EC312 Twelve Week Exam Spring 2015
March 30, 2015
United States Naval Academy
Electrical and Computer Engineering Department
EC312 - 12 Week Midterm (Version A) – Spring 2015
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Do a page check: you should have 6 pages including this cover sheet.
You have 50 minutes to complete this exam.
An FE-approved calculator may be used for this exam.
This exam is closed book and closed notes. You may use two single-sided hand-written pages of notes.
Turn in your two single-sided hand-written pages of notes with your exam.
This exam may be given as a makeup exam to several midshipmen at a later time. No communication is
permitted concerning this exam with anyone who has not yet taken the exam.
Name:
___SOLUTIONS___________
Instructor:
____________________
Page 1 of 6
EC312 Twelve Week Exam Spring 2015
March 30, 2015
Question 1. Consider the signal below on the left.
Time-Domain
Amplitude
(V) here.
Type
equation
15
Voltage (V)
10
Frequency-Domain
10
5
0
-5
-10
-15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
0.7
0.9
1
2
Time (µs)
Frequency (MHz)
1) (5 pts) Write out the time-domain equation for the signal: 𝑉𝑉(𝑡𝑡) = 10 sin(2𝜋𝜋(2000000)𝑡𝑡) 𝑉𝑉
𝑇𝑇 = .5𝜇𝜇𝜇𝜇, i.e. 𝑓𝑓 =
1
𝑇𝑇
=
1
.5𝜇𝜇𝜇𝜇
= 2 𝑀𝑀𝑀𝑀𝑀𝑀
2) (5 pts) On the axes to the right, plot the frequency domain representation of the signal. (Make sure to
label axes and relevant values.)
Question 2. (5 pts) Circle all of the following which are advantages of using modulation. (Could be more than
one answer.)
(i) Systems can have smaller antennas due to higher carrier frequencies
(ii) Modulated signals have smaller bandwidth than baseband signals
(iii) Signals can be deconflicted by modulating at different carrier frequencies (i.e. Frequency Division
Multiplexing)
(iv) Modulation enables a more extensive use of the EM spectrum since higher frequencies are available
(v) Modulation makes it possible to transmit digital signals via free space, which would otherwise be
impossible since antennas can’t send DC signals
Question 3. (6 pts) What is the length of the driven element in a 300 MHz Yagi antenna?
Find wavelength: 𝜆𝜆 =
𝑐𝑐
𝑓𝑓
=
3×108 𝑚𝑚/𝑠𝑠
300×106 𝐻𝐻𝐻𝐻
= 1𝑚𝑚
𝜆𝜆
Driven element of a Yagi is a dipole, which has length =
2
1𝑚𝑚
2
= .5𝑚𝑚
Page 2 of 6
EC312 Twelve Week Exam Spring 2015
March 30, 2015
Question 4. (12 pts) Given the frequency domain representation of an AM signal shown below, write out the
equation for the carrier signal 𝑣𝑣𝑐𝑐 (𝑡𝑡) and the equation for the baseband signal 𝑣𝑣𝑚𝑚 (𝑡𝑡), including units.
𝑣𝑣𝑐𝑐 (𝑡𝑡) = _____ 10 sin(2𝜋𝜋(150000)𝑡𝑡) 𝑉𝑉 _____________________________
𝑣𝑣𝑚𝑚 (𝑡𝑡) = _____ 4 sin(2𝜋𝜋(5000)𝑡𝑡) + 6 sin(2𝜋𝜋(2500)𝑡𝑡) 𝑉𝑉 _______________________
Amplitude (V)
10
5
1
~
145
147.5 150.0 152.5
155
Frequency(kHz)
Question 5. (8 pts) Suppose that the AM signal from the previous question was provided as the input signal
𝑉𝑉𝑆𝑆 (𝑡𝑡) for the low-pass filter shown below. If R = 1kΩ and C = 1.07nF, draw the frequency domain
representation for the output 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 in the box below, labelling all relevant values and axes. (Note: assume that
all frequencies in the passband of the filter are passed without attenuation.)
Amplitude (V)
10
1
2𝜋𝜋𝜋𝜋𝜋𝜋
= 148.742 𝑘𝑘𝑘𝑘𝑘𝑘
All frequencies above the cut-off
frequency are filtered out and all
frequencies below the cut-off
frequency are passed, so we are left
with the two frequencies at 145 kHz
and 147.5 kHz.
5
1
~
𝑓𝑓𝑐𝑐𝑐𝑐 =
145
147.5 150.0 152.5
155
Frequency(kHz)
Draw your answer in this box, with
appropriate labels
Page 3 of 6
EC312 Twelve Week Exam Spring 2015
March 30, 2015
Question 6. (8 pts) You are designing an amplifier for a sensor system, which must take 5 mW as input power
and amplify it to produce an output of 14 dBm (+/- .5dBm). You check with the lab techs, and they have the
three amplifiers 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 (shown here to the right) available. Select which amplifiers you would use (in which
arrangement) and draw them into the box below. Show all work for full credit.
𝑃𝑃𝑖𝑖𝑖𝑖
𝑃𝑃𝑖𝑖𝑖𝑖 = 5𝑚𝑚𝑚𝑚
= 10 log(5) 𝑑𝑑𝑑𝑑𝑑𝑑
≈ 7 𝑑𝑑𝑑𝑑𝑑𝑑
𝐴𝐴1 = 3 𝑑𝑑𝑑𝑑
𝐴𝐴2 = 4 𝑑𝑑𝑑𝑑
𝑃𝑃𝑜𝑜𝑜𝑜𝑜𝑜
𝑃𝑃𝑜𝑜𝑜𝑜𝑜𝑜 = 14𝑑𝑑𝑑𝑑𝑑𝑑
Required amplification in dB is 𝑃𝑃𝑜𝑜𝑜𝑜𝑜𝑜 [𝑑𝑑𝑑𝑑𝑑𝑑] − 𝑃𝑃𝑖𝑖𝑖𝑖 [𝑑𝑑𝑑𝑑𝑑𝑑] = 7 𝑑𝑑𝑑𝑑 = 𝐴𝐴1 + 𝐴𝐴2
Can also do this by converting 𝑃𝑃𝑜𝑜𝑜𝑜𝑜𝑜 to mW, finding required amplification as a unitless gain, and then
converting 𝐴𝐴1 and 𝐴𝐴2 to unitless gain and considering their product.
Question 7. (4 pts) If an RLC bandpass filter has the correct resonant frequency but is not selective enough (i.e.
it passes frequencies that should be attenuated), explain how could you make the filter more selective without
changing the resonant frequency.
2𝜋𝜋𝑓𝑓 𝐿𝐿
1
To make it more selective, we want a higher Q (i.e. a smaller bandwidth). Since 𝑄𝑄 = 𝑅𝑅𝑟𝑟 , and 𝑓𝑓𝑟𝑟 = 2𝜋𝜋√𝐿𝐿𝐿𝐿, the
only way to increase Q without changing the resonant frequency 𝑓𝑓𝑟𝑟 is to decrease the resistance R.
Question 8. (3 pts) Circle all of the following which are true about the frequency domain representation of the
square wave shown to the right:
(i) The frequency domain representation has a fundamental
frequency at 𝑓𝑓0 = 1/𝑇𝑇
(ii) The frequency domain representation will also look like a
square wave
(iii) The frequency domain representation will have equally
spaced integer multiples of the fundamental frequency
Question 9. Consider the antenna radiation pattern shown to the
right.
1) (3 pts) What is the half-power beamwidth for the main lobe?
Half-power (i.e. -3dB) occurs at 330 deg. and 30 deg., so the
half-power beamwidth is 60 deg.
2) (3 pts) What is the Side-Lobe-Level (SLL) for the rear lobe?
SLL(dB) = 𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 (𝑑𝑑𝑑𝑑) - 𝐺𝐺𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 (𝑑𝑑𝑑𝑑) = 0dB – (-12dB) = 12dB
Page 4 of 6
EC312 Twelve Week Exam Spring 2015
March 30, 2015
Question 10. The radio tower for WNAV AM 1430 in Annapolis is approx. 386 feet tall.
1) (2 pts) What is the nominal radio horizon line-of-sight distance for this radio tower (in miles)?
𝑑𝑑[𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚] = �2ℎ[𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓] = �2(386) = 27.8 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
2) (2 pts) A friend in Gaithersburg (approx. 40 miles from Annapolis) claims that she regularly hears
WNAV broadcasts at all hours of the day. Which radio propagation phenomenon would be the MOST
likely reason for this?
Ground wave propagation. (This is farther than line-of-sight, but two close for sky waves.)
3) (2 pts) Another friend who lives in New York City (approx. 200 miles from Annapolis) claims that one
night he heard a radio broadcast from WNAV, but your friend in Philadelphia (approx. 100 miles from
Annapolis) didn’t hear it. Which radio propagation phenomenon would be the MOST likely reason for
this?
Sky wave propagation. (This is farther than ground waves would travel, and it would explain why the
transmission skipped Philadelphia but arrived in NYC.)
Question 11. (10 pts) You and your lab partner stand 100m apart from each other, each holding a standard
dipole antenna. Your lab partner transmits a signal with a transmit power of 10W, and you receive .153mW.
Assuming an ideal propagation environment, at what frequency is your lab partner transmitting?
For a dipole, gain g=1.64.
2
Rearranging the Friis free space equation, we have 𝜆𝜆 =
𝑐𝑐
Solving, we have 𝜆𝜆 ≈ 3𝑚𝑚, i.e. 𝑓𝑓 = = 100𝑀𝑀𝑀𝑀𝑀𝑀
𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟 (4𝜋𝜋𝜋𝜋)2
𝑃𝑃𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑔𝑔2
=
.000153𝑊𝑊 �4𝜋𝜋(100)�
10𝑊𝑊 (1.64)2
2
𝜆𝜆
Question 12. (6 pts) A communication system uses the QPSK digital modulation scheme illustrated by the
constellation diagram to the right. A signal is received (below left) with the indicated phase changes. Write out
the bitstream that has been received.
01
90°
00
180°
0°
10
11
00 01 11 10 00 11 01 10
270°
Page 5 of 6
EC312 Twelve Week Exam Spring 2015
Highest frequency
component is 22Hz
March 30, 2015
Question 13. The figure below is a graph of the signal
𝑉𝑉𝑠𝑠 (𝑡𝑡) = sin( 2𝜋𝜋(22)𝑡𝑡) + 2 sin( 2𝜋𝜋(15)𝑡𝑡) + 5 sin(2𝜋𝜋(2)𝑡𝑡)
8
111
110
6
101
Voltage (V)
4
101
2
100
0
011
010
-2
-4
001
-6
-8
000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Suppose the signal is sampled at a 20Hz sampling rate and quantized with a 3-bit quantizer with a range of -8V
to +8V.
1) (4 pts) Circle the 5th sample point.
Sampling period is 1/20 = .05 seconds. First sample is at time 0, so fifth sample is at time 0.2.
2) (4 pts) Next to the 5th sample point, write the binary number that will be assigned to that sample.
Since we’re using a 3-bit quantizer and the range is -8V to +8V, we have 2^3=8 possible levels, as
shown above. The fifth sample point falls in the sixth quantization level, i.e. it is assigned a value of
101.
3) (4 pts) Will the conversion from digital back to analog suffer from aliasing? Briefly explain why or why
not.
To avoid aliasing, we have to sample at higher than the Nyquist frequency.
The Nyquist frequency = Two times the highest frequency component = 2*22Hz = 44Hz.
Since our sampling rate is 20 Hz which is less than the Nyquist frequency, we will have aliasing.
Question 14. (4 pts) Two Systems Engineering majors both want to use the “free space” communication
channel at the same time for their Capstone projects. List two different ways they can simultaneously use this
communication channel without interfering with each other.
List two of the following:
- Time Division Multiplexing (e.g. take turns transmitting)
- Frequency Division Multiplexing (e.g. transmit at different carrier frequencies)
- Spatial Multiplexing (e.g. use low power transmission and separate them physically)
Turn in your equation sheets with your exam!
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