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EE 350
PROBLEM SET 5
DUE: 16 October 2007
Laboratory sections will meet during the week of October 8.
Reading assignment: Lathi Sections 3.1 and 3.2
Exam II
is scheduled for Thursday, October 18 from 8:15 pm to 10:15 pm in 121 Sparks (all sections). The second exam
covers material from Problem Sets 4 and 5, and Laboratory #2. The date and location of a review session will be announced
on the EE 350 web page.
Problem 22:
(16 points)
In Problem Set 4, Problem 19, you showed that
f
(
t
)
∗
δ
(
t
−
T
)=
f
(
t
−
T
)
.
Using this relationship you will now
•
investigate the relationship between causal signals and causal systems, and
•
simplify the task of calculating the zerostate response in situations where either the derivative of either
f
(
t
)o
r
h
(
t
)
yields a sum of weighted impulses.
1. (6 points) Consider a LTI system with input
f
(
t
), impulse response
h
(
t
), and zerostate response
y
(
t
).
(a) (1 point) Using the convolution integral, show that the input
f
(
t
δ
(
t
) results in the zerostate response
y
(
t
h
(
t
).
(b) (3 points) Suppose that
h
(
t
u
(
t
+1)
−
u
(
t
−
1) and
f
(
t
δ
(
t
).
•
Is
h
(
t
) a causal or noncausal signal?
•
Find the zerostate response
y
(
t
).
•
Based on the relationship between the given input
f
(
t
) and the resulting zerostate response
y
(
t
), is the system
causal or noncausal?
(c) (2 points) Consider an arbitrary LTI system with input
f
(
t
), impulse response function
h
(
t
), and the zerostate
response
y
(
t
Z
∞
−∞
f
(
τ
)
h
(
t
−
τ
)
dτ.
What restriction must be placed on
h
(
t
) so that the system is causal? Justify your answer.
2. (10 points) Suppose we apply the input
f
(
t
)=2
u
(
t
−
2
u
(
t
)
to a LTI system that has the impulse response function
h
(
t
u
(
t
−
1)
−
u
(
t
−
3)
.
Neither
f
(
t
)or
h
(
t
) is expressed directly as a sum of weighted impulse functionals. However, careful inspection of
f
(
t
)
reveals that its derivative with respect to time can be expressed as the sum of two impulse functionals.
•
(2 points) Find an expression for
df / d t
in terms of impulse functionals and sketch
.
•
(2 points) Let
g
(
t
) denote the response of the system to the input
,thatis
g
(
t
df
dt
∗
h
(
t
)
.
Calculate and sketch
g
(
t
).
•
(3 points) Using the result from Problem 19 part 4, show the zerostate response of the system to
f
(
t
)canbe
expressed as
y
(
t
Z
t
−∞
g
(
τ
)
dτ.
•
(3 points) Using the last two results, calculate and sketch
y
(
t
).
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View Full DocumentProblem 23:
(14 points)
1. (9 points) Consider three linear timeinvariant systems whose impulse responses
h
(
t
) are specifed as
•
h
1
(
t
)=
e
t
u
(
−
t
)
•
h
2
(
t
u
(
t
+1)
•
h
3
(
t
)=[
e
−
2
t
−
5
e
−
10
t
]
u
(
t
)
(a) (3 points) ClassiFy each impulse response as either a causal or noncausal
signal
. In order to receive credit, justiFy
your answer.
(b) (3 points) ClassiFy each
system
, corresponding to the impulse Functions considered above, as either causal or
noncausal. In order to receive credit, justiFy your answer.
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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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