389.063, VU Mobile Kommunikation
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Institut für Nachrichtentechnik
und Hochfrequenztechnik
Midterm Exam:
When answering the following 37 questions, always remember that there is
someone who has to grade them. So please use legible handwriting.
- You can answer the questions in German and/or English.
- There is a total of 56 points in this exam.
TestResult = floor(sum(Points)/5.1);
- Please leave the rest of this page empty.
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389.063, VU Mobile Kommunikation
Problem 1:
(17 points)
Assume that you just got a new job in a testing facility for directional radio
receivers that can be modeled in the following way:
Noise
(Gaussian and real valued)
ML
Detector
Antenna
Detected
Symbols
The corresponding directional transmitter is mounted some kilometers away.
It is constantly broadcasting a 2 PSK modulated signal (with equally probable
symbols):
s2
d
s1
0
d
You start measuring and find out that the average√symbol error rate, at the
receiver under your investigation, is P{Es } = Q ( x), where x is some real
number known to you.
Express your results in terms of x when answering the following questions:
1 p.
1 What is the probability P(s 6= s1 |s1 ) that symbol s1 is transmitted but not
received?
1 p.
2 What is the probability P(s = s1 ) that symbol s1 is received?
1 p.
3 You use a remote control program to modify the transmitter in the following way: Symbol s2 is now transmitted twice as often as Symbol s1 . What
is the new probability P(s 6= s1 |s1 ) that symbol s1 is transmitted but not
received?
1 p.
4 What is the new average symbol error probability P{Es }?
2 p.
5 You use your remote control program again to revert the receiver back to
transmitting equally probable symbols but change the transmitter symbol
constellation to the following:
s2
s1
0.5 d 0
0.5 d
What is the new probability P(s 6= s1 |s1 ) that symbol s1 is transmitted but
not received?
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389.063, VU Mobile Kommunikation
1 p.
6 You use your remote control program again to change the transmitter
symbol constellation to the following:
s2
s1
0
d
What is the new probability P(s 6= s1 |s1 ) that symbol s1 is transmitted but
not received?
1 p.
7 You found a new menu item in your remote control program that allows
you to add a new symbol to the transmitter symbol constellation.
s3
s2
s1
d
0
d
All three symbols are transmitted equally probable. What is the new
probability P(s 6= s1 |s1 ) that symbol s1 is transmitted but not received?
2 p.
8 What is the new probability P(s 6= s2 |s2 ) that symbol s2 is transmitted but
not received?
2 p.
9 What is the new average symbol error probability P{Es }?
5 p.
10 Finally, you add some additional new symbols to the transmitter symbol
constellation.
s5
s4
s3
s2
3d
d
0
d
s1
3d
All five symbols are transmitted equally probable. What is the average
probability of symbol error P{Es }?
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389.063, VU Mobile Kommunikation
Problem 2:
(17 points)
Assume that the marketing1 department decided to cancel all this cumbersome measuring because workers in other departments get their job done much
faster. You were told that they are using a tool called “Matlab” that is not
only easier and cheaper to obtain, but also works “ideally”. In order to save
time, you organized some source code from the other department that looks
the following:
01
02
03
xvAlphabet = [-3,-1,0,1,3];
xvTXSymbols = randsrc(1,100000,[1:length(xvAlphabet)]);
xvSNRVec
= -10:2:16;
04
xvTXSamples
05
06
for xvSNRLoop=1:length(xvSNRVec);
xvRXSamples = xvTXSamples + ...
10^(-xvSNRVec(xvSNRLoop)/20)*randn(1,size(xvTXSamples,2))/sqrt(2);
07
= xvAlphabet(xvTXSymbols);
[xvNotNeeded,xvRXSymbols] = min(abs(...
xvAlphabet.’*ones(1,length(xvTXSamples)) - ...
ones(length(xvAlphabet ),1)*xvRXSamples ...
));
09
10
xvCollect_SER(xvSNRLoop)
end;
= sum(xvTXSymbols~=xvRXSymbols)/size(xvTXSamples,2);
11
12
13
semilogy(xvSNRVec,xvCollect_SER,’-ko’,’LineWidth’,2,’MarkerSize’,6,’MarkerFaceColor’,’w’);
ylim([5e-5 1]);
grid on;
1 p.
11 Sketch a scatter plot of the signal transmitted.
1 p.
12 Sketch a scatter plot of the input signal to the ML detector.
1 p.
13 How many symbols are transmitted in total during one execution of the
program?
2 p.
14 Explain line 07 in detail. How does this line of code work, and what is
its function?
2 p.
15 Does line 07 work for arbitrary (even complex) symbol alphabets? Is it
optimal in the sense of minimizing the average SER2 ?
1 p.
16 Is line 07 optimal in the sense of minimizing the average SER, if the
transmitted symbols are not equally probable and this fact is not known
to the receiver? Why?
1 p.
17 Explain line 09 in detail. How does this line of code work, and what is
its function?
1 p.
18 Sketch (only qualitatively) the output of this program (the figure that is
plotted). In addition, label the ordinate.
1
2
One of the shortest definitions of marketing is “meeting needs profitably.”
SER . . . Symbol Error Rate
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389.063, VU Mobile Kommunikation
5 p.
19 Label the abscissa of the plot. (Hint: “average SNR at the receiver” is a
slightly wrong answer).
1 p.
20 Modify the code such that 100 symbols are transmitted per SNR set.
1 p.
21 Sketch the new output of the program (the figure that is plotted). (Hint:
the answer is substantially different from [18]) — Remember: xvSNRVec
ranges from -10 to 16 with 100 symbols being transmitted.
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389.063, VU Mobile Kommunikation
Problem 3:
(7 points)
To simulate chance occurrences, a computer can’t literally toss a coin or roll
a die. Instead, it relies on special numerical recipes for generating strings of
shuffled digits that pass for random numbers. Such sequences of pseudorandom
numbers play crucial roles not only in computer games but also in simulations
of physical processes3 .
Matlab uses a modified version of Marsaglia’s “subtract with borrow” algorithm (the default in Matlab versions 5 and later) to generate random
numbers using rand().
1 p.
22 You start Matlab 7.2, type rand(), and press Return. The Result is
−0.4326, always! How can this happen?
1 p.
23 In an article you read: “Almost all random-number generators calculate
a new random number from preceding values. The only true randomness
in such schemes is in the choice of the seed which is only a few hundred
bits at most.” What does “the seed” mean?
1 p.
24 Assume while debugging Matlab code you want rand() to generate the
same sequence of “random” numbers every time the program is executed.
Is this possible, and how to achieve this? Explain what “random” means
in the sense of rand().
(Answer in sentences, the exact Matlab source-code is not required.)
Marsaglia once said: “Random numbers are like sex: ... Even when they are
bad they are still pretty good.”
25 Assume that your new computer is so incredibly fast that Matlab can
generate one billion (= 109 ) random numbers a second. Will the random
numbers ever start repeating? If yes, will this ever happen? When?
2 p.
Algorithm guru Donald Knuth of Stanford University once advised: “. . . random numbers should never be produced by a random method. Some theory
should be used.”
26 So you decided to use Marsaglia’s “subtract with borrow” algorithm (the
one Matlab has already implemented) in your Monte Carlo simulation.
Do you ever have to pay attention on the fact that the use of particular
methods for generating random numbers can produce misleading results
in simulations? Can this happen in reality if assuming simulation times
of less than a day? If yes, why?
2 p.
3
Quote from Science News, Week of Sept. 27, 2003; Vol. 164, No. 13
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389.063, VU Mobile Kommunikation
Problem 4:
(9 points)
Note: Erlang tables are attached at the end of this test.
Consider a GSM operator in the 900 MHz band. Every operator owns 8 MHz of
bandwidth (2×8 MHz: for up- and downlink; 200 kHz of bandwidth is allocated
for a single radio channel, one radio channel has 8 time slots. One time slot
requires to one logical channel). One time slot (logical channel) per BS is
dedicated for one BCCH (signalization)
- The signal to interference ratio (SIR) for satisfactory call quality is 18 dB.
- The fading margin is 9 dB, consider the fading margin only once.
- The power decrease is assumed to follow the d−4 law.
You now have to design a radio access network satisfying the following assumptions and constraints:
- Less than 3% of the calls are blocked.
- Your network has 100 000 subscribers.
- One third of the subscribers are active with an average call duration of 90
seconds per hour.
1 p.
27 What is the reuse distance?
1 p.
28 What is the cluster size for hexagonal layout?
2 p.
29 What is the maximum traffic load which can be served pro cell?
1 p.
30 How many frequency channels and how many logical channels are required per cell?
1 p.
31 How many subscribers can be maximally assigned to one cell?
1 p.
32 How many cells should your radio access network have if you guarantee
that no more than 3% of the calls will be blocked on average?
2 p.
33 What is the traffic load which can be served pro cell in order to get
maximum C/I ratio?
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389.063, VU Mobile Kommunikation
Problem 5:
(6 points)
Compute the probability that the output power of selection diversity is 5 dB
lower than the mean power of each branch.
1 p.
34 Calculate the threshold in linear scale.
2 p.
35 Calculate the probability cdf (γ) for Nr = 1, 2, 8 antennas.
Consider now that Nr = 2 and that the mean powers in the branches are 1,5 γ
and 0,5 γ.
2 p.
36 Calculate the new probability cdf (γ).
1 p.
37 Is the diversity more or less efficient when the average branch powers are
different?
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