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

8 t he poles a t 05 and 08 a re enclosed within t he

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e inverse z -transform o f ( e- 2 - 2)z F [z] = ~(z--,_c-e--""'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 kz-k k=O 1 1. 7 -3 1 1.6-1 1 1.6-2 < e- 2 . D etermine t he z ero-state 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 l1.6-1 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 ampled-data s ystem i n F ig. P ll.6-2, 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 1.6-3 ( 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 11.7-3, determine t he z ero-state response if t he i nput F or t he s ampled-data s ystem in Fig. P ll.6-3, 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 11.7-3, determine t he z ero-state response if t he i nput J[k] = e - 2k [_(k + 1)] , 717 12.1 Frequency Response o f Discrete-Time 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 discrete-time systems, which is very similar to t he frequency response o f continuous-time systems with some significant differences. 1 2.1 Frequency Response o f Discrete-Time Systems For (asymptotically stable) continuous-time 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 discrete-time 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 continuous-time systems. In Sec. 9.4-2 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 (...
View Full Document

Ask a homework question - tutors are online