EE 350 Signals and Systems Spring 2007 Final Exam

Question # 00613737 Posted By: Prof.Longines Updated on: 11/06/2017 10:30 AM Due on: 11/06/2017
Subject Engineering Topic Cellular Engineering Tutorials:
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Final Exam
One 8 ½ x 11” sheet of notes, and a calculator are allowed during the exam.
Write all answers neatly and show your work to get full credit.
Rationalize all complex fractions.
A calculator is allowed during the exam, but should not be needed.
Write neatly and WATCH YOUR ALGEBRA!
Problem 1 /15
Problem 2 /30
Problem 3 /35
Problem 4 /20
Total /100
Course grade
EE350 Spring 2007 - Final Exam Name: ___________________________
9 May 2007
EE350 – Final Exam, Spring 2007 page 2/8
1) (15 points) For the questions in this problem, no explanation is necessary
Consider the following three systems
a) y(t) = x(t+2)sin(wt+2), where w ? 0
b) y(t) = e-3t (x(t) + 1)
c) h(t) = ( 2) 3
u t ? t
x(t) is the system input, y(t) is the system output, h(t) is the system impulse response and u(t)
is the unit step function.
Circle YES or NO for each of the following questions for each of these three systems.
System (a) System (b) System (c)
Is the system linear ?
YES NO YES NO YES NO
Is the system time invariant ?
YES NO YES NO YES NO
Is the system memoryless?
YES NO YES NO YES NO
Is the system causal ?
YES NO YES NO YES NO
Is the system stable ?
YES NO YES NO YES NO
EE350 Spring 2007 - Final Exam Name: ___________________________
9 May 2007
EE350 – Final Exam, Spring 2007 page 3/8
2) (30 points total) You want to expand table 4.2 to include the transform pair for a triangular
pulse, x(t):
x(t)
t
F
X(j?)=?
-A A
1
and
Y(j?)
?
F
y(t)?
-A A
1
a) (10 points) One approach to solving this problem is by knowing that a square pulse
convolved with itself is a triangle pulse with a base width twice the length of the square
pulse widths. Use this and your knowledge of the transform relations to determine X(j?)
from x(t). Note: you need to know the amplitude of the triangular pulse that results when
you convolve two square pulses.
b) (10 points) Use duality to determine y(t) from the transform pair x(t) and X(j?).
c) (10 points) Using the tables and an approach similar to that used in (a), calculate y(t)
from Y(j?).
EE350 Spring 2007 - Final Exam Name: ___________________________
9 May 2007
EE350 – Final Exam, Spring 2007 page 4/8
3) (35 points) Consider the following system:
sin(?bt)
x(t)
X X
sin(?ct)
y(t)
H(j?) X X
cos(?bt) cos(?ct)
ys(t)
yc(t)
as(t)
ac(t)
xs(t)
xc(t)
H(j?)
+
1
-?b ?b
H(j?)
?
The Fourier Transform of x(t), X(j?) has the real and imaginary parts given below:
1
-?b ?b ?
ex {X(j?)} \Å {X(j?)}
? -?b ?b
1
-1
Note: These plots are Real vs. ? and Im vs. ?, not Im vs. Real.
(a) (20 points) Draw fully labeled sketches of the real and imaginary parts of Xc(j?), Xs(j?),
Ac(j?), As(j?) and Ys(j?). Axis for your answers are on a future page. Show your work
on this and the next page for full or partial credit.
EE350 Spring 2007 - Final Exam Name: ___________________________
9 May 2007
EE350 – Final Exam, Spring 2007 page 5/8
Workspace for Problem 3
EE350 Spring 2007 - Final Exam Name: ___________________________
9 May 2007
EE350 – Final Exam, Spring 2007 page 6/8
Problem 3a (cont.)
?
e x {Xc(j?)} \Å {Xc(j?)}
?
?
e x {Xs(j?)} \Å {Xs(j?)}
?
?
e x {Ac(j?)} \Å {Ac(j?)}
?
?
e x {As(j?)} \Å {As(j?)}
?
?
e x {Ys(j?)} \Å {Ys(j?)}
?
EE350 Spring 2007 - Final Exam Name: ___________________________
9 May 2007
EE350 – Final Exam, Spring 2007 page 7/8
Problem 3 (cont.)
b) (5 points) Yc(j?) has real and imaginary parts as shown below
?
ex {Yc(j?)} \Å {Yc(j?)}
-?c ?c
?
1/4
-1/4
??c
1/4
?c
?c-?b ?c+?b ?c-?b ?c+?b
Draw the labeled sketches of the real and imaginary parts of Y(j?). This is equivalent to a
Single Side Band/Upper Side Band modulation scheme. At what frequency is the signal
modulated?
?
e x {Y(j?)} \Å {Y(j?)}
?
c) (10 points) What small changes do you need to make to the original block diagram to
create a lower side band modulation? What would the output of the system be under the
LSB configuration? At what frequency is the signal modulated?
?
e x {Y(j?)} \Å {Y(j?)}
?
EE350 Spring 2007 - Final Exam Name: ___________________________
9 May 2007
EE350 – Final Exam, Spring 2007 page 8/8
4) (20 points) Consider the causal discrete-time system characterized by the following inputoutput
relationship:
y[n ? 1] + 5y[n] + 6y[n + 1] = 2x[n]
(a) (5 points) Using iterative methods (Chapter 2), determine and plot the impulse response of
this system for three non-zero terms.
(b) (5 points) What is the transfer function for this system?
(c) (5 points) Using the Power Function (long division) method, determine the first 3 non-zero
terms of h[n].
(d) (5 points) Invert the transform found in (b) using tables to determine the impulse response for
this system. What ROC must it have to be causal and stable?
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