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Unformatted text preview: impulse components and add them t o yield the system response t o t he input t T here is t he p ossibility of a n i mpulse in a ddition t o c haracteristic modes. 164 2 T ime-Domain Analysis of Continuous-Time Systems f (t). T he s um o f t he responses t o t he i mpulse components is in t he form of a n
i ntegral, known a s t he convolution integral. T he s ystem response is o btained as t he
c onvolution of t he i nput f (t) w ith t he s ystem's i mpulse response h (t). T herefore,
t he knowledge o f t he s ystem's impulse response allows us t o d etermine t he s ystem
response t o any a rbitrary i nput. LTIC s ystems have a very special relationship t o t he e verlasting exponential
signal est b ecause t he response of an LTIC system t o such a n i nput s ignal is t he
s ame signal w ithin a m ultiplicative constant. T he response of a n L TIC system t o
t he e verlasting e xponential i nput est is H (s)e st , w here H (s) is t he t ransfer function
of t he s ystem.
Differential equations of LTIC systems c an also be solved by t he classical
m ethod, w here t he response is o btained as a s um of n atural a nd forced response.
These are n ot t he s ame as t he z ero-input a nd z ero-state components, a lthough t hey
s atisfy t he s ame equations respectively. Although simple, this m ethod suffers from
t he fact t hat i t i s a pplicable t o a r estricted class of i nput signals, a nd t he s ystem
response c annot b e e xpressed as a n explicit function of the i nput. T his l imitation
makes i t useless i n t he t heoretical s tudy o f systems.
A linear s ystem is in a zero s tate if all initial conditions are zero. A system in
a zero s tate is i ncapable of generating a ny r esponse in the absence of a n e xternal
i nput. W hen s ome i nitial conditions are applied t o a s ystem, t hen, if t he s ystem
eventually goes t o zero s tate i n t he a bsence of a ny e xternal i nput, t he s ystem is said
t o b e a symptotically s table. In c ontrast, if t he s ystem's response increases w ithout
b ound, i t is u nstable. I f n either t he s ystem goes t o zero s tate n or t he r esponse
increases indefinitely, t he s ystem is marginally stable. T he s tability criterion in
t erms o f t he l ocation of a s ystem's c haracteristic roots c an be summarized as follows:
1. A n L TIC s ystem is asymptotically s table if, a nd only if, all t he c haracteristic
roots are in t he LHP. T he r oots may b e r epeated o r unrepeated. P roblems 165 References
1 . L athi, B .P., Signals and Systems, B erkeley-Cambridge P ress, C armichael, C a.,
2 . K ailath, T ., L inear S ystem, P rentice-Hall, Englewood Cliffs, N.J., 1980. 3 . L athi, B .P., Modern Digital and Analog Communication Systems, T hird E d.,
O xford U niversity P ress, New York, 1998. Problems
2 .2-1 An LTIC system is specified by the equation
(D2 + 5 D + 6) y(t) = (D + l )f(t) ( a) Find the characteristic polynomial, characteristic equation, characteristic roots,
and characteristic modes of this system.
( b) Find ya(t), t he zero-input component of the response y (t) for t 2: 0, if t he initial
conditions are ya(O) = 2 and ya(O) = - l.
2 .2-2 Repeat Pr...
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