Colorado State University, Ft. Collins
Spring 2016
ECE 312: Linear Systems Analysis II (Signal and Systems)
Homework 4
Assigned on: 03/22/2016, Due by: 04/07/2016
4.1
Determine the z-transform for each of the following sequences. Indicate the region
of convergence. Indicate whether or not the Fourier transform of the sequence exists.
(a)  n  5
(b)  0.5 un 
n
1
(c)  
3
n2
un  2
n
1
(d) 2 un    un  1
4
n
4.2
Given the following unilateral z-transforms,
(i)
(ii)
(iii)
(iv)
0.5z 2
z  1z  0.5
0.5 z
X z  
z  1z  0.5
0.5
X z  
z  1z  0.5
z
X z   2
z  z 1
X z  
(a) Find the inverse unilateral z-transform of each function
(b) Evaluate each xn in part (a) for the first three nonzero values.
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(c) Use the final-value property to evaluate x for each function. Compare the
results with those obtained from xn in part (a), explain why or why not they
agree with each other.
(d) Use the initial-value property to evaluate x0 for each function. Compare the
results with those obtained from xn in part (a), explain why or why not they
agree with each other.
4.3
For each of the following difference equations and associated input and initial
conditions, determine the zero-input and zero-state responses by using the unilateral
z-transform.
(a) yn  3 yn  1  xn
n
1
xn    un
2
y 1  1
(b) yn 
1
1
yn  1  xn  xn  1
2
2
xn  un
y 1  0
(c) yn 
1
1
yn  1  xn  xn  1
2
2
xn  un
y 1  1
4.4
If X z  denotes the unilateral z-tranform of xn , determine, in terms of X z  , the
unilateral z-transform of:
2
(a) xn  3
n
(b) xn  3
(c)
 xk 
k 0
4.5
The following is known about a discrete-time LTI system with input xn and
output yn :
1. If xn   2 for all n , then yn  0 for all n .
n
 2 un for all n , then yn for all n is of the form
2. If xn  1
n
n
1
yn   n  a  un
4
where a is a constant.
(a) Determine the value of the constant a .
(b) Determine the final value of the output if the input is xn  un.
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