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Unformatted text preview: em response t o a n a rbitrary i nput f (t). 2 T imeDomain Analysis of ContinuousTime Systems 120 T his display shows t he i nputoutput p airs in Figs. 2.3b, c, a nd d, respectively. T he
l ast p air r epresents t he s ystem response t o only one of t he i mpulse components
of f (t). T he t otal r esponse y(t) is o btained by summing all such components as
depicted in Fig. 2.3e. Summing on b oth sides of t he above display yields (with !:::.r > 2.4 System Response t o E xternal I nput: T he Z eroState Response t his r eason t his i ntegral is given a special name: t he c onvolution i ntegral. T he
c onvolution integral of two functions h (t) a nd fz (t) is d enoted symbolically by
h (t) * f z(t) a nd is defined as 0) lim ~T_O * fz(t) == h (t)
00 00 " f(n!:::.r)8(t  n!:::.r)!:::.r ==*
L....i n =oo . lim .67"+0 " f(n!:::.r)h(t  n!:::.r)!:::.r
L....t n =oo . The output y{ t} T he input I {t} v v I" I {t} 1: h (T)fz(t  T) dr (2.30) Some i mportant p roperties of t he c onvolution integral are given below.
1. T he Commutative Property: C onvolution o peration is commutative; t hat is,
h (t)*fz(t) = f z(t)*h(t). T his p roperty c an b e proved by a change of variable .
In Eq. (2.30), if we let x = t  T so t hat r = t  x a nd dT =  dx, we o btain' T he l efthand s ide is t he i nput f (t) r epresented as a s um of all t he i mpulse components i n a m anner i llustrated in Fig. 2.3a. T he r ighthand side is t he o utput y(t)
r epresented as a s um o f t he o utput c omponents as shown in Fig. 2.3e. B oth t he
l efthand side a nd t he r ighthand side, by definition, are integrals given b yt 1 :f(r)8(tr)dT==* 1 : f (T)h(tT)dT 121 h (t) * fz(t) == (2.28) J~oo f z(x)h(t  x) dx 1 : fz (x)h(tX)dX = f z(t) * h (t) (2.31) I 2. T he Distributive Property: A ccording t o t his property: y {t) T he l efthand s ide expresses t he i nput f (t) as m ade u p o f t he i mpulse components
in a m anner d epicted i n Fig. 2.3a. T he r ighthand expresses t he o utput as m ade u p
o f t he s um o f t he s ystem responses t o all t he i mpulse components of t he i nput as
illustrated in F ig. 2.3e. To summarize, t he ( zerostate) response y(t) t o t he i nput
f (t) is given b y
(2.29) h (t) * [fz(t) + h (t)] = h (t) * fz(t) + h (t) * h (t) (2.32) 3. The Associative Property: According t o t his p roperty: h (t) * [fz(t) * h (t)] = [ h (t) * fz(t)] * h (t) (2.33) T he p roofs of (2.32) a nd (2.33) follow directly from t he d efinition of t he convolution integral. T hey a re left as a n exercise for t he r eader. 4. T he Shift Property: I f
T his is t he r esult we seek. We have o btained t he s ystem response y(t) t o i nput
f (t) i n t erms o f t he u nit i mpulse response h(t). Knowing h(t), we c an d etermine
t he r esponse y (t) t o a ny i nput. Observe once again the allpervasive nature o f the
s ystem's characteristic modes. The s ystem response t o any input is determined
b y the impulse response, which in turn is made up o f characteristic m odes o f the
s ystem....
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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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