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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.
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...
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