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ELEC6258: Simulation of Mobile Communications: Baseband Modulation Assignment

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 Tueday 15/3/2022, to the electronic handin system:


Implementation 30%

How well are the baseband system and adaptive equalisation scheme implemented in Matlab? How well are the questions posed in the assignment answered?

Presentation 20%

How does the presentation show the understanding? Do you include all correct labels? Do you use the right label units? Do you explain your results when requested?

Interpretation 20%

How well are the obtained results discussed and interpreted? How well are the concepts of baseband system model and channel equalisation understood?

Accuracy 30%

Are the obtained results accurate and correct? Is the 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/noteswiki/w/ELEC6258/Resources Introduction         Some general feedbacks will be given in the feedback lecture.

If you notice any mistakes in this document or have any queries about it, please email me at [email protected]

Michael Ng

Baseband System Simulation

In this assignment, the following 8PSK baseband communication system is simulated, where k indicates the symbol-spaced sampling quantity.



sampling at

symbol rate

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 8PSK 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 8PSK constellation in Figure 2.

z1  = 001

z6  = 110

Figure 2: 8PSK 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.

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 σ 2  = 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 σ 2   =  {0.5, 1}.   What are the corresponding outage probabilities, for σ 2   = {0.5, 1}, if the minimum required received signal amplitude for successful detection is 0.7?

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 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) 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{(t)} and quadrature-phase Im{(t)} of the complex-value signal (t). Show the amplitude versus time plots of the transmitted symbol sequence x[k] and the received sampled signal [k], using the Matlab stem function. Explain your results.

b) Non-ideal Channel. The channel is non-ideal with an impulse response of c(t) = δ(t) + 0.3δ(t − Tsyb ), where Tsyb  = 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{(t)} and quadrature-phase Im{(t)} of the complex-value signal (t). Show the amplitude versus time plots of the transmitted symbol sequence x[k] and the received sampled signal [k], using the Matlab stem function. Explain your results.

c) Compare and discuss the eye diagrams (for both the real and imaginary parts) of the ideal and non-ideal channels above.

d) Equalisation. For the case of non-ideal channel in c), implement an equaliser y[k] = [k] − 0.3[k − 1] + 0.09[k − 2] − 0.027[k − 3] + 0.0081[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(b) 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 (t), and show the amplitude versus time plots of the transmitted symbol sequence x[k] and the received sampled signal [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 (t), and show the amplitude versus time plots of the transmitted symbol sequence x[k] and the received sampled signal [k]. Explain your results.

c) Equalisation. For the case of non-ideal channel in b), implement an equaliser

y[k] = [k] − 0.3[k − 1] + 0.09[k − 2] − 0.027[k − 3] + 0.0081[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. Explain your results.

4 Adaptive Equalisation [12 marks]

The symbol-rate sampled channel output [k] is given by

[k] = h1 x[k] + h2 x[k − 1] + e[k]

where the two complex-valued channel paths are given by:

h1  = −0.1 + 0.8i  and  h2  = 0.7

while x[k] is the transmitted 8PSK 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] =      a[k − τ] = a0(←)[k] + a 1(←)[k − 1] + a2(←)[k − 2]

τ =0

is used to detect the transmitted symbol x[k].

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] − y[k]|2] .

What is the corresponding MMSE value? 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]) [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 for 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.