350 ECE Signals and Systems
Spring 2011
Sample 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. Tables 3.1, 4.1, 4.2, 9.1 and 9.2 and the table of common
Fourier series will be provided. This test is cumulative, so also review problems from sample exam #1.
1.
Determine the (exponential) Fourier series for
x(t) = cos(2t) + 2sin(3t) - cos(5t) - 1
What is T0? What is 0?
2.
For a periodic function with period T=4, the non-zero Fourier series coefficients are
a0 = 1, a1 = a-1 = 2, a3 = a*-3 = j
a. Express the function x(t) in the form
i.e., a sum of cosines with varying magnitudes and phases
x(t) = Akcos(kt+k)
b. Graph the magnitude of ak versus .
c. Graph the phase (angle) of ak versus .
d. If x(t) is passed through a real filter h(t) with the positive portion of the magnitude and phase shown below
|H(jw)|
H(jw)
2
3
0
3
0
what are the Fourier series coefficients of y(t) (call them bk)?
e. What is y(t) for these Fourier series coefficients?
3.
Determine the (exponential) Fourier series of x(t) as shown below using the Fourier integral.
x(t)
2
0
4.
t
6
Given the signal
x(t)
1 1 < t < 1
x(t ) =
else
0
1
t
1
-1
a.
Determine the Fourier transform of x(t) both by equation 4.9 and by using the tables.
Use the results of (a) to determine the Fourier Transform of the following signals.
1
1
t
t
b.
-1
-1
c.
2
1
1/2
1
3/2
t
-1/2
d.
3
1
-1
3
-1
e.
-1
2
t
2
1
t
-1
1
1
-1
f.
5.
6.
7.
1
-1
g.
t
Use the property of Duality and your results from problem #4 to find the inverse transform of y(t) given the
transform
1 1 < < 1 .
Y ( j ) =
else
0
Determine value the of y(t) when
x(t)=te-2tu(t)
h(t)=e-4tu(t)
a. By convolving x(t) and h(t)
b. By taking the Fourier Transform of x(t) and h(t).
A system has the impulse response
H ( j ) =
3(3 + jw)
(4 + jw)(2 + jw)
a.
b.
c.
d.
8.
Determine h(t).
Is this system causal? Is this system stable? Why or why not?
Draw the straight line Bode plot for this system.
What type of filter is this?
For the following system
y" + 3y' + 2y = x(t)
a.
b.
c.
d.
e.
9.
Determine the impulse response by first finding the unit step response, ys(t).
Find the frequency response H(j) using eigen function methods.
Find H(j) using Fourier transforms, then invert the transform to find the impulse response.
Draw the Bode plot describing this system.
What type of filter is this?
The relationship between the input and output voltages of following circuit
R
L
C
R
x (t)=Vin(t)
y(t)=Vout(t)
is given by the following differential equation
dx(t )
dy (t ) L d 2 y (t ) y (t )
=2
+
+
.
dt
dt
R dt 2
RC
As derived from Kirchoffs voltage and current laws as well as the rules for voltage and current around
resistors, capacitors and inductors:
ic = C
a.
b.
c.
d.
dVc
,
dt
Vl = L
dil
,
dt
and
Vr = i r R .
Verify that you get this differential equation relating the input and output voltages
What is the frequency response of this system?
If R=50.005 C=0.1 L=0.1, what is the impulse response of the system?
Draw the straight line Bode plot approximation for the magnitude describing this system. What type of
filter is this?
e. If x(t) = 3cos(10t + 15), what is y(t)?
10. Explain in a short paragraph (with proper grammar and sentence structure) what are the three main points you
should know from Chapter 5, and what is significant about them.