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

V 84 i ntroduction to signals and systems 1 7

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Unformatted text preview: is a f inite-memory s ystem with a memory of T seconds. Networks containing inductive a nd capacitive elements generally have infinite memory because the response of such networks a t any instant t is determined by their inputs over the entire past ( -00, t ). T his is t rue for the R C circuit of Fig. 1.26. In this book we will generally examine dynamic systems. Instantaneous systems are a special case of dynamic systems. 1. 7- 4 t- Causal a nd Noncausal S ystems A c ausal (also known as a p hysical or n on-anticipative) s ystem is one for which t he o utput a t a ny instant to depends only on the value of the input f (t) for t ::; to. In other words, the value of the o utput a t t he present instant depends only on t he p ast a nd p resent values of the input f (t), n ot on its future values. To p ut i t simply, in a causal system the o utput c annot s tart before the input is applied. I f t he response s tarts before the input, i t means t hat t he system knows t he i nput i n t he future a nd a cts o n this knowledge before t he i nput is applied. A system t hat violates the condition of causality is called a n oncausal (or a nticipative) system. Any practical system t hat operates in real t imet must necessarily be causal. We do not yet know how t o build a system t hat c an respond t o future inputs (inputs not yet applied). A noncausal system is a prophetic system t hat knows t he future input and acts o n i t in the present. Thus, if we apply an input starting a t t = 0 t o a noncausal system, the o utput would begin even before t = o. As a n example, consider the s ystem specified by y(t) = f (t - 2) + f (t + 2) (1.46) tIn real-time operations, the response to an input is essentially simultaneous (contemporaneous) with the input itself. o 2 4 t- Fig. 1 .30 A noncausal system and its realization by a delayed causal system. For the input f (t) i llustrated in Fig. 1.30a, t he o utput y(t), as computed from Eq. (1.46) (shown in Fig. 1.30b), s tarts even before t he i nput is applied. Equation (1.46) shows t hat y (t), t he o utput a t t , is given by t he s um of t he i nput values two seconds before and two seconds after t ( at t - 2 and t + 2 respectively). B ut if we are operating the system in real time a t t, we do not know what t he value of t he i nput will be two seconds later. T hus i t is impossible t o implement this system in real time. For this reason, noncausal systems are unrealizable in r eal t ime. W hy S tudy N oncausal Systems? From t he above discussion i t may seem t hat noncausal systems have no practical purpose. This is n ot the case; they are valuable in t he s tudy of systems for several reasons. First, noncausal systems are realizable when t he i ndependent variable is o ther t han "time" (e.g., s pace). Consider, for example, an electric charge of density q(x) placed along t he x-axis for x ?: O. This charge density produces an electric field E (x) t hat is p resent a t every point on the x-axis from x = - 00 t o 00. I n this case the input...
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