Homework 8 EE235, Spring 2012 Solutions 1. Find the (unilateral ...

(a) Use (unilateral) Laplace transforms to find the output y(t).
d2y(t) − y(t)
dt2 =
↓
4x(t)
s 2 Y (s) − d
dt x(t)|t=0 − sx(t)|t=0 − Y (s) = 4X(s)
Y (s)(s 2 − 1) = 4X(s)
Y (s) =
4
s2 s
− 1 s2 + 25
Y (s) = 2 s
13 s2 2 s
−
− 1 13 s2 + 25
Y (s) = 2 1 1 2 s
( + ) −
26 s − 1 s + 1 13 s2 ↓
+ 25
y(t) = 2
26 (et + e −t )u(t) − 2
13 cos(5t)u(t)
(b) State if the system is stable/unstable/marginally stable. Justify your
answer.
The system is unstable because there is a pole at s = 1.
5. Consider the system with the impulse response
h(t) = δ(t) + e −3t u(t) + 2e −t u(t)
(a) Find the transfer function of the inverse system.
Want to find G(s) such that G(s) = 1
H(s) .
H(s) = 1 + 1 2
+
s + 3 s + 1
H(s) =
(s + 3)(s + 1)
(s + 3)(s + 1) +
(s + 1)
(s + 3)(s + 1) +
2(s + 3)
(s + 3)(s + 1)
H(s) = s2 + 4s + 3 + 2s + 6 + s + 1
(s + 3)(s + 1)
H(s) = s2 + 7s + 10
(s + 3)(s + 1)
G(s) =
(s + 3)(s + 1)
s2 + 7s + 10
G(s) =
(s + 3)(s + 1)
(s + 2)(s + 5)
4