GEORGIA INSTITUTE OF TECHNOLOGY
SCHOOL of ELECTRICAL and COMPUTER ENGINEERING
ECE 2025
Summer 2008
Problem Set #9
Assigned:
Week of 14Jul08
Due Date:
22/23Jul08
Please check the Tsquare “Announcements” daily. All official course announcements will be
posted there.
ALL
of the
STARRED
problems should be turned in for grading.
To give students in the Monday recitations some breathing room after Quiz 3,
students with
recitations on Monday may turn in this homework at the start of lab on Wednesday.
Students in Tuesday recitations should turn it in at the start of their recitation as usual.
PROBLEM 9.1
:
Try parts (b) and (c) of Problem 11.6 on p. 342 of
Signal Processing First
.
PROBLEM 9.2
:
Try Problem 118 on p. 343 of
Signal Processing First
.
PROBLEM 9.3
:
Suppose a mysterious function
f
(
a
) is defined by
f
(
a
) =
integraldisplay
10
0
t
exp (
−
jat
)
dt
+
integraldisplay
20
10
(10
−
t
) exp (
−
jat
)
dt
Sketch the function
x
(
t
) =
1
2
π
integraldisplay
∞
∞
f
(
a
) exp (
jat
)
da
(I often put questions like this on test to make sure you recognize the form of the Fourier
transform and the inverse Fourier transform.)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
PROBLEM 9.4
*:
The homework problems that one gets in engineering and math classes often feel unrelated to
anything in the real world. To try to make this material more “real,” the exposition in this problem
and the next problem “dresses up” some standard ECE2025 questions with some background
information about where these things actually show up in real life! (If you don’t care about the
background, you need not pay close attention to it. You can just work the problems, but hopefully
you’ll find the background interesting.)
(a) The “ramp filter” is often used in medical imaging applications such as Xray computeraided
tomography. It has a frequency response given by
H
(
jω
) =
braceleftBigg
jω
for

ω

< ω
0
0
otherwise
Find the impulse response,
h
(
t
), of this filter. (You may need to go back to your old calculus
text if you’ve forgotten how to take derivatives of quotients.) Also make a plot of the impulse
response (this is pretty complicated, so you should feel free to use MATLAB or one of those
fancy graphing calculators to help you make the plot.) If you’ve ever had a CAT scan done,
you’ve run across this filter in practice!
A hint on making the plot:
We’re interested in the overall shape of the curve; you can
pick whatever
ω
0
you find the most interesting. I found that the following bit of MATLAB
code made a nice plot. You should feel free to use it:
omega_0 = 2*pi;
period = 2*pi/omega;
t = 5*period:period/100:5*period;
h = [fill in code that goes here]
h([t == 0]) = [fill in value that goes here]
plot(t,h);
You may ask yourself what the
h([t == 0]) =
business is all about. Well, it turns out that
h
(
t
) is indeterminite at
t
= 0, i.e. we wind up dividing zero by zero. To find a meaningful
value for
h
(0), use L’Hopital’s rule. Alternatively, you can probably guess what
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 JUANG
 Signal Processing, LTI System

Click to edit the document details