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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, timeinvariant (LTI) system. Linearity allowed us t o use
t he p rinciple of superposition, a nd t imeinvariance 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 erostate 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 timeinvariant system. I f t he s ystem is timevarying,
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) 2.41 * 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 TimeDomain Analysis of ContinuousTime 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 eroState R esponse h (11:) = 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)/(tr)dr
(2.37)
In deriving Eq. (2.37), we assumed t he s ystem t o b e linear a nd t imeinvariant.
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...
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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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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