Unformatted text preview: 3 I f t he two systems are t o b e equivalent, y (kT) in Eq. (12.29c) must be equal t o y[k] i n Eq. (12.30). T herefore y (t)
(a) h[k] = lim T ha(kT)
T _O (12.31) This is t he t imedomain criterion for equivalence of t he two systems. according t o
t his criterion, h[k], t he u nit impulse response of H[z] in Fig. 12.8a, must be equal
t o T t imes t he s amples of halt), t he u nit impulse response o f t he s ystem in Fig.
12.8b, assuming t hat T .... O. T his is known as t he i mpulse i nvariance c riterion
of filter design. t I{. (s) (b) 1 2.42
F ig. 1 2.8 A nalog f ilter r ealization w ith a d igital f ilter. 12.8b. Therefore y (kT), t he samples of t he o utput in Fig. 12.8b, are identical t o
y[k], t he o utput of H[z] in Fig. 12.8a.
T he o utput y(t) of t he s ystem in Fig. 12.8b i st T he FrequencyDomain Equivalence Criterion I n Sec. 2.43 [Eq. (2.47)J, we proved t hat for a n analog system with transfer
function Ha(s), t he s ystem response y(t) t o t he everlasting exponential i nput f (t) =
e st is also a n e verlasting exponential
(12.32) y(t) = i :f(r)ha(tr)dr Similarly, in Eq. (9.57a), we showed t hat for a discretetime system with transfer
function H[z], t he s ystem response y[kJ t o a n everlasting exponential i nput f[k] = zk
is also a n everlasting exponential H[zJz k : An integral is a sum in t he limit. Therefore, t he above equation can be expressed
as y[k] = H[z]zk 00 y(t) = lim . 6.7_0 " f(ml::.r)ha(t  ml::.T)l::.r
L t (12.29a) m =oo (12.33) I f t he s ystems in Figs. 12.8a a nd 12.8b are equivalent, t hen t he response of b oth s ystems t o a n e verlasting exponential i nput f (t) = e st s hould be t he same. A
continuoustime signal f (t) = e st s ampled every T seconds results in a discretetime signal F or o ur purpose i t is convenient t o use t he n otation T for l::.r in t he above equation.
S uch a change of notation yields f[kJ = e skT = zk w ith z = e sT
00 y(t) = lim T
T O " f (mT)h,,(t  mT) ~ (12.29b) m =oo T his discretetime exponential zk is applied a t t he i nput of H[zJ i n Fig. 12.8a, whose
response is y[kJ = H[zJzklz=e.T T he response a t t he k th sampling i nstant is y (kT) o btained by setting t = k T in
t he above equation: y (kT) = lim T
T O " ~ f (mT)h,,[(k  m)TJ (12.29c) = H [esT]eskT (12.34) Now, for t he s ystem in Fig. 12.8b, y (kT), t he k th sample o f t he o utput y(t) in Eq.
(12.32), is m =oo I n Fig. 12.8a, t he i nput t o H[zJ is f (kT) = f[kJ. I f h[kJ is t he u nit impulse response
o f H[zJ, t hen y[k], t he o utput of H[z], is given b y (12.35)
I f t he two systems are t o b e equivalent, a necessary condition is t hat y[kJ in Eq. (12.34) must be equal t o y (kT) in Eq. (12.35). This condition means t hat
00 y[kJ = L f[m]h[k  mJ (12.30) (12.36) m =oo t For t he s ake o f generality, we a re a ssuming a noncausal system. T he a rgument a nd t he results
a re a lso valid for causal systems. This is t he frequencydomain criterion for equivalence of t he two sys...
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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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