EECE 5610: Homework #3

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Problem 1
(a) Find e(0), e(1), and e(10) for
E(z) = 0.1
z(z − 0.9)
using the inversion formula.
(b) Check the value of e(0) using the initial-value property.
(c) Check the values calculated in part (a) using partial fractions.
(d) Find e(k) for k = 0, 1, 2, 3, 4 if Z[e(k)] is given by
E(z) = 1.98z
(z
2 − 0.9z + 0.9)(z − 0.8)(z
2 − 1.2z + 0.27)
(e) A continuous time function e(t), when sampled at a rate of 10 Hz (T = 0.1s), has the following ztransform E(z) = 2z
z−0.8
. Find function e(t).
(f) Repeat part (e) for E(z) = 2z
z+0.8
.
(g) From parts (e) and (f), what is the effect on the inverse z-transform of changing the sign on a real pole?
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EECE 5610 (Professor Milad Siami ): Homework #3 Problem 2
Problem 2
Consider the system described by
x(k + 1) = 
0 1
0 3 
x(k) + 
1
1

u(k),
y(k) =
−2 1
x(k).
(a) Find the transfer function Y (z)/U(z).
(b) Using any similarity transformation, find a different state model for this system.
(c) Find the transfer function of the system from the transformed state equations.
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EECE 5610 (Professor Milad Siami ): Homework #3 Problem 3
Problem 3
Given the MATLAB program
that solves the difference equation of a digital controller.
(a) Find the transfer function of the controller from input e(.) to output m(.).
(b) Find the z-transform of the controller input {e(k)}∞
k=0.
(c) Use the results of parts (a) and (b) to find the inverse z-transform of the controller output.
(d) Run the program to check the results of part (c). Please attach your MATLAB code/result (from the
command window) to your report.
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EECE 5610 (Professor Milad Siami ): Homework #3 Problem 4
Problem 4
Problem 3.7-7 of the textbook (page 113).
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