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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
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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