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

# T his p roperty c an b e proved by a change of

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: I t is i mportant t o keep in mind t he a ssumptions used in deriving Eq. (2.29). We assumed a linear, time-invariant (LTI) system. Linearity allowed us t o use t he p rinciple of superposition, a nd t ime-invariance m ade i t possible t o express t he s ystem's r esponse t o 8(t - n!:::.T) a s h (t - n!:::.r). h (t) t hen * f z(t - T) h (t - T) * f z(t) T he z ero-state r esponse y(t) o btained in Eq. (2.29) is given by a n i ntegral t hat o ccurs frequently in t he physical sciences, engineering, a nd m athematics. For t I n d eriving t his r esult we have assumed a time-invariant system. I f t he s ystem is time-varying, t hen t he s ystem r esponse t o t he i nput 6 (t - n6T) c annot b e expressed as h (t - n6r), b ut i nstead has t he form h (t, n6T). Using t his form will modify Eq. (2.28) as y (t) = 1: f (T)h(t, T )dr where h (t, r ) is t he s ystem r esponse a t i nstant t t o a u nit i mpulse i nput l ocated a t (2.28n) T. (2.34a) = crt - T) (2.34b) - T2) = crt - Tl - T2) (2.34c) a nd h (t - Til * f z(t P roof: We are given * h (t) = Therefore The Convolution Integral = crt - T) h (t) i l(t) 2.4-1 * fz(t) = e(t) h (t) * fz(t 1: 1: h (T)fz(t - r) dT = crt) - T) = h (T)fz(t - T - T) dT = c (t - T) E quation (2.34b) follows from (2.34a) a nd t he c ommutative p roperty o f convolution; Eq. (2.34c) follows directly from (2.34a) a nd (2.34b). 5. Convolution with an Impulse: C onvolution o f a f unction f (t) w ith a u nit impulse results in t he f unction f (t) itself. B y d efinition of convolution 122 2 Time-Domain Analysis of Continuous-Time S ystems I (t) * 8(t) = 1: l (r)8(t - r) dr 2.4 S ystem R esponse t o E xternal I nput: T he Z ero-State R esponse h (1-1:) = 0 (2.35) p; Because 8(t - r) is a n impulse located a t r = t, according t o t he s ampling p roperty of t he i mpulse [Eq. (1.24)], t he i ntegral in t he above e quation is t he value of I (r) a t r = t, t hat is, I (t). T herefore I (t) * 8(t) = I (t) 1 <0 o 1:- (b) F ig. 2 .5 Limits of convolution integral. y (t) = I (t) t- Fig. 2 .4 Width property of convolution. 1: I (r ) h(t - r) dr t~O r)dr t <O t y(t) = l h(r)/(t-r)dr (2.37) In deriving Eq. (2.37), we assumed t he s ystem t o b e linear a nd t ime-invariant. T here were no o ther r estrictions e ither o n t he s ystem or on t he i nput s ignal I (t). I n p ractice, most s ystems a re causal, so t hat t heir response cannot begin before t he i nput s tarts. F urthermore, m ost i nputs a re also causal, which means t hey s tart a t t = O. C ausality r estrictions o n b oth signals a nd s ystems further simplify t he l imits of integration in E q. (2.37). By definition, t he response of a causal system c annot b egin before i ts i nput begins. Consequently, t he c ausal system's response t o a u nit i mpulse 8(t) ( which is located a t t = 0 ) c annot b egin before t = o. T herefore, a causal system's u nit impulse response h (t) i s a causal signal. I t is i mportant t o r emember t hat t he i ntegration in Eq. (2.37) is performed w ith r espect t o r ( not t). I f t he i nput I...
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