Unformatted text preview: e inverse z transform o f ( e 2  2)z
F [z] = ~(z,_ce""'2),.(z'2""'") Realize a s ystem whose t ransfer f unction is given by
6 w hen t he region o f convergence is
( a) Izl > 2 ( b) e  2 < Izl < 2 ( c) Izl H[z] = L kzk
k=O 1 1. 7 3
1 1.61 1 1.62 < e 2
. D etermine t he z erostate response o f a s ystem h aving a transfer function D etermine y[kJ, t he o utput s amples for the s ystem d epicted i n Fig. P l1.61 if the
i nput J (t) = e  2t u(t). H[z] Izl > 0.8 = (z + 0.2;(Z  0.8) For t he s ampleddata s ystem i n F ig. P ll.62, show t hat t he z transfer function is
a nd a n i nput J[k] g iven b y Tz _
G[z]
[ ]  l+GH[z]
w here t he t ransfer function GH[z] c orresponds t o G (s)H(s) i n Table 12.1.
1 1.63 ( a) J[k] + 1)]
( c) J[k] = eku[k] + 2 ku[(k + 1)]
1 1. 7 4 F or t he s ystem i n P roblem 11.73, determine t he z erostate response if t he i nput F or t he s ampleddata s ystem in Fig. P ll.63, show t hat t he o utput Y[z] is given by Y[z] = FG[z]
I +GH[z] N ote t hat FG[z] c orresponds t o t he e ntry for F(s)G(s) i n T able 12.1. I t is n ot t he
s ame as F[z]G[z]. I n t his case, i t is n ot possible t o e xpress t he o utput a s Y[z] =
T[z]F[z]. Consequently, t he z transfer function o f s uch s ystems does n ot e xist, a nd
t heir analysis is little more complicated. = eku[k] ( b) J[k] = 2 ku[(k J[k] = 2 ku[k]
1 1. 7 5 + u [(k + 1)] F or t he s ystem i n P roblem 11.73, determine t he z erostate response if t he i nput J[k] = e  2k [_(k + 1)] , 717 12.1 Frequency Response o f DiscreteTime Systems cos O k = =} IH[eifljl cos (Ok + LH[e iflj ) (12.5) In other words, t he s ystem response y [kj t o a sinusoidal i nput cos O k is given by
(12.6a)
Following t he s ame argument, t he s ystem response t o a sinusoid cos ( Ok
y [kj = F requency Response and Digital Filters
F iltering characteristics of a system are specified by its frequency response. For
t his reason it is i mportant t o s tudy frequency response o f discretetime systems,
which is very similar to t he frequency response o f continuoustime systems with
some significant differences. 1 2.1 Frequency Response o f DiscreteTime Systems For (asymptotically stable) continuoustime systems we showed t hat t he system
response to a n i nput ejwt is H { jw ) ejwt, a nd t hat t he response t o a n i nput cos w t is
IH (jw)1 cos [wt + L H(jw)j. Similar results hold for discretetime systems. We now
show t hat for a n ( asymptotically stable) LTID system, t he system response to an
i nput ejnk is H [einjejflk a nd t he response to a n i nput cos O k is IH[ejfljl cos ( Ok +
L H[e ifl ]). T he p roof is similar t o t he one used in continuoustime systems. In Sec. 9.42
we showed t hat an LTID system response to a n (everlasting) exponential z k is also
a n (everlasting) exponential H [zjzk. I t is helpful t o r epresent this relationship by
a directed arrow notation as
Zk = =} H [zjzk
(12.1)
S etting z = e±jfl in t his relationship yields
ejflk = =} e jflk = =} [eiflj eiflk
H [e jflje  iflk
H (...
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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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