EE 350 EXAM III 10 November 2011 Last Name (Print): .sold::.; onS First Name (Print): ID number (Last 4 digits): _ Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO -- Problem Weight Score -- 1 I------ 25 2 25 3 25 4 25 Total 100 -- Test Form A INSTRUCTIONS 1. You have 2 hours to complete this exam. 2. This is a closed hook exam. You may use one 8.5" x 11" note sheet. 3. Calculators are not allowed. 4. Solve each part of the problem in the space following the question. If you need more space, continue your solution on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO credit will be given to solutions that do not meet this requirement. 5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. 6. The quality of your analysis and evaluation is as important as your answers. Your rea.<;oning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your
Transcript
EE 350 EXAM III 10 November 2011
Last Name (Print): .sold::.; onS
First Name (Print):
ID number (Last 4 digits): _
Section:
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
- Problem Weight Score
-
1 I-----
25 2 25 3 25 4 25
Total 100 -
Test Form A
INSTRUCTIONS
1. You have 2 hours to complete this exam.
2. This is a closed hook exam. You may use one 8.5" x 11" note sheet.
3. Calculators are not allowed.
4. Solve each part of the problem in the space following the question. Ifyou need more space, continue your solution on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO credit will be given to solutions that do not meet this requirement.
5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned.
6. The quality of your analysis and evaluation is as important as your answers. Your rea.<;oning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your wo~
Problem 1: (25 points)
1. J(1(2)points) ~ig.ure 1 shows a periodic signal J(t). Determine the trigonometric Fourier series representation of t by speclfymg the parameters (ao ' an, bn).
f(t} To.::: 2:tT ~ WIQ::: ~ ::=. J
rD
Figure 1: Periodic signal J(t).
Bect<l..v~e. .f(-t:)::. - f(--t:') IS afl oJl.J- -{")L~On I .f:,meJ
(b) (5 points) Compose a MATLAB m-file that calculates and plots y(t) using the first 500 harmonics over the time interval -2 S t [sec] S 2 with a time vector that contains 500 uniformly spaced points.
Consider the communications system in Figure 3 with input f(t) and output y(t).
a(t) b(t) y(t)f(t) )-----+-1 HI (0)) )-----.1
cos(SOt) cos (lOOt)
F(O))
-10 0 10
-1 -
-L_....L.......L.....J..._....L...--I~ OJ
-50 -40 40 50 -100 100
Figure 3: Communication system with input f(t) and output y(t).
1. (5 points) Sketch the Fourier transform A(Jw) of the signal a(t). To receive credit, carefully label important features of the graph.
A(m)
---....-+-.....J'-----....J..----J---ir---\......----i~m -60 -So .. '{O o 'fO So 60
Figure 4: Fourier spectra of a(t) .
f}-CW) .J.- Fe lJJ - seJ) + 2.
13
2. (5 points) Sketch the Fourier transform B(Jw) of the signal b(t). To receive credit, carefully label important features of the graph.
B(@)
.!Z.
----__.........-=t-----~---~I____...----- .... 0)o ~o So
Figure 5: Fourier spectra of b(t) .
.'3. (5 points) Sketch the Fourier transform C(Jw) of the signal c(t). To receive credit, carefully label important features of the graph.
C(@)
1.. '1
o-/50 -(L{O .so 60
Figure 6: Fourier spectra of c(t).
CCt.) ::: b( -t,) COS C{a 01:.)
((w) ::. .L C> (e.v ... Lo~ + 2
4. (5 points) Sketch the Fourier transform Y(Jw) of the signal y(t). To receive credit, carefully label important features of the graph.
Y(m)
o -----....L..---+---.....L---.:....-~-----___t~ OJ
Figure 7: Fourier spectra of y(t).
5. (5 points) Determine the impulse response h2 (t) of the filter with frequency response function H 2 (Jw). Express your answer in terms of real-valued sinusoidal and sinc functions.
W :; I O~~l-th [00 tQ~-t C~ ')J:::-? 200t s;;' c. (leo *")
-- --
15
EE 350 EXAM III 10 November 2011
Last Name (Print):
First Name (Print):
ID number (Last 4 digits):
Section:
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
Problem Weight Score
1 25
2 25
3 25
4 25
Total 100
Test Form A
INSTRUCTIONS
1. You have 2 hours to complete this exam.
2. This is a closed book exam. You may use one 8.5” × 11” note sheet.
3. Calculators are not allowed.
4. Solve each part of the problem in the space following the question. If you need more space, continue your solutionon the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO
credit will be given to solutions that do not meet this requirement.
5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and agrade of ZERO will be assigned.
6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be preciseand clear; your complete English sentences should convey what you are doing. To receive credit, you must
show your work.
1
Problem 1: (25 points)
1. (12 points) Figure 1 shows a periodic signal f(t). Determine the trigonometric Fourier series representation off(t) by specifying the parameters (ao, an, bn).
Figure 1: Periodic signal f(t).
2
3
2. (13 points) Consider the periodic signal f(t) in Figure 2. As indicated in the figure, the shape of f(t) in theinterval −π/4 < t < π/4 is cos(at).
Figure 2: Periodic signal f(t).
(a) (2 points) Determine the fundamental frequency of the periodic signal f(t).
(b) (3 points) What is the average value of f(t)?
(c) (8 points) Determine the complex exponential Fourier series coefficients Dn for f(t).
4
5
Problem 2: (25 Points)
1. (6 points) Consider the following three periodic signals represented using an exponential Fourier series
x1(t) =
100∑
k=0
(
1
2
)k
ek10t
x2(t) =
100∑
k=−100
cos(kπ) ek20t
x3(t) =
100∑
k=−100
sin
(
kπ
2
)
ek30t.
(a) (3 points) Which signals, if any, are real-valued? Justify your answer in one or two short sentences.
(b) (3 points) Which signals, if any, are an even function of time? Justify your answer in one or two shortsentences.
6
2. (10 points) A periodic signal f(t) with fundamental frequency ωo and the exponential Fourier series represen-tation
Dfn =
A
nπ[1 − cos(nπ)]
is passed through a linear time-invariant system with frequency response representation
|H(ω)| =
{
2π ω0 < |ω| < 4ωo
0 otherwise
6 H(ω) =
{
ωωo
π2
−4ω0 < ω < 4ωo
0 otherwise
to produce a periodic response y(t).
(a) (2 points) What type of filter is the linear time-invariant system? Justify your answer in one or two shortsentences.
(b) (4 points) Specify all nonzero exponential Fourier series coefficients Dyn of the periodic signal y(t).
(c) (4 points) Determine an expression for the response y(t) using real-valued sinusoids.
7
8
3. (9 points) A periodic signal f(t) with fundamental period ωo = 10 rad/sec and trigonometric Fourier seriesrepresentation
f(t) =
∞∑
n=1
2
ncos(nπ) sin(nωot)
is passed through a linear-time invariant system represented by the frequency response function
H(ω) =5
ω
to produce a periodic response y(t).
(a) (4 points) Determine the compact trigonometric Fourier series representation of y(t).
(b) (5 points) Compose a MATLAB m-file that calculates and plots y(t) using the first 500 harmonics overthe time interval −2 ≤ t [sec] ≤ 2 with a time vector that contains 500 uniformly spaced points.
9
Problem 3: (25 points)
1. (8 points) Determine, by direct integration, the Fourier transform of the signal
f(t) = 2e−|t| [u(t + 1) − u(t − 1)] .
10
2. (9 points) For second-order linear time-invariant system, the input
f(t) = e−2tu(t)
results in the zero-state responsey(t) =
(
e−3t − e−4t)
u(t).
(a) (5 points) Determine the frequency response function representation, H(ω), of the system and expressyour answer in the standard form
H(ω) =bm(ω)m + bm−1(ω)m−1 + · · ·+ b1(ω) + b0
(ω)n + an−1(ω)n−1 + · · ·+ a1(ω) + a0
.
(b) (2 points) Determine the ODE representation of the system and express your answer in the standard form
dny
dtn+ an−1
dn−1y
dtn−1+ · · ·+ aoy = bm
dmf
dtm+ bm−1
dm−1f
dtm−1+ · · ·+ bof.
(c) (2 points) Is the system asymptotically stable, marginally stable, or unstable?
11
3. (8 points) Suppose that f(t) is a real-valued function whose Fourier transform is F (ω). Show that
Ef =
∫ ∞
−∞
f2(t)dt =1
2π
∫ ∞
−∞
|F (ω)|2dω,
which is Parseval’s theorem for Fourier transforms.
12
Problem 4: (25 points)
Consider the communications system in Figure 3 with input f(t) and output y(t).
Figure 3: Communication system with input f(t) and output y(t).
1. (5 points) Sketch the Fourier transform A(ω) of the signal a(t). To receive credit, carefully label importantfeatures of the graph.
Figure 4: Fourier spectra of a(t).
13
2. (5 points) Sketch the Fourier transform B(ω) of the signal b(t). To receive credit, carefully label importantfeatures of the graph.
Figure 5: Fourier spectra of b(t).
3. (5 points) Sketch the Fourier transform C(ω) of the signal c(t). To receive credit, carefully label importantfeatures of the graph.
Figure 6: Fourier spectra of c(t).
14
4. (5 points) Sketch the Fourier transform Y (ω) of the signal y(t). To receive credit, carefully label importantfeatures of the graph.
Figure 7: Fourier spectra of y(t).
5. (5 points) Determine the impulse response h2(t) of the filter with frequency response function H2(ω). Expressyour answer in terms of real-valued sinusoidal and sinc functions.
