EEEM062: Applied Mathematics for Communication Systems Semester 1 2022/3
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Faculty of Engineering & Physical Sciences
Department of Electrical and Electronic Engineering
Undergraduate Programmes in Electrical and Electronic Engineering
EEEM062: Applied Mathematics for Communication Systems
FHEQ Level 7 Examination
Semester 1 2022/3
Q1.
(a) Let there be a multiple-input multiple-output communication (MIMO) where either linear precoding is used at the transmitter, i.e. r=HW1s + n, or linear detection is used at the receiver, i.e. y=W2 (Hs + n ), where r is a received signal, y is a detected signal, H is the MIMO channel, s is the transmit signal, n is the noise and W1 as well as W2 are linear maps.
(i) Explain which of the two techniques (i.e., linear precoding or linear detection) is more likely to provide the best bit error performance, as well as the benefits and drawbacks of both techniques (in a maximum of 150 words). [15 %]
(ii) Let s= [2 − 1]T , H= 1 2(−2) 1(3)T and n= [ − 1 −4 2]T , where [.]T is the
「 1 −0.2]
transpose operator. Determine the value of r , when W1 = |L0.3 2 」| .
Justify your answer by providing all the details of your calculation. [15%]
(iii) Calculate the value of y when the minimum mean square error (MMSE) detection method is applied at the receiver, for a signal-to-noise ratio of 0 dB. Justify your answer by providing all the details of your calculations. [15%]
(iv) Let the value of the possible transmitted signals be either [2 − 1]T or [1.5 − 1.5]T. Determine values of the elements of W1 for which r is correctly detected when the transmitted signal is [2 − 1]T and the maximum likelihood (ML) detection method is applied at the receiver. Justify your answer by providing all the details of your calculations. [20%]
(b) Explain, in your own words (in max. of 100 words), the main factors that can affect the probability of error when performing detection at a receiver. [10%]
(c) Let us assume that a symbol s, s ∈ {− 12,1, 2.5, 6} , is transmitted over a wireless channel, where a noise “ is added at the receiver side, such that r = s + n . The noise follows a Gaussian probability distribution with a mean of x and a variance of 1, i.e. n → (x,1) . Assuming that all the possible values of s are equiprobable,
(i) determine the probability of detection error for the symbol s by using the Q function for the decision threshold values, 0, 1.5, and 3 when x= 0. Justify your answer by providing all the details of your calculation. [15 %]
(ii) will the probability of detection error for the symbol s increase or decrease if x= − 14 ? Justify your answer without calculation (in a max. of 100 words). [10 %]
Q2.
(a) Explain, in your own words (in a max. of 100 words), the concept of matrix rank and the meaning of “full rank” for a matrix. Then, explain and how this concept relates to MIMO communication. [10 %]
(b) Explain, in your own words (in a max. of 100 words), which concept of linear algebra is related to invariance under a linear map and how it can help with the design of techniques that mitigate the effect of small scale fading, in multi-antenna communications. [10 %]
「a b ]
(c) A linear map that is characterised by the following matrix A = |Lc d」| , outputs the
「−4] 「 1 ]
vector |L −3」| when the vector |L − 1」| is used as an input for it. Similarly, this linear map 「− 1] 「1]
outputs the vector |L 3 」| when its input is the vector |L2」| . Determine the values of a,
b, c, d. Justify your answer by providing all the details of your calculations. [20%]
(d) A multiple-input multiple-output (MIMO) system uses the Eigen decomposition
「− 1 3 ]
process to parallelise the MIMO channel H = |L 5 −2」| . The Eigen decomposition process is performed at the receiver, where one part of the decomposition is fed back at the transmitter side to precode the transmit signal s = [2 + j 1 − 2j]T and another part is used at the receiver side to equalise the received signal. Note that both parts are unit normalised. Assuming that the noise is such that n = [ −4 4]T , calculate the value of the received signal after the channel parallelisation process. Justify your answer by providing all the details of your calculations. [30%]
(e) Explain, in your own words (in a max. of 50 words), the particularity of an upper or lower triangular matrix in terms of eigenvalues. Provide a proof for the case of a generic 3 by 3 upper triangle matrix. [15%]
(f) Let A and B be two distinct invertible square matrices. Determine through derivation if the inequality A(B + A )− 1 B ≠ A - A (A + B)− 1 A is correct. Justify your answer by providing all the details of your derivation. [15%]
Q3.
(a) In probability and statistics,
(i) explain in less than 100 words what the probability density function (pdf) is and how it can be used to calculate the probability of a random variable getting values in a specific range. [10%]
(ii) If you have two random variables X and Y that are both uniformly distributed but the range of values for X is wider than the range of values for Y, what can you conclude about the maximum value of their pdf? Justify your answer in less than 100 words. [10%]
(b) In estimation theory,
(i) give two characteristics a good estimator should have and explain in your
own words why these characteristics are required. Justify your answer in less than 100 words. [10%]
(ii) explain in your own words what the Cramer Rao Bound is and how it can be used to decide how good an estimator is. [10%]
(d) Assume a telecommunication system transmitting a sequence of randomly and
independently produced 4-QAM symbols (i.e., -1-j, -1+j, 1-j, 1+j) with equal probability over at the transmission channel h = 0.5 + 0.2j, and that at the receiver there is complex-valued Additive White Gaussian Noise (AWGN) of variance σ2 = 1.
(i) Calculate the signal-to-noise ratio (SNR) at the receiver as a function of the power of the transmitted symbol Es to the power of the noise and explain the corresponding calculation steps. [15%]
(ii) Calculate the SNR at the receiver as a function of the power of the transmitted symbol Es to the power of the noise when the channel h is not of a specific value, but it is instead modelled as identically distributed samples of a zero-mean, normal distribution of variance σh(2) = 0.5. Explain all calculation steps.
Also, in less than 100 words, discuss the reasoning behind the two channel modelling approaches (i.e., h being of a specific value vs. being a random variable) and explain when each approach can be used. [15%]
(iii) Assume that the maximum variance of the transmitted signal (that is related to the maximum transmitted power) cannot be larger than one. Explain what kind of processing should take place at the transmitter to fulfil this constraint. [15%]
(iv) For the transmitted symbol of part (iii), calculate the SNR at the receiver as a function of the power of the transmitted symbol s to the power of the noise, when the AWGN variance at the receiver is one, and when the channel can be modelled in Matlab as h=0.5*(randn+j*randn). [15%]
2024-01-18