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Unformatted text preview: ur basic signal components. There, we show t hat
any arbitrary i nput signal can be expressed as a weighted sum of sinusoids (or
exponentials) h aving various frequencies. T hus a knowledge of t he s ystem response
t o a sinusoid enables us t o d etermine t he s ystem response t o a n a rbitrary i nput f (t). Systems whose parameters do n ot change with time are t ime-invariant (also
c onstant-parameter) systems. For such a system, if t he i nput is delayed by T
seconds, t he o utput is t he s ame as before b ut delayed by T (assuming identical
initial conditions). This p roperty is expressed graphically in Fig. 1.28.
I t is possible t o verify t hat t he s ystem in Fig. 1.26 is a t ime-invariant system.
Networks composed of R LC elements a nd o ther commonly used active elements
such as transistors are time-invariant systems. A system with a n i nput-output
r elationship described by a linear differential equation of t he form (1.44) is a linear
time-invariant (LTI) system when t he coefficients a i a nd bi of such equation are
constants. I f these coefficients are functions of time, t hen t he s ystem is a linear
t ime-varying system. T he s ystem described in exercise E1.12 is a n example of a
linear time-varying system. Another familiar example of a time-varying system is
t he c arbon microphone, in which t he resistance R is a function of t he mechanical
pressure generated by sound waves on t he c arbon granules of t he microphone. An
equivalent circuit for t he microphone appears in Fig. 1.29. T he response is t he
c urrent i(t), a nd t he e quation describing t he circuit is L di(t) + R (t)i(t) = f (t) dt O ne of t he coefficients in this equation, R (t), is time-varying.
,0, E xercise E l.14
Show t hat a system described by t he following equation is time-varying parameter system: y (t) = (sin t) i (t - 2)
Hint: Show t hat t he s ystem fails t o satisfy t he time-invariance property. 'V 84 I ntroduction to Signals and Systems 1. 7 Classification of Systems L / (t) f (t) R (t) o
F ig. 1 .29 An example of a linear time-varying system. 1 .7-3 (a) I nstantaneous a nd Dynamic Systems As observed earlier, a system's o utput a t any instant t generally depends upon
the entire p ast i nput. However, in a special class of systems, t he o utput a t any
instant t depends only on its input a t t hat i nstant. In resistive networks, for example, any o utput o f t he network a t some instant t depends only on t he i nput a t t he
i nstant t . I n t hese systems, past history is irrelevant in determining t he response.
Such systems are said t o b e i nstantaneous or m emory l ess systems. More precisely, a system i s s aid t o b e instantaneous (or memoryless) if its o utput a t any
instant t depends, a t most, on the strength of its input(s) a t t he same instant b ut
n ot on any past o r f uture values of the input(s). Otherwise, t he system is said t o b e
d ynamic (or a s ystem with memory). A system whose response a t t is completely
determined by t he i nput signals over the past T seconds [interval from (t - T) t o
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