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

6 e xercise e 112 show that the system described by

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ) 'V M ore C omments o n Linear S ystems Almost all systems observed in practice become nonlinear when large enough signals are applied t o t hem. However, many systems show linear behavior for small signals. T he analysis of nonlinear systems is generally difficult. Nonlinearities can arise in so many ways t hat describing t hem w ith a common mathematical form is impossible. Not only is each system a category in itself, b ut even for a given 82 1 I ntroduction t o Signals a nd Systems 1. 7 system, changes i n initial conditions or i nput a mplitudes may change t he n ature of t he problem. O n t he o ther hand, t he s uperposition property of linear systems is a powerful unifying principle which allows for a general solution. T he s uperposition p roperty ( linearity) greatly simplifies t he analysis of linear systems. Because of t he d ecomposition property, we c an evaluate separately the two components of t he o utput. T he z ero-input component can be computed by assuming t he i nput t o b e zero, a nd t he z ero-state component can be computed by assuming zero initial conditions. Moreover, if we express a n i nput f (t) as a sum of simpler functions, 83 Classification of Systems y ( I) f (l) (a) then, by virtue o f linearity, t he response y(t) is given by y (l - T) f (l - T) (1.45) where Yk(t) is t he z ero-state response t o a n i nput fk(t). T his apparently trivial observation has profound implications. As we shall see repeatedly in l ater c hapters, it proves extremely useful a nd opens new avenues for analyzing linear systems. (b) f (l) F ig. 1 .28 f(l) T ime-invariance p roperty. 1.7-2 Time-Invariant and Time-Varying Parameter Systems 1- (a) F ig. 1 .27 !J.I (b) S ignal r epresentation i n t erms of i mpulse a nd s tep c omponents. As a n e xample, consider an a rbitrary i nput f (t) such as t he one shown in Fig. 1.27a. We c an a pproximate f (t) w ith a sum of rectangular pulses of width At a nd of varying heights. T he a pproximation improves as A t ...... 0, when t he r ectangular pulses become impulses spaced A t seconds a part (with A t ...... 0). T hus, a n a rbitrary i nput c an be replaced by a weighted sum of impUlses spaced A t ( At ...... 0) seconds a part. Therefore, if we know t he s ystem response t o a u nit impulse, we c an immediately determine t he s ystem response t o a n a rbitrary i nput f (t) by adding t he s ystem response to each impulse component of f (t). A similar situation is d epicted in Fig. 1.27b, where f (t) is a pproximated by a sum of s tep functions of varying m agnitude a nd spaced At seconds a part. T he a pproximation improves as A t becomes smaller. Therefore, if we know t he s ystem response t o a u nit s tep input, we c an compute t he s ystem response t o a ny a rbitrary i nput f (t) w ith relative ease. Time-domain analysis of linear systems (discussed in Chapter 2) uses this approach. In C hapters 4,6,10, a nd 11 we employ t he same approach b ut i nstead use sinusoids or exponentials as o...
View Full Document

Ask a homework question - tutors are online