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

# 7 1 linear and nonlinear systems the concept o f

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online