liear system. geeral, it is difficult to solve Eq. (2.81) directly for hi [), give h[). However, i Chapter 3 we will see that the z-trasform provides a straightforward method of fidig a iverse system. 2.5 LNEAR CONSTANT-COEFFCENT DFFERENCE EQUATONS A importat subclass of liear time-ivariat systems cosists of those systems for which the iput x[ ) ad the output y[) satisfy a Nth-order liear-costat-coefficiet differece equatio of the form N M L aky[ - k) = L bmx[ - m). (2.82) k=o The properties discussed i Sectio 2.4 ad some of the aalysis techiques itroduced there ca be used to fid differece equatio represetatios for some of the liear time-ivariat systems that we have defied. Example 2.14 Differece Equatio Represetatio of the Accumulator m=o A example of the class of liear costat-coefficiet differece equatios is the accumulator system defied by y[] = L x[k]. To show that the iput ad output satisfy a differece equatio of the form ofeq. (2.82), ote that we ca write the output for - 1 as -l y[ - 1] = L x[k]. k=-x By separatig the term X[l] from the sum, we ca rewrite Eq. (2.83) as f)- (2.83) (2.84) y[rz] = x[] + )' x[k]..:=-x (2.85)