Unformatted text preview: ystems are invertible and which are noninvertible. F or·
t he invertible systems, find t he i nput-output r elationship of the inverse system.
( a) yet) Show t hat t he system satisfies the homogeneity property b ut n ot t he a dditivity property.
1 . 7 -5 103 Problems 30 +
Y 2(r) f it) IF f (t) IH D, D3 D2 Fig. P I.S·l Fig. P1.8-1 f ( I) +
2R 2R R For t he c ircuit depicted in Fig. P1.8-1, find t he differential equations relating o utputs
YI(t) a nd Y2(t) t o t he input J(t). 1 .8-2 R epeat P rob. 1.8-1 for t he c ircuit in Fig. P1.8-2.
F ig. P 1.1-5
1 .1-6 T he i nductor L a nd t he c apacitor C in Fig. P1.7-6 are nonli!lear, which makes t he
circuit nonlinear. T he r emaining 3 elements are linear. Show t hat t he o utput yet) of
this nonlinear circuit satisfies the linearity conditions with respect t o t he i nput J(t)
a nd t he i nitial conditions (all t he initial inductor currents a nd c apacitor voltages).
Recognize t hat a c urrent source is a n o pen circuit when the current is zero. q
, 0.1 H 1 .8-3
+ IF y (t) 2Q F ig. P 1.1-6 1. 7 -7 For t he s ystems described by the equations below, with the i nput J(t) a nd o utput
yet), d etermine which of t he systems are causal a nd which are noncausal.
( a) yet) = J (t - ( b) yet) = J (-t) 2) h F ig. P 1.8-3 c f it) J t ( c) yet) = J(at)
( d) yet) = J(at) a> 1
a<1 W ater flows into a t ank a t a r ate of qi u nits/s a nd flows o ut t hrough t he outflow valve
a t a r ate of qO u nits/s (Fig. P1.8-3). D etermine t he e quation relating the outflow qO
t o t he i nput qi. T he outflow r ate is p roportional t o t he head h. T hus qo = R h where
R is t he valve resistance. Determine also the differential equation relating the head
h t o t he i nput qi.
(Hint: T he n et inflow of water in time 6 t is (qi - qo)6t. T his inflow is also A 6h
where A is t he cross section of t he t ank.) 2.1 T ime-Domain Analysis o f
C ontinuous-Time S ystems
I n t his b ook we consider two m ethods o f a nalysis of linear time-invariant (LTI)
systems: t he t ime-domain m ethod a nd t he f requency-domain m ethod. I n t his c hapter we discuss t he t ime-domain a nalysis o f l inear, time-invariant, continuous-time
(LTIC) systems. 2.1 Introduction I ntroduction 105 T heoretically t he p owers m a nd n in t he above e quations c an t ake o n any
value. P ractical noise considerations, however, require m ::; n . Noise is a ny un-.
desirable signal, n atural o r m anmade, which interferes w ith t he d esired signals in
t he s ystem. S ome of t he sources of noise are: t he e lectromagnetic r adiation from
s tars, r andom m otion o f e lectrons in s ystem c omponents, interference from n earby
r adio a nd television s tations, t ransients p roduced by automobile ignition systems,
fluorescent lighting, a nd so on. We show in C hapter 6 t hat a s ystem specified by
Eq. (2.1) behaves as a n (m - n)th-order d ifferentiator o f h igh-frequency signals if
m > n . U nfortunately, noise is a w ideband s ignal containing c omponents o f all frequencies from 0 t o 0 0...
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