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

# 84 where yn t is a linear combination of the

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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., 1987. 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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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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