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

# 127a we c an a pproximate f t w ith a sum of

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: 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 t]...
View Full Document

Ask a homework question - tutors are online