ECE220
Signals and Information
Spring 2008
Homework 8: Due Monday, April 21st,
at 10:08pm
Drop your homework in the collection box marked
”ECE220 Spring 2008, homework”
, located
on the second floor of Phillips at the south entrance to 219 Phillips.
Print your
name, NetId, and lab section
in the top right corner on all pages.
NOTE: This homework is based on lectures 2066 and Chapter 12 in the textbook.
Problem 1
A periodic signal
x
(
t
) with a period
T
s
= 10

3
seconds is input to a bandpass filter with the
frequency reponse
H
(
jω
) = 1 for 3000
π <

ω

<
5000
π
.
(a)
The Fourier series for
x
(
t
) has coefficients
a
k
=
braceleftbigg
10
k
= 0
2
j
k
k
negationslash
= 0
Find the output
y
(
t
).
(b)
Now assume that
x
(
t
) is bandlimited with
X
(
jω
) = 0 for

ω

>
3000
π
. Suppose that
x
(
t
)
is modulated by cos(4500
πt
) and next bandpass filtered with
H
(
jω
). Make a plot of the
Fourier transform of
Y
(
jω
).
Problem 2
Consider the following sampling and reconstruction system:
x
(
t
)
−→
⊗
↑
p
(
t
)
x
s
(
t
)
−→
LTI system
H
r
(
jω
)
x
r
(
t
)
−→
(a)
Plot the Fourier transform
X
(
jω
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 Spring '05
 JOHNSON
 Digital Signal Processing, Lowpass filter, sampling rate

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