EEEM062: Applied Mathematics for Communication Systems 2019/20
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EEEM062: Applied Mathematics for Communication Systems
2019/20
Q1.
(a) In probability theory and statistics,
(i) explain what the cumulative distribution function (cdf) ofa real-valuedrandom variable describes.
(ii) Give the equation relating the cumulative distribution function (cdf) of a real-valued
random variable to its probability density function (pdf).
(iii) Show how the expected value and the variance of a real-valued continuous random
(iv) Assume the following function
(
) = 
2 0 ≤ 
< 1
Can 
(
) be a cumulative distribution function? Explain why.
(b) In Matlab we use the function “rand” to produce samples of a uniformly distributed random
(i) Write a Matlab script that uses the function “rand” to produce a sequence of randomly generated - 1 or 1 symbols, that are statistically independent and their probability of appearance of - 1 is P(- 1)=0.25. Explain your solution.
(ii) Calculate the mean theoretical value ofthe random variable Y=3X.
(iii) Ifwe produce random samples by using the Matlab function 4*rand- 1, calculate the pdf
of these random samples and calculate the probability of producing a sample with a value smaller than 0.
Q2.
(a) In estimation theory,
(i) explain ifthere is any minimum in the variance of unbiased estimators.
(ii) Explain what a Minimum Variance Unbiased Estimator (MVUB) is and explain if a
MVUB always meets the Cramer-Rao Lower Bound.
(iii) Explain what the Best Linear Unbiased Estimator (BLUE) is and what prior knowledge
is needed in advance to determine it.
(iv) Assume that you want to estimate an unknown parameter by using a noisy received
answer.
(v) For the estimation problem of part (iv) explain if your estimator reaches the Cramer- Rao Lower Bound.
(b) Consider the observations 
[
] = 
+ 
[
] with 
= 0, … , 
− 1, where 
is a parameter to
be estimated and 
[
] is an additive white Gaussian noise ofvariance 
2 .
(i) Write the probability density function of the vector 
= [
0 … 
−1]
for a given value of 
.
(ii) Calculate the Cramer-Rao Lower Bound for this estimation problem of finding 
from
[
].
Q3.
(a) Explain (in 2 or 3 sentences) the difference between a deterministic and a random signal.
(b) Let s (t ) be a time domain signal and α be a parameter, such thatα > 1 . Match each of the
following modified mathematical definition ofs(t) (1st column ofTable Q3.b) with their correct corresponding effect on s (t ) (2nd column of Table Q3.b).
Table Q3.b
|
|
Effects |
|
a) αs (t ) |
1) Compression |
|
b) s (t ) |
2) Amplification |
|
c) s (αt ) |
3) Reflection |
|
d) s (t |
4) Expansion |
|
e) s ( −t ) |
5) Attenuation |
(c) Explain (in 1 or 2 sentences) what filtering is, in signal processing.
1, t <−7 and t ≥−1
(d) Let u (t ) = 
− (t + 6) , −7 ≤ t <−6 be a time domain signal.
(t + 2) , −2 ≤ t <−1
(i) Draw the signal u (t ) .
(ii) Draw the signal u (t − 4)and determine if it is an even signal. Justify your answer.
(e) Let rect (t ) = 
1, − 1
2 ≤ t ≤ 1
2 be a time domain signal. Derive its auto-correlation function.
(f) Does shifting a signal in the time domain modifies the content of its frequency information.
Justify your answer.
(g) Determine the type of signal that will result from the convolution of a sinus cardinal waveform
2022-01-11