Colorado State University, Ft. Collins
Spring 2016
ECE 312: Linear Systems Analysis II (Signal and Systems)
Homework 5
Assigned on: 04/05/2016, Due by: 04/21/2016
5.1
(a) Given the discrete-time function
 1  n
  ,
f n   2 
0,

 10  n  20
otherwise
express the bilateral z-transform of this function in closed form (not as a power
series).
(b) Find the region of convergence of the transform of part (a)
(c) Repeat parts (a) and (b) for the discrete-time function
 1  n
  ,
 2 
 1 n
 
f n    ,
 4 
0,


 10  n  10
n  21
otherwise
(d) Repeat parts (a) and (b) for the discrete-time function
 1  n
  ,
 2 
 1 n
 
f n    ,
 4 
0,


 10  n  0
1  n  10
otherwise
1
5.2
Consider and LTI system with input xn and output yn for which
yn  1 
5
yn  yn  1  xn
2
The system may or may not be stable or causal.
By considering the pole-zero pattern associated with the preceding difference
equation, determine three possible choices for the impulse response of the system.
Show that each choice satisfies the difference equation.
5.3
We are given the following five facts about a discrete-time signal xn with Ztransform X z  :
1. xn is real and right-sided.
2. X z  has exactly two poles.
3. X z  has two zeros at the origin.
4. X z  has a pole at z 
5. X z  1 

1 j3
e
2
8
.
3
Determine X z  and specify its region of convergence.
5.4
Consider the discrete-time LTI system with input xn and output yn for which
yn  1 
10
yn  yn  1  xn
3
The system is stable. Determine the impulse response of the system.
2
5.5
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.
3