EE 350
PROBLEM SET 5
DUE: 13 October 2008
Reading assignment: Lathi Sections 3.1 through 3.3
Exam II
is scheduled for Thursday, October 16 from 8:15 pm to 10:15 pm in 102 Forum (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.
Recitations sections will meet during the week of September 29.
Problem 24:
(12 points)
Using the relationship
f
(
t
)
*
δ
(
t

T
)=
f
(
t

T
)
(1)
derived in lecture and in Problem 21, part 1, evaluation of the convolution integral
y
(
t
f
(
t
)
*
h
(
t
±
∞
∞
f
(
τ
)
h
(
t

τ
)
dτ
is simple when either
f
(
t
)or
h
(
t
) is a sum of weighted impulses. This problem extends this result to the case where
either the derivative of either
f
(
t
h
(
t
) yields a sum of weighted impulses.
1. (3 points) As as an example of the utility of equation (1), suppose that
y
(
t
f
(
t
)
*
h
(
t
) where
f
(
t

2
δ
(
t

4)
h
(
t
u
(
t
+1)

u
(
t

1)
.
Determine
y
(
t
), and sketch
f
(
t
),
h
(
t
), and
y
(
t
) on a single plot.
2. (9 points) Now suppose we apply the input
f
(
t
u
(
t
)

u
(
t

1)
to a LTI system that has the impulse response function
h
(
t
)=2
u
(
t

2)

2
u
(
t

3)
.
Neither
f
(
t
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/dt
in terms of impulse functionals and sketch
df/dt
.
•
(3 points) Let
g
(
t
) denote the response of the system to the input
df/dt
, that is
g
(
t
df
dt
*
h
(
t
)
.
Calculate and sketch
g
(
t
).
•
(1 point) Using the result from Problem 21 part 4, show the zerostate response of the system to
f
(
t
) can
be expressed as
y
(
t
±
t
∞
g
(
τ
)
dτ.
•
(3 points) Using the last two results, calculate and sketch
y
(
t
).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentProblem 25:
(12 points)
In lecture it was shown that when the impulse response function
h
(
t
) is a causal signal, then the system is causal.
Conversely, if the system impulse response is noncausal then the system is noncausal. To illustrate this important
concept, consider two LTI systems that are represented by the impulse response functions
System 1:
h
1
(
t
)=
u
(
t
+1)

u
(
t

1)
System 2:
h
2
(
t
u
(
t
)

u
(
t

2)
.
1. (2 points) Sketch
h
1
(
t
) and
h
2
(
t
), and specify whether or not each impulse response function is a causal or
noncausal signal.
2. (2 points) Specify whether or not each system is casual.
3. (8 points) Using convolution, Fnd and sketch the zerostate unitstep response of each system. By comparing
a sketch of the zerostate unitstep response to a sketch of the input
f
(
t
u
(
t
), explain why which system, if
any, is noncausal.
Problem 26:
(12 points)
1. (8 points) Consider three linear timeinvariant systems whose impulse responses
h
(
t
) are speciFed as
•
h
1
(
t
e

t
u
(
t
)
•
h
2
(
t
tu
(
t
)
Classify each system, corresponding to the impulse functions considered above, as either BIBO stable or not
BIBO stable. In order to receive credit, justify your answer.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB
 RLC, LTI system theory, Impulse response, frequency response function

Click to edit the document details