Colorado State University, Ft. Collins
Spring 2016
ECE 312: Linear Systems Analysis II (Signal and Systems)
Homework 3
Assigned on: 03/01/ 2016, Due by: 03/24/2016
3.1
Determine the causality and the stability for the systems with the following impulse
responses:
(a)
(b)
(c)
(d)
(e)
(f)
hn  sin  nun
hn  e  n u n
hn  e n un
hn  sin nun
hn  ne  n un
hn  e  n sin n un
3.2
Find the responses for systems described by the following difference equation with
the initial conditions given:
(i)
(ii)
(iii)
(iv)
5
yn  1  2 n un , y 1  0
6
yn  0.7 yn  1  e  n un , y 1  0
yn  3 yn  1  2 yn  2  3un, y 1  0 , y 2  0
yn  0.7 yn  1  cosnun m y 1  1
yn 
3.3
Consider a causal system with each of the subsequent system characteristic equations.
(i)
z  0.6  0
1
(ii)
(iii)
(iv)
(v)
(vi)
z 2  1.5 z  1  0
z2 1  0
z 3  2 z 2  1.5 z  0.5  0
z  0.63  0
z  0.7z  3z  0.2  0
(a)
(b)
Give the national response for each of the systems.
Determine the stability of the systems.
3.4
You are given a system that is known to be time-invariant
When input to the system, a discrete time signal x1 n produces an output y1 n
where
x1 n   n   n  1   n  2
And
y1 n  2 n  1  2 n  2 n  1
A second discrete time signal x2 n produces an output y2 n where
x2 n   n  1   n  2
And
y2 n  2 n  2 n  1
A third discrete time signal x3 n produces an output y3 n where
x3 n   n   n  1  2 n  2   n  3
And
y3 n  2 n  1  2 n  3 n  1  2 n  2
Determine whether or not the system is linear. Justify your answer.
2
3.5
Consider a causal LTI system whose input xn and output yn are related through
the block diagram representation shown below.
xn
yn
D
2
3
6
D

1
9
8
(a) Determine a difference equation relating yn and xn .
(b) Is this system stable? Explain why or why not.
3