Sample final exam

UNIVERSITY OF AKRON
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
4400: 341
INTRODUCTION TO COMMUNICATION SYSTEMS - Spring 2015
SAMPLE FINAL EXAM
TIME: 1 hour 30 minutes
INSTRUCTIONS:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Write your name on each page
Clearly show all your work, in steps, in the space assigned for each question.
Answers without reasoning will not receive any credit.
With some thinking, there are always short cuts to the answer. If you are taking too
much time to answer a question it is advisable to leave it and return to it after you
have attempted the rest of the problems.
If you feel that you do not understand a question or it is ambiguous, please ask
about it.
The list of the needed equations and formulae is attached to the end of this
booklet.
Relax and Perform.
GOOD LUCK
Question
Total Grade
1
20
2
20
3
30
4
10
Total
80
Your Score
Name:_______________________________________
Problem 1 (20 points)
−2t
Consider the signal x(t) = e u(t) .
a) Calculate its energy in time and frequency domains and verify the Parseval’s theorem
for it.
b) Calculate its autocorrelation and ESD functions.
c) Calculate the essential bandwidth of this signal where the essential bandwidth must
contain 95% of the signal energy.
d) If we pass x(t) through a system whose impulse response is h(t) = δ (t − 4) , Find the
output signal ESD and its energy.
Name:_______________________________________
Problem 2 (20 points)
a) The signal shown in the figure is supposed to be transmitted using DSB+C
modulation. Calculate the required modulation index and maximum power efficiency of
this modulation.
b) We have a superheterodyne AM receiver with a cheap non-ideal local oscillator. In
addition to fLO, this local oscillator always generates another frequency tone at
fLO-10KHz. The intermediate frequency fIF is set to be at 200KHz. Plot the structure of
the superheterodyne receiver. Assuming that the receiver is tuned to receive a channel at
center frequency fc = 350KHz, calculate fLO. Calculate possible image frequencies.
Name:_______________________________________
Problem 3 (30 points)
An angle modulated signal with frequency fc = 150KHz is described by the equation,
ϕ FM (t) = 10 cos(2π fc t + 0.01sin 2000π t)
a) Find the power of the modulated signal and its frequency deviation. Estimate the
bandwidth of the modulated signal.
b) Design and sketch the block diagram of an Armstrong indirect FM modulator that
generate a WBFM signal with carrier 96.3MHz and frequency deviation 20.48KHz using
the above signal. Only frequency doublers are available as frequency multipliers. In
addition, an oscillator with adjustable frequency from 13MHz to 14MHz is also available
for mixing, along with bandpass filters of any specifications.
c) To demodulate the WBFM signal generated in part (b), we use a slope detector as
shown in the figure which is in fact a high pass filter. We know that the 3dB frequency of
this filter is at 1/RC. Assuming R = 1KΩ, find a suitable value for the capacitor, C.
Problem 4 (10 points)
Consider the signal 𝑥 𝑡 = cos 10𝜋𝑡 + 2cos (30𝜋𝑡).
a) Find out the Nyquist sampling rate for this signal.
b) If we sample this signal at 25Hz, and then from the sampled signal try to reconstruct
the original signal by passing in through a LPF with bandwidth 18Hz, find out the signal
in time domain at the output of the LPF. Sketch the spectrum of this signal. Is it a perfect
reconstruction? Explain.
e ± jx = cos x ± j sin x
sin 2 x =
cos x =
1
(1 − cos 2x )
2
cos 2 x =
1
T →∞ T
Signal Power: Pg = lim
Signal Energy: Eg =
∫
∞
−∞
∫
T /2
−T /2
e jx + e− jx
2
e jx − e− jx
sin x =
2j
1
(1 + cos 2x )
2
x(t)x * (t)dt =
x(t)x * (t)dt =
∫
∞
−∞
∫
∞
−∞
∫a
2
dx
1
x
= ( ) tan −1 ( )
2
a
a
+x
Sg ( f )df = Rg (0)
Eg ( f )df = Rg (0)
Fourier Transform and Inverse Fourier Transform:
X( f ) =
x(t) =
∞
∫−∞ x(t)e− j 2πft dt
∞
∫−∞ X( f )e j 2 πft df
mmax − mmin
2A + mmax + mmin
usefulpower
Ps
AM power efficiency: η =
=
totalpower
Pc + Ps
Δf
FM and PM modulation index: β =
where B is the message bandwidth
B
AM modulation index: µ =
Δf =
k f mp
2π
•
for FM and Δf =
kp m p
2π
for PM
FM and PM bandwidth: BFM = 2B(β + 1)
Nyquist Rate is 2B (B is the analog signal bandwidth).
m 2p
Quantization Noise Power: N = 3L2