EEEM062: Applied Mathematics for Communication Systems 2018/9
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EEEM062: Applied Mathematics for Communication Systems
2018/9
Q1.
(a) Explain (in 1 or 2 sentences) what a vector is, and its main difference with a scalar number.
(b) Explain (in 1 or 2 sentences) the relationship between vectors and bases.
(c) Explain (in 1 or 2 sentences) what a linear map is, and provide an example of a linear map.
(d) Let A and B be two matrices. Which of the following identities are true or false in the general case?
1. AB = BA , 2. A B = B A , 3. ABH = AHBH , 4. A 1 + BT = A 1 A + BBT , 5. AT 1 = A 1 T . Note that . T and . H stand for the transpose and transpose conjugate operators, respectively.
(e) Let u 1 2j 3 4j j 1 jT be a vector with complex elements. Compute its
transpose conjugate, its Euclidean norm, and its inverse. Then, compute the outer product of u and v 1 j 2 jT .
1 1 1
(f) Let A 1 1 1 . Determine the null space of the linear map associated to A .
0 1 1
(g) Determine if the dimension of the null space of any full-rank matrix A can be greater than
one. Justify your answer.
Q2.
(a) Explain (in 1 or 2 sentences) the meaning and aim of detection in the context of
communication.
(b) Match each of the following categories of detection methods (1st column of Table Q2.b) to the
statement that best describes it (2nd column of Table Q2.b).
Table Q2.b
Detection Methods |
Descriptions |
a) Soft |
1) Iterative and list-based detections are part of this category |
b) Coherent |
2) Detection outcomes belong to a finite set |
c) Linear |
3) Assign a probability (likelihood) to an outcome |
d) Non-linear |
4) Require prior knowledge |
e) Hard |
5) Relies on matrix theory |
(c) Explain (in 1 or 2 sentences) what the Q function is and what its main usage in communication systems is.
(d) Explain (in 2 or 3 sentences) the difference between Zero-Forcing (ZF) and the Minimum- Mean-Square-Error (MMSE) detection methods.
4.2 0.4
1.8 3.1
noise n= 1.6 3.7T at the receiver, where s belongs to the following set of possible transmit vectors: s 1 1 T , 1 1 T , 1 1 T , 1 1 T .
(i) Determine the value ofthe received vector r Hs n .
(ii) Determine the value ofthe detected signal when applying the ZF detection method.
(iii) Determine the value ofthe detected signal when applying the Maximum-Likelihood
(ML) detection method.
(f) Let us assume that a symbol s,s, is transmitted over a wireless channel, where noise
is added to it at the receiver such that r s+ n . The noise follows a Gaussian distribution
with a mean of 0 and a variance of β, i.e. n 0, . In which of these two cases will the
probability of detection error for the symbol s be the lowest (when assuming a decision threshold set to 0): a) 2 and 1 , or b) 1and 12 . Justify your answer.
Q3.
(a) In probability theory and statistics,
(i) show how the expected value and the variance of a real-valued continuous random variable can be computed as a function of the probability density function.
(ii) Give the mathematical definition of independent events and explain, in a sentence or
two, the meaning of statistical independence.
(iii) If two events are mutually exclusive, can they also be statistically independent?
Justify your answer.
(b) In estimation theory,
(i) explain what an unbiased estimator is.
(ii) Explain in words what the Cramer-Rao Lower Bound is.
(iii) When you want to estimate a parameter in the presence of noise, is it possible to
find the same estimator by using a different estimation method? Justify your answer.
(c) In probability theory and statistics, let be a complex random variable consisting of the real random variables and so that , with and being statistically independent, of zero mean and having a variance of two. Calculate the mean value and the variance of .
(d) In a telecommunication system assume that you have received ten samples of a signal, where the -th sample is
with being samples of zero-mean, uncorrelated noise having a variance of one, and
being an unknown, small (of some degrees per sample) frequency offset. Using the principles of the Linear Least-Squares Estimation, find an estimator for the frequency offset .
2022-01-11