EECE 340 Homework, Spring 2008

EECE 340-Signals and Systems Homework #8 Problem #1 The system shown below is LTI. Let the system input and impulse response be respectively given by: x(n) = u(n) -u(n-N) h(n) = a" u(n) x(n) hin) y(n) Determine the system output, y(#), using the inverse Z-transform. Problem #2 A causal, linear and shift-invariant discrete-time system is described by the following difference equation: y(n) = y(n-1) + yp —-2)+x(n-D. a) Find H (2), plot the poles and zeros of H (z) and indicate the ROC. b) Find the unit sample (impulse) response of the system. c) Find a stable and non-causal unit sample response that satisfies the given difference equation. Problem #3 Consider a causal, linear and shift-invariant discrete-time system with transfer function, H(z), given by “1-1 A(z)= ira where ais real. I-az"' a) For what range of values of a the system is stable. b) For 0<a<1 plot the pole-zero diagram and specify the ROC. c) Show graphically in the z-plane that this system is an all pass; i.e.,|H(@)|= constant for all a. Page 1 of 2 Problem #4 Consider the following difference equation representing a causal discrete-time system: y(n) — 2r cosOy(n—-1) 4+ r? y(n — 2) = x(n). Determine the system output, p(7), if its input is given by x(”) = @"u(n). Problem #5 A discrete-time system has the following transfer function: 1-2" A(z) = ——__—__—— @) 1-221 +0.752° a) Determine the impulse response, /(n), of the system if it is assumed causal. b) Determine the impulse response, h(n) , of the system if it is assumed stable. Page 2 of 2 EECE 240 smework # & oR 3 y) A ln) = Bie (Wy, «on = &€C1/ ~ 4 (h—N) = %, Ch) — XL, [A) [t(2) = , [el>/a/ fu az27 x, (2) = i » 18/>! Y (2) =H h (CE = (i- eC aay Z* Boyle e my -felymax(al) YY =H = Z C-#) (-a 27) ate any (2 -~ / ~ le | >raz(e, 1) ji) = Zz gn! d2 a) (2-1) (@ -+) ~ y(n) =_4 =! a | Q—} Q—| A-éf a —M of-/ =a — A-t/ _ at ~ Q-/ fo= y(n) _ alts 2 fill = gln—-1) + YCn~2) + oo a) Via = FV 4E VYOLEMY (@ll-# 27] = 2X H(2 = LO _ Zz! X(@ ogee = z Z 2-1 Z ~ /2/> ee l Zz” d 49 J, @~ BE) (2-EE) 4 A =F, z= (2 12/5) (#- 124) Za Ieys B= [+s Ns <) 4A stihl. (n00 = Casa ALO? one Lowe FR O.0, ca thcde phe unt civals , Ther, H(z) = fe De ie SE </alc tHe zm At) =z. i pen @-405), oe = <7, ay a | n =-= (55), m0 Hf pY) sf P+ PH! -—h A (n) == Cp- EE = —/4/s — ae (si om 7" {> ZZ 2g YL <O B3 y yi — I ena4ln-P 40° Gh - 2 =z (h). Wie [1—zrene Z'4 (+37) = Xlz) Ha) = LE | X(® - l—- zfo@mo Ey eget Zz i 2 - —2tfwme Fe pf