Question

3. (Oppenheim Willsky) Determine the z-transform for each of the following sequences. Sketch the pole-zero plot and indicate the region of convergence. Indicate whether or not the discrete-time Fourier transform of the sequence exists. (a) 8[n +5] (b) (-1)"u[n] (c) (-3)”u[-n – 2] (d) 27u[n] +(4)”u[n – 1]

Answer 1

x, [n] = sants] Apply the Z-transform. n=- X,625 = 4,00) 2" .$[n+] z" = 745) [: stnts]e 4 o; otherwis h=- 0 plot the There are 5 zeros at origin. porte-zero plot as shown below. z-plane As The $[n+5]z" is convergeble at na-5. region of convergence is entire z-plane. As the signal is bounded or having convergence, the discrete-time fourier transform of the sequence exists.

X₂ [n] = f)" uin] Apply the z-transform. n=-06 X2(z) = = 422 [n] in 6413" ucn) zi" po n=-* - (-)"z" since, urmat"n. since, usn]=(1, n o 10, otherwise This sum is convergable if and only if 1 실시 2.12171 - Region of convergence. -- X2(Z) = 1+) + ) + ... . ;-(-22) iaza con z 1 poles: 2+1=0 Z=-1 tros: 2=0 Sketch the pole-zero plot. +Roc:12171 two values ine, As com uin] is bounded between I and al, the discrete-time Fourier transform of the sequence exists.

az [n] = (-3)* u(-n-a] Apply the z-transform. ns- xsce = f (m) 25" - CYB)"uf-n-2] 2h n=-. lins-2 z-n since, uf-n-2] = since, uf-n-2 o others n=-D 2 n=- * This sum has convergence of and only if 732l</ 12/</ Region of convergence X, (c) - 8 (35” - [(-32) + ] n=- +32-1 (+32 1 + (32-1) (32+i) 32+1 - It 92²-1 = 92 32+1 32+1 Zeros: 2=0 =) Two zups of origin. poles: 32+1 = 2-3 sketch the pole-zero plot 1 Im Roc:121 <3 he Here, zin] has no convergence. Therefore, the discrete-time fourier transform of the sequence does not exist.

ay (m3 = 2ucn] + (%) "urn-1). Apply the Z-transform. Xyles @ "ucn) z* + s ."ufn1] " hood na-a n=-00 na 20 h ) no - summation exists for Summation exists 1 1 1 1 for 12 <1 12/<2 1217lly .. Overall region of convergence is, Roc: 4[2] < 2. Xyle) = 1 / 1 t = 2-(2-2)(42-1) + 42(2-2) (2-2)(42-1) 2-(82-2-42²+2) +82-42² = (2-2) (97-1) = 4-2 poles: 2-2803) Z=2 42-1-02 tuos: 4-2CD)=4 (2-2) (Y2-1) Roc. sketch the pole-tuo plot: Thus, the discrete-time The sequence has convergence. Pourier transform exists.