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ECE 130A: Signal Analysis and Processing
Prof. Michael Liebling
Fall 2010
TAs: Jingning Han, Hari Sivakumar
Distribution: Monday, 11 October 2010
Homework Due: Tuesday, 19 October 2010, 6:00pm
University of California, Santa Barbara
Homework 3
Problem 1: Matched ﬁlter (20 points)
The matched ﬁlter for a signal
s
(
t
) is deﬁned as an LTI system with
impulse response
h
(
t
)
=
s
(

t
).
(a)
For
s
(
t
)
=
I
[3,6]
(
t
), sketch the matched ﬁlter impulse response.
(b)
Find and sketch the convolution of
s
(
t
) with its matched ﬁlter.
(c)
Repeat (a) and (b) for
s
(
t
)
=
I
[

1,1]
(
t
)

2
I
[1,2]
(
t
).
(d)
Based on your observations in (a)–(c), provide an intuitive justiﬁcation for the following statement:
A signal passed through its matched ﬁlter gives a signal that has its maximum at the origin.
Problem 2: Complex exponential through LTI system (20 points)
(a)
Determine the output
y
(
t
) of a system with impulse response
h
(
t
)
=
I
[

1,2]
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