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

For the input f t i llustrated in fig 130a t he o

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Unformatted text preview: [Le., t he charge density q(x)] s tarts a t x = 0, b ut i ts o utput [the electric field E (x)] begins before x = o. Clearly, this space charge system is noncausal. This discussion shows t hat only temporal systems (systems with time as independent variable) must be causal in order t o b e realizable. T he t erms "before" a nd "after" have a special connection t o causality only when t he i ndependent variable is time. This connection is lost for variables other t han time. Nontemporal systems, such as those occurring in optics, can be noncausal and still realizable. Moreover, even for t emporal systems, such as those used for signal processing, the s tudy of noncausal systems is important. In such systems we may have all input d ata prerecorded. (This often happens with speech, geophysical, a nd meteorological signals, and with space probes.) In such cases, the input's future values are available t o us. For example, suppose we h ad a set of input signal records available for t he system described by Eq. (1.46). We c an t hen c ompute y(t) since, for any t , we need only refer t o t he records t o find t he i nput's value two seconds before and two seconds after t . Thus, noncausal systems can be realized, although not in real time. We 86 I ntroduction t o Signals a nd Systems 1. 7 Classification of Systems 1 .7-5 Noncausal systems are realizable with time delay! may therefore b e able t o realize a noncausal system, provided t hat we are willing t o a ccept a t ime delay in t he o utput. Consider a system whose o utput y(t) is t he same as y(t) in Eq. (1.46) delayed by two seconds (Fig 1.30c), so t hat y(t) = y(t - 2) = f (t - 4) + f (t) Here t he value o f t he o utput y a t a ny instant t is t he s um of t he values of t he i nput f a t t a nd a t t he i nstant four seconds earlier [at (t - 4)]. In this case, t he o utput a t any i nstant t does not depend on future values of t he i nput, a nd t he s ystem is causal. T he o utput of this system, which is y(t), is identical t o t hat in Eq. (1.46) or Fig. 1.30b e xcept for a delay of two seconds. Thus, a noncausal system may be realized or satisfactorily approximated in real time by using a causal system with a delay. A t hird r eason for studying noncausal systems is t hat t hey provide a n u pper bound o n t he p erformance of causal systems. For example, if we wish t o design a filter for s eparating a signal from noise, t hen t he o ptimum filter is invariably a noncausal system. Although unrealizable, this noncausal system's performance acts as t he u pper l imit o n what can be achieved a nd gives us a s tandard for evaluating t he p erformance of causal filters. At first glance, noncausal systems may seem inscrutable. Actually, there is n othing mysterious about these systems a nd t heir approximate realization through using physical s ystems w ith delay. I f we want t o know w hat will happen one year from now, we h ave two choices: go t o a p rophet (an unrealizable pe...
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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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