Spring EE350 2005 - Exam 2
18 March 2005
Name: _Solutions _______________
EE 350 Signals and Systems
Spring 2005
Exam #2
One 8 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
Problem 2
Problem 3
/20
/40
/40
Total
/100
Homework average to date:
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Course grade to date:
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EE350 Spring 2005 - Exam 2
18 March 2005
Name: _Solutions _______________
1. (20 points) You are given a signal, x(t), that is a half-rectified sinusoid
x(t)
X0
t
-T0
0
T0
2T0
Confirm the results from the table of Fourier Series for Common Signals that the
Fourier Series coefficients for the sawtooth are as they appear in the table. Use the
Fourier Series integral for this problem. Hint: ej2k=1.
EE350 Spring 2005 - Exam 2
Name: _Solutions _______________
18 March 2005
2. (40 points) For the following system
y" + 9y' + 8y = 2x(t) + 6x(t)
a. (7 points) Find H(j) using Fourier transforms. Show your steps.
b. (10 points) Invert the transform to find the impulse response.
c. (10 Draw points) the magnitude Bode plot for this system.
0
- 2 .5
-2 0
8
0 .1
= 1 *1 0 -1
1
= 1 *1 0 0
3
10
= 1 *1 0 1
d. (5 points) What type of filter is this? Explain why you gave this answer.
This is a Low pass filter. The gain at low frequencies is near 0dB and is much higher than
the gain at higher frequencies where the slope drops at 20 dB/decade.
e. (8 points) If x(t) = e-2tu(t), what is y(t).
EE350 Spring 2005 - Exam 2
18 March 2005
Name: _Solutions_______________
3. (40 points) A signal is
sin(3t )
(1 + cos(4t ))
t
(a) (10 points) What is the equation describing X(j)? Show your work and explain
your steps clearly!
x(t ) =
(b) (10 points) Draw & fully label X(j).
+
+
=
1.5
1
0.5
-7
-3
-1 0
1
3
7
EE350 Spring 2005 - Exam 2
18 March 2005
Name: _Solutions _______________
(c) (10 points) According to Nyquist, what is the smallest sampling frequency, s, that
may be used to sample x(t) so that full reconstruction is possible? Explain clearly
how you got your answer. What is the Nyquist rate? What is the Nyquist
frequency?
max = 7 rad/sec
therefore, s > 2max = 14 rad/sec
(d) (10 points) If this signal is sampled at twice the Nyquist rate you derived above,
sketch and label the spectrum of the sampled signal, x(nT).
1.5
1
0.5
-56
-28
-7 0 7
28
56