Question

3. For each of the following discrete-time sequences: (i) Find the Z-transform (ZT), if it exists, and plot the region of convergence (ROC) in the Z-plane (ii) Find the poles and zeros and plot them in the 2-plane (iii) Determine whether the DTFT of the sequence exists (a) x[n] = 8[n – 1] + 28[n – 3] (b) [n] = (0.9e-j*)" u[n + 2] – 2-ul-n - 1] (c) x[n] = 2-" un + 1]

Answer 1

Ô mm) = $(n-1) + 2 :(n-3) TX (2) = 3 + 2 33 - Roc Intire 2-plane except to (1) дос 8121=1 Z-plane whole 2-plane except to x = 314273 242 23 zens 237220 = 727 121 222 Poles = 20 =0,oo olzl= 82 21=1 Real & real Polex Zeros e feane) 2-Reane 22 Time circule circule All Poles lige Inside Unit Radive rence stable Analode Roc 121=1 rence DYRT is defined.

① xM) = (0.g En unt2) - Ajn utny o- O 10.g eVitja uh) & 27. z 12170.g. bution (o.gelighth untry & 27 : 2 2-o.get ; 121700 loigē unaye+3j 127 11 )? 73 -o.gěst - wen-)) <3:12 Zone 3 121 <0.5 - 0. 5 ③ Roc 127o.gq ² Roc 3 121 < 0.5 Mo common Кос fence & Transform not exist © 3 ; h) on 2M) = 5 464+) ir um 21. thronel @jual untu) *Zuip rs 121705 am) 2 unt)) <Z7y x) = 284, 13170.5 Hozzo.5

ii) X(2) - 22. 2 zenosz²=0 220,0 Polop 7 22 207 121=2 12/24 2 =0,0 z-flone 771) 212015 L121- Since Poles lies 3 noide MAA05) - Gre Kno preT 21=1 Grue then exist. Note ② Pob liep inside 12121 circill then DTFT exist or ROC Include/ contain 121=2 circule. ề an uch) & 2-7. 2 ) Roc 1217191 - a un punt 3 Roc ; 21419) | x & ritg X2) aneh kelig 2 XG)