Question

Question 1 10 points Using the definition of the transform, determine the transforms for each of the following signals. Sketch the pole-zero plot and indicate the region of convergence. (a) (5 points) (-3)"[n-2 () (5 points) "0(9) 15 points transforms for each of Question 2... Using 3-transform pairs and properties tables, determine the the following signals. (a) (5 points) un-un-2 (b) (5 points) -- [n - 2 (e) (5 points) nyin-1 ... 10 points Question 3 Find the inverse (a) (5 points) transform of the following: 2(3-4) 2 - 53 +6 (b) (5 points) (3-1) EE 312 - Summer 2019 HW 6 Due: Jul. 25, 2019 ......... 20 points Question ... Consider an LTI system for which +0.2) (-0.8) (a) (10 points) Find the zero-state response of an LTID system with transfer function (b) (10 points) Write the difference equation relating the output y[n) to input[n]. Question 5..... 35 points Consider an LII system for which [] +0.25yin-1 -0.125yn - 2 = x (a) (15 points) Find the system function () = Ye for this system, (b) (10 points) Plot the poles and zeros of H (3) and indicate the region of convergence, assuming that the system is causal (e) (10 points) Determine yn) if [n] = "u[m]

Answer 1

ا ا = (2)۷ ا 0 . 57 ها 1+ 0 .5 0 . 257 - Inverse zT m bls د ا فر ا ن ن ن (5) = (5) ۷ u ( 8 ) + ( - os و ((ره) + (5) ( D. Given H(z)= z. (2+02) (z-o-8) (+52) (7-5.8) و R A + کا 7+ 2 0 .2 مرہ - 2 +0.2 (3) 0.8 + 0.2 - 2+ 0.2 + 7.0, 72) 0-2 + 008 X(7) 2+ 0.2 7 -0.8 Given 8(x) = etum) ح . ع , د ( e 1 ج - }

ا ا = (2)۷ ا 0 . 57 ها 1+ 0 .5 0 . 257 - Inverse zT m bls د ا فر ا ن ن ن (5) = (5) ۷ u ( 8 ) + ( - os و ((ره) + (5) ( D. Given H(z)= z. (2+02) (z-o-8) (+52) (7-5.8) و R A + کا 7+ 2 0 .2 مرہ - 2 +0.2 (3) 0.8 + 0.2 - 2+ 0.2 + 7.0, 72) 0-2 + 008 X(7) 2+ 0.2 7 -0.8 Given 8(x) = etum) ح . ع , د ( e 1 ج - }

- b) Y(7)= 0.2 *(7) + 08 *(7) Z+0.2 Z-0.8. Y(z) = 0.2.7. Z×(z) 10.02ZX(7) Z+0.2 Z-0.8 y (2) Inverse Z-T on bls we get Y(n) = 0.2 60.2)x (-1) +0.8 (0.8)’z(n-1) => Y() = x(n-1){0-2(-0,2) 70.8 (0.8)"}| p)"

3) a H(2) 225 +6 H(②) _ 74 (2-3)(7-2) ) = _ H(2) Inverse Z.T on bls | Ho) = (dt) - yuko) | (7) 3 73 | H(2) 22+, 글 글 + J - Y(2) = H(2) - 1 22 + 23 Inverse Z-T on bis | yo) = (n-1) - 2 x (n -) + 3 x() -3 )

2) a) un) - u (n-2) Tel. let Hycom)= u(n) – u (n-2) Z.T on bls |4(2): 1-3 b) HEAD= y1=34(1-2) 2т от Бs H(Z) = Z ' Z-V 1-8Z s) HOO) = nyºu(n-1] = non-itu (n-1) HN) = n.. Onlu(n-1) z transform in bls 46=** ] firmitas a can be done in this was

=> 5) Given y[n] +0.254(n-1) - 0.125 y (n-2] – x[n] both sides By Z-transform on y (7) +0.25 7'y(7) - 0.1257%() = x(z) Y(z) (1+0.257'-0.12522] = x(z) Y(7) *(z) = (+(z)= 170-252-0.1252? 2 z?+0.257 -0.125 H (Z) = b) Y(Z) 10.257'-0.125 x(2) 72 (Z+0.5) (7-0.25) -1 -05 025/ Iz>0.25 der zeregin of a convergene z>10.5)

Given x(n) = (sun) | x(7)= _7 . 1 -0,57 - Y(7) _ I +0.257'-0,12572 X (Z) Y(z) = (tosa)(1-0257') <1-osa) | y(x)= (T-0.2523) (1-0253)| Y(Z) = (+057)(-osa) (10,257') + + LA 170.57 B 1-0.57' - 1-0.257