389.

157 WC1 SS 2018

Midterm Exam
VU Wireless Communications 1, 389.157, SS 2018,

Important for getting a grade:

• Write your name and matriculation number on every sheet you hand in!
• Answer all questions tagged with boxes such as XY short and precise, and
state the question number next to the solution.
• Sit alone at a table and try to maximize the distance to your nearest neigh-
bor (min. 2 seats).
• It is strictly forbidden to talk to other students.
• You are allowed to use lecture notes and books, handwritten material is
allowed only if it is written by yourself.
• It is prohibited to use any electronic device (e.g., cellphone, laptop, notepad,
ebook reader, etc.), the only thing you might use is a pocket calculator
(without any wireless connectivity).
• In case questions arise, raise your hand and wait until a supervisor is coming
to you.
• You are not allowed to leave the room during the exam except for going to
the bathroom. In this case you must ask for permission and leave all your
papers at the supervisors desk.

1 Cellular Principles
Consider a cellular system operating at a carrier frequency of 900 MHz that offers
both voice call and broadband data transfer services. All cells have hexagonal
shapes of the same size and the base stations are located exactly in the center
of the cells. Both the transmitter and the receiver use isotropic antennas and
the transmit power of the base station is 10 W. The measured path-loss exponent
(including shadowing) is n = 3.8.
4% 1 Assume that each user generates on average 5 connection requests per hour,
of which 60 % are data transfer requests. The average connection duration is 4,
and 0.8 minutes for broadband data service and voice calls, respectively. Find
the average load per user in Erlang.
4% 2 We are considering a cell in the business district. The user density varies over

E. Zöchmann, T. Blazek, M. Rupp, C. Mecklenbräuker 1

 389.157 WC1 SS 2018

the time of day, and within one day is given as

 
2πt
ρuser (t) = 190 − 51 cos in users/km2 (1)
24h

Find the cell radius R so that each cell is assigned with 35 channels and the
queuing probability is guaranteed to be not larger than 2 %. Hint: The area of
a triangle is given as T = 12 bh, where b is the length of the base of the triangle,
and h is the altitude of the triangle.
4% 3 Find the reuse distance D, if the receive power at the base station location of
a co-channel cell is measured to be −152.03 dBm.
5% 4 Considering only interfering base stations in the first tier, calculate the signal-
to-interference (SIR) ration in dB.
5% 5 The provider changes its contracts. Instead of a maximum queuing probability,
it now offers a median queuing probability of 2%. This median is calculated over
the full day cycle. How many servers does it need? What is the maximum queuing
probability in that case? Hint: If you did not derive a cell area in Question 2,
assume the area to be 0.4 km2
5% 6 Due to your job, you have to frequently call a call center. You call so often,
that you collect the following data: The probability of being queued is 40 %, if
you are queued, you wait an average of 5 minutes until you get a line, and the
operator takes 15.3 minutes on average to deal with your issue. Assume your
experience is representative for everyone. How many people are serving calls in
this call center simultaneously? (Round all results to 2 decimal digits)

2 Moments of the Power Delay Profile

A local spatial average of a power delay profile Pr (τ ), measured at 2.1 GHz, is
shown in Figure 1.
3% 7 Calculate the mean power Pm .
3% 8 Find the mean delay Tm for this channel.
3% 9 For this channel calculate the RMS delay spread Sτ .
3% 10 A particular modulation format provides suitable BER performance when-
ever STτs ≤ 0.1. Determine the shortest symbol period Ts that can be sent through
the RF channel shown in Figure 1, without using an equalizer.
3% 11 Determine the highest symbol rate that may be sent through the RF channel
of Question [10].
3% 12 You would like to increase the data rate from the previous question, yet
keeping a simple equalizer (one tap equalizer). What modulation format would
you propose?

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 389.157 WC1 SS 2018

Figure 1: Indoor channel response

3 OFDM
Consider a 20 MHz OFDM system. The length of one OFDM symbol including
the cyclic prefix is 4 µs. 20% of this is made up of the cyclic prefix itself.
3% 13 What subcarrier spacing is used? How many subcarriers does the system
use?
4% 14 We now analyze the transmission of data packets. A packet consists of
2 OFDM symbols used as preamble (not used for data transmission) and 100
OFDM symbols of data. However, 4 subcarriers are NOT used for data trans-
mission as they are needed as pilots. Given that all other subcarriers are used
for data transmission, calculate the number of bits that can be transmitted in
one packet, if 4-QAM, 16-QAM or 64-QAM is used. Furthermore, calculate the
maximum achievable throughput when continously transmitting packets. (Data
in bits per second.)
If you did not calculate the number of subcarriers, use 60 as number of subcarriers
instead.
3% 15 What is the maximum channel delay that can be corrected by the cyclic
prefix?
3% 16 Due to the strong sidelobes of OFDM, we now demand a guard band, a
frequency range of 1.25 MHz at both ends of the spectrum to be kept empty.
How many subcarriers do we lose. Calculate the throughput that we can now
achieve (preamble and pilots stay the same).
4% 17 To combat outdoor channels, we half the bandwidth to 10MHz, but leave the
number of subcarriers and percentage of cyclic prefix exactly as they are. How
long is the cyclic prefix now? How much longer can the longest multipath be
than the line of sight?

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 389.157 WC1 SS 2018

Do you think this is appropriate? Give reasons.

3% 18 The OFDM System is equipped with 2 × 2 MIMO, and each subcarrier
experiences the following channel:
 
1 −2
H=
2 −4

What is the maximum achievable multiplexing gain?

3% 19 Consider a highly time variant, but frequency flat channel, and the number of
subcarriers is constant as before. Is it better to have a large or a small bandwidth
to combat the channel? Give reasons for your answer.

4 Moving Users and Doppler Spread

3% 20 Users of a communication system at 2.1 GHz are moving with 150 km/h.
Find νmax and the coherence time. (Hint: use the Fleury uncertainty relationship
1
Tcoh ≤ 2πS ν
, and Sν = ν√
max
2
).
3% 21 How does the coherence time influence a communication system?
3% 22 Looking at the Fleury uncertainty relationship, explain how the RMS Doppler
spread influences the coherence time.
3% 23 Assume that one pilot symbol is enough to estimate the channel correctly for
a duration of Tcoh . How many symbols in one frame have to be pilot symbols for
channel tracking? Use the coherence time calculated in Question [20] and assume
a frame duration of a)Tf = 700 µs, b)Tf = 7.5 ms.
3% 24 Assume you know the channel perfectly, you would like to employ a system
that exploits time-diversity. In order to to so, the transmit symbol is repeated
multiple times. Do you prefer a channel with a long or a short coherence time?

5 Approximations of the Rice Distribution

5% 25 Rician fading is an appropriate model for propagation conditions, where one
path with peak amplitude s is much stronger than the others. The probability
density function for the amplitude values r is given by
 2
r + s2
  
r rs
fR (r) = 2 exp − 2
I0 2 , r ≥ 0 , (2)
σ 2σ σ

where r is the instantaneous magnitude of the received voltage signal and σ 2 is

the power of the received signal. The modified Bessel function of the first kind
and 0th order is denoted by I0 .

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 389.157 WC1 SS 2018

For our calculations we will approximate the Bessel function via I0 (x) ≈ exp(x)
√
2πx
, x
0. Show that
√ r the approximate probability density functions can be written as
(r−s)2

fR (r) ≈ √2πσs 2 exp − 2σ2 .
s
7% 26 Show that the mode of the approximate distribution is given as r = 2
1+
q
s2
1 + K1 , where K = 2σ


2 is the Rician K-factor.

5% 27 The approximate PDF looks almost like a Gaussian PDF. The mode of
the approximate PDF is also known from the previous question. For sake of
2

1
simplicity you’d like to approximate further to fR (r) ≈ √2πσ 2
exp − (r−s)
2σ 2
.
This approximation leads to a Gaussian distribution. Calculate the minimum
Rician K-factor, for which the mode of the Gaussian pdf deviates only by 5%.

E. Zöchmann, T. Blazek, M. Rupp, C. Mecklenbräuker 5