Signal Processing and Linear Systems-B.P.Lathi copy

7 1 linear and nonlinear systems the concept o f

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Unformatted text preview: ffect also increases k-fold. Thus, if /(r) dr (1.36d) to From Eq. (1.36a), t he o utput voltage y(t) a t a n i nstant t c an be c omputed i f we know t he i nput c urrent flowing i n t he c apacitor throughout its entire p ast ( -00 t o t). A lternatively, if we know t he i nput c urrent / (t) from some moment to onward, then, using Eq. (1.36d), we c an still calculate y(t) for t ~ to from a knowledge of t he i nput c urrent, provided we know v c(to), t he i nitial capacitor voltage (voltage a t to). T hus v c(to) c ontains all t he relevant information a bout t he c ircuit's entire p ast ( -00 t o to) t hat we need t o c ompute y(t) for t ~ to. Therefore, t he response of a system a t t > to c an b e d etermined from its input(s) during t he i nterval to t o t a nd from c ertain i nitial c onditions a t t = to. I n t he p receding example, we n eeded only one initial condition. However, in more complex systems, several initial conditions may b e necessary. We know, for example, t hat i n passive R LC networks, t he i nitial values of all inductor currents C - ---> e kc - ---> ke t hen for all real or imaginary k (1.39) T hus, l inearity implies two properties: homogeneity (scaling) and additivity$. B oth t hese properties can b e combined into one p roperty ( superposition), which is expressed as follows: I f a nd t hen for all values of constants kl a nd k 2, t S trictly speaking, independent inductor currents a nd c apacitor voltages. t O ther classifications, such as deterministic a nd probabilistic systems, are beyond t he scope of this t ext a nd a re n ot considered. $ A l inear system must also satisfy t he a dditional condition of s moothness, where small changes in t he s ystem's inputs m ust r esult in small changes in its o utputs. 2 80 1 I ntroduction to Signals a nd Systems (1.40) T his is t rue for a ll q a nd C2. I t m ay a ppear t hat a dditivity implies homogeneity. Unfortunately, there are cases where homogeneity does not follow from additivity. See t he case in Exercise E1.11 below. 6 These facts can be readily verified from Eq. (1.42) for t he R C circuit in Fig. 1.26. For instance, if we double t he i nitial condition vc(O), t he zero-input component doubles; if we double t he i nput J (t), t he z ero-state component doubles. • E xample 1 .9 Show t hat the system described by the equation dy dt E xercise E LI! Show that a system with the input (cause) c(t) and the output (effect) e(t) related by e(t) = Re{ c(t)} satisfies t he additivity property but violates the homogeneity property. Hence, such a system is not linear. Hint: show that Eq. (1.39) is not satisfied when k is complex. 'V 81 1. 7 Classification of Systems + 3y(t) = J (t) (1.43) is linear. Let the system response to the inputs h (t) and h (t) be Yl(t) and Y2(t), respectively. Then R esponse o f a L inear System For t he sake of simplicity, we discuss below only s ingle-input, s ingle-output ( SISO) s ystems. B ut t he discussion can be readily extended t o...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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