Lab Exercises for the MSc Course B31SI2:

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Lab Exercises for the MSc Course B31SI2: Principles of Mobile Communications 7
Exercise 3
A flat fading channel simulator can be constructed by using the sum-ofsinusoids
method. With this method, a baseband complex Gaussian random
signal can in general be modelled by
g(t) = g1(t) + jg2(t), (3.1)
where
gi(t) = X
Ni
n=1
ci,n cos(2πfi,nt + θi,n) , i = 1, 2 . (3.2)
The gains ci,n, discrete Doppler frequencies fi,n, and phases θi,n can be
determined by using the so-called method of exact Doppler spread (MEDS)
and are given as follows
ci,n =
vuut
2b
Ni
, (3.3a)
fi,n = fm sin [ π
2Ni
(n 12)] , (3.3b)
θi,n = 2π
n
Ni + 1
. (3.3c)
In (3.3), b stands for the variance of ?gi(t) and fm denotes the maximum
Doppler frequency. Note that this channel simulator is of deterministic
nature since all the model parameters are kept constant instead of random
during simulations. Fig. 3.1 shows the structure of the corresponding
deterministic simulation model for Rayleigh fading channels, where ?α(t)
indicates the received signal envelope in the complex baseband.
c o s ( 2 t )
1 1
pf,1 N
+ q ,1 N
tf +q ,2 N
Fig. 3.1: A deterministic Rayleigh fading channel simulator.
8 Lab Exercises for the MSc Course B31SI2: Principles of Mobile Communications
3.1 For the given number of sinusoids N1 = 9 and N2 = 10, compute the
simulation model parameters ci,n, fi,n, and θi,n according to (3.3a)–
(3.3c) by using b = 1 and fm = 91 Hz. Write your results in Table 3.1
given at the end of Exercise.
3.2 Develop a Matlab function m-file to simulate the deterministic processes
gi(t) for i = 1, 2. Use the quantities ci,n, fi,n, θi,n, and t as input
arguments.
3.3 Write a Matlab script m-file to carry out the simulation of the channel
amplitudeα(t) = |g(t)| = |g1(t) + jg2(t)| and plotαdB(t) = 20logα(t).
Solve this problem by making use of your function m-file developed in
Exercise 3.2. The simulation model parameters ci,n, fi,n, θi,n, and Ni
are given as listed in Table 3.1. Use the simulation time Tsim = 0.4 s
and the sampling frequency fs = 270.8 kHz, which corresponds to the
symbol rate used in GSM. If a vehicle drives with a speed of v = 109.2
km/h, what is the distance it covers in the specified simulation time?
Hint: The discretization of the time t can be expressed as t = 0 :
1/fs : Tsim.
3.4 Simulate the deterministic functions g1(t),g(t), andα(t) by using the
sampling frequency fs = 50 kHz and the simulation time Tsim = 20 s.
Determine the mean value and variance of g1(t), g(t), andα(t). Determine
the probability density function (PDF) of g1(t) andα(t) by
using histograms. Which distributions do they approximate? Compare
the PDF of g1(t) with the theoretical result given in (1.1), where
mμ = 0 and σμ = 1. Compare the PDF ofα(t) with the theoretical
result pα(x) given in (2.4), where p = 2, and K = 0.
Hint: Use the Matlab functions mean, std, and hist.
3.5 Determine a part of the signal g1(t)|t∈I from the simulation results of
g1(t) in the interval I = [10 000 /fs , 20 000 /fs], where fs = 1000 Hz.
Find the autocorrelation function (ACF) φ?
g1g1
(τ ) of g1(t)|t∈I and plot
it. What is the value of φ
g1g1
(0)
Hint: Use the Matlab function xcorr with the option biased.
Lab Exercises for the MSc Course B31SI2: Principles of Mobile Communications 9
3.6 The ACF φ?
g1g1
(τ ) of g1(t) can be expressed by
φg1g1
(τ ) = X
N1
n=1
cos (2π f1,n τ ) . (3.4)
Compute φ?
g1g1
(τ ) over the interval 0 ≤ τ ≤ 0.08 s and sketch your
result in the same figure as in Exercise 3.5. For the sake of comparison,
the ideal ACF φg1g1
(τ ) in (2.1) is required to be shown in this figure
as well. Use the value 2 for p and 91 Hz for fm.
10 Lab Exercises for the MSc Course B31SI2: Principles of Mobile Communications
i n fi,n (Hz) ci,n θi,n (rad)
1 1
1 2
1 3
1 4
1 5
1 6
1 7
1 8
1 9
2 1
2 2
2 3
2 4
2 5
2 6
2 7
2 8
2 9
2 10
Table 3.1: Parameters of the simulation model (fm = 91 Hz, b = 1, N1 = 9, N2 = 10).

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Lab Exercises for the MSc Course B31SI2:

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