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

84 where yn t is a linear combination of the

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: 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...
View Full Document

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

Ask a homework question - tutors are online