ELEC6258: Simulation of Mobile Communications:
Baseband Modulation Assignment
Dr Soon Xin Ng (Michael) [email protected]
Submission Details
This coursework forms your assessment for the Baseband Modulation part of ELEC6258: Simu-
lation of Mobile Communications. This coursework contributes 30% of your mark for ELEC6258
and completing it should require up to 41 hours of work outside the scheduled lectures. You
should make sure that you get started on this coursework early enough to finish it before the
deadline. There’s no way that you can complete a 41-hour coursework in only the week before
the deadline.
Although you may verbally discuss your ideas with your classmates, you should not show
them your Matlab code or results. When you are finished, you should copy and paste your
Matlab code and results into a Word document, which you should then save as a .pdf file.
Then you should submit the .pdf version of your Word document before 4pm on Friday
12/3/2021, to the electronic handin system:
https://handin.ecs.soton.ac.uk/handin/2021/ELEC6258/4/
Implementation How well are the baseband system and adaptive equalisation
30% scheme implemented in Matlab? How well are the questions
posed in the assignment answered?
Presentation How does the presentation show the understanding? Do you
20% include all correct labels? Do you use the right label units?
Do you explain your results when requested?
Interpretation How well are the obtained results discussed and interpreted?
20% How well are the concepts of baseband system model and channel
equalisation understood?
Accuracy Are the obtained results accurate and correct? Is the
30% formulation or derivation correct? Are the plots accurate?
Do you include all required plots?
Table 1: Marking Scheme.
The marking scheme (out of 100%) of this assignment is shown in Table 1. The marks
distribution for each question (out of 30 marks) is shown next to the question number. Lecture
notes and related files for this assignment can be downloaded from the following website:
https://secure.ecs.soton.ac.uk/notes/elec6238/baseband/
Some general feedbacks will be given in the feedback lecture at the end of the semester.
If you notice any mistakes in this document or have any queries about it, please email me
at [email protected]
Michael Ng
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Baseband System Simulation
In this assignment, the following 8-Star-QAM baseband communication system is simulated,
where k indicates the symbol-spaced sampling quantity.
Rx
filter
x(t) x[k]
sampling at
symbol rate
^ ^ y[k]
equaliser
8-Star-QAM
symbols
x[k]
Tx
filter
x(t)
channel
c(t) +
e(t)
Figure 1: Baseband communication system.
You may assume a symbol rate, e.g. fsyb = 8 kHz (kSymbols/sec) and hence a bit rate of
fb = 24 Kbps since we have 3 bits for each 8-Star-QAM symbol. We will use complex number
to represent our baseband signals, where the in-phase component is represented by the real-
part and the quadrature-phase component is represented by the imaginary-part. Each of the
transmitted complex-valued symbol x[k] takes one of the eight possible values as shown in the
Gray-labelled 8-Star-QAM constellation in Figure 2.
b
−b
a
−a
a−a−b b
Imag
Real
x1 = 001x2 = 010
x6 = 110 x0 = 000
x4 = 100x7 = 111
r = b/a
x3 = 011
x5 = 101
Figure 2: 8-Star-QAM constellation.
Simulation of analogue signal is performed using a sufficiently high sampling rate, e.g. 2
MHz (actually 400 kHz would be sufficient as this is a baseband system). The transmit filter
is a properly designed square root raised cosine filter, and the receive filter is identical to the
transmit filter. You need to consider the correct sampling instances.
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1 Fading Channel Modelling [4 marks]
a) Small-scale fading. Briefly explain flat fading and frequency selective fading, in terms
of signal bandwidth and channel coherence bandwidth.
b) Rayleigh distribution. Plot and describe the theoretical Probability Distribution Func-
tion (PDF) of Rayleigh distribution and the corresponding histogram (empirical PDF) for
variance σ2 = 1/2 and number of samples N = {1000, 100000}.
c) Flat Rayleigh fading. Given that the received signals experience flat Rayleigh fading,
plot the corresponding Cumulative Distribution Function (CDF) of the received signal envelope
for variances of σ2 = {0.5, 1}. What are the corresponding outage probabilities, for σ2 =
{0.5, 1}, if the minimum required received signal amplitude for successful detection is 0.8?
d) Correlated flat Rayleigh fading. Plot and describe the magnitude v/s time curves of
the received signals that experience correlated flat Rayleigh fading, when the maximum Doppler
frequencies are fm = {1000, 100} Hz, respectively.
2 Noise-Free System [5 marks]
In this question, the amplitude ratio of the 8-Star-QAM is r = 1.5. In a noise-free system, we
assume that the noise e(t) in Fig. 1 is absent. Show both the in-phase and quadrature-phase
components when plotting any complex-valued signal.
a) Normalized 8-Star-QAM. What is the average transmission power for the 8-Star-QAM
scheme of Figure 2 if we have a = 1 and an amplitude ratio of r = b/a = 1.5? Find out the
values of a and b that would give a unit average transmission power for r = 1.5.
b) Ideal Channel. The channel is ideal with an impulse response of c(t) = δ(t).
Plot the eye diagrams of both the in-phase Re{xˆ(t)} and quadrature-phase
Im{xˆ(t)} of the complex-value signal xˆ(t). Show the amplitude versus time plots of
the transmitted symbol sequence x[k] and the received sampled signal xˆ[k], using
the Matlab ’stem’ function. Explain your results.
c) Non-ideal Channel. The channel is non-ideal with an impulse response of c(t) = δ(t) +
0.4δ(t−Tsyb), where Tsyb = 1fsyb is the symbol period. We assume that the imaginary component
of the channel is not present.
Plot the eye diagrams of both the in-phase Re{xˆ(t)} and quadrature-phase
Im{xˆ(t)} of the complex-value signal xˆ(t). Show the amplitude versus time plots of
the transmitted symbol sequence x[k] and the received sampled signal xˆ[k], using
the Matlab ’stem’ function. Explain your results.
d) Compare and discuss the eye diagrams (for both the real and imaginary parts) of the
ideal and non-ideal channels above.
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e) Equalisation. For the case of non-ideal channel in c), implement an equaliser
y[k] = xˆ[k]− 0.4xˆ[k − 1] + 0.16xˆ[k − 2]− 0.064xˆ[k − 3] + 0.0256xˆ[k − 4]
to equalise your received signals.
Show the amplitude versus time plot of y[k] after the equalisation and compare
it with the one in Q2(c) before equalisation.
3 Noisy System [9 marks]
Assume that the Additive White Gaussian Noise (AWGN) e(t) is present, which has a total
variance of N0 = 0.01 for both complex-dimensions (this corresponds to a variance of N0/2 =
0.005 per dimension). Show both the in-phase and quadrature-phase components when plotting
any complex-valued signal.
a) Ideal Channel. The channel is ideal with an impulse response of c(t) = δ(t).
Plot the eye diagram of xˆ(t), and show the amplitude versus time plots of the
transmitted symbol sequence x[k] and the received sampled signal xˆ[k]. How does
this eye diagram compared to that in Q2(b)? How does this eye diagram compared
to that in Q2(c)?
b) Non-ideal Channel. The channel is non-ideal with an impulse response of c(t) = δ(t) +
0.5δ(t− Tsyb).
Plot the eye diagram of xˆ(t), and show the amplitude versus time plots of the
transmitted symbol sequence x[k] and the received sampled signal xˆ[k].
c) Equalisation. For the case of non-ideal channel in b), implement an equaliser
y[k] = xˆ[k]− 0.4xˆ[k − 1] + 0.16xˆ[k − 2]− 0.064xˆ[k − 3] + 0.0256xˆ[k − 4]
Show the amplitude versus time plot of y[k]. Explain your results.
d) Bit Error Ratio (BER) Plots. For the case of non-ideal channel in b), plot the BER
of the system after the five-tap equaliser for SNR ranging from 0 dB to 18 dB.
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4 Adaptive Equalisation [12 marks]
The symbol-rate sampled channel output xˆ[k] is given by
xˆ[k] = h1x[k] + h2x[k − 1] + e[k]
where the two complex-valued channel paths are given by:
h1 = 0.8− 0.2i and h2 = 0.7i
while x[k] is the transmitted 8-Star-QAM symbol and the noise e[k] is the equivalent noise in
the digital domain, which has a total variance of N0 = 0.4. A three-tap adaptive equaliser
y[k] =
2∑
τ=0
a∗τ xˆ[k − τ ] = a∗0xˆ[k] + a∗1xˆ[k − 1] + a∗2xˆ[k − 2]
is used to detect the transmitted symbol x[k − 1].
a) Optimal MMSE solution: Compute the optimal Minimum Mean Square Error (MMSE)
solution of a = [a0 a1 a2]
T that minimises the MSE cost function:
J(a) = E[|x[k − 1]− y[k]|2] .
You can compute the above solution by calculations or by using a Matlab program. Remember
to show your calculations or your Matlab codes in your report.
b) Implement the LMS adaptive algorithm using Matlab:
aj[k] = aj[k − 1] + 2µ (x∗[k − 1]− y∗[k]) xˆ[k − j], 0 ≤ j ≤ 2
You should choose an appropriate small positive number for adaptive gain (or step size) µ and
you can use the initialisation (0.0, 0.0, 0.0) for the equaliser’s weights, i.e. aj[0], 0 ≤ j ≤ 2.
Explain and document your Matlab codes.
c) Investigate the convergence behaviour of this LMS algorithm. Try two µ values
and remember to look at both the real and imaginary parts of the equaliser weights. Plot also
the MSE evolution curves. Explain your results and observations.
d) Investigate the steady-state MSE performance of this LMS algorithm. Try two
µ values and remember to look at both the real and imaginary parts of the equaliser weights.
Plot also the MSE evolution curves. Explain your results and observations.
e) Comparisons: Compare the advantages and disadvantages of the LMS algorithm, the
steepest descent algorithm and the Wiener-Hopf method. There is no need to compute the
curves for the steepest descent algorithm.
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