Page 114 - Fundamentals of Communications Systems

3.28 Chapter Three

Problem 3.14. For two independent random variables X 1 and X 2 are transformed
into a third random variable Y = X 1 + X 2 . Show

.
(a) m Y = m X 1 + m X 2
2
2
2
(b) σ = σ X 1 + σ .
Y
X 2
Problem 3.15. Consider two random samples, X 1 at the input to a communication
system, and X 2 , at the output of a communication system and model them
as jointly Gaussian random variables. Plot the conditional probability density
(x 1 |x 2 ) for
function, f X 1 |X 2
(a) m 1 = m 2 = 0, σ 1 = σ 2 = 1, and ρ = 0
(b) m 1 = m 2 = 0, σ 1 = 2 σ 2 = 1, and ρ = 0
(c) m 1 = m 2 = 0, σ 1 = σ 2 = 1, and ρ = 0.8
(d) m 1 = 1 m 2 = 2, σ 1 = σ 2 = 1, and ρ = 0
(e) m 1 = 1 m 2 = 2, σ 1 = 2 σ 2 = 1, and ρ = 0.8
(f) Rank order the five cases in terms of how much information is available in
the output, X 2 , about the input X 1 .

Problem 3.16. Consider a uniform random variable with the PDF given in Eq.
(3.16) with a = 0 and b = 1.

(a) Compute m Y .
(b) Compute σ X .
(c) Compute P (X ≤ 0.4).
(d) Use rand in Matlab and produce 100 independent samples of a uniform
random variable. How many of these 100 samples are less than or equal to
0.4? Does this coincide with the results you obtained in (c)?.
Problem 3.17. The received signal strength in a wireless radio environment is of-
ten well modeled as Rayleigh random variable (see Eq. (3.25)). For this problem
assume the received signal strength at a particular location for a mobile phone
is a random variable, X, with a Rayleigh distribution. A cellular phone using
frequency modulation needs to have a signal level of at least X = 20mV to op-
erate in an acceptable fashion. Field tests on campus have shown the average
signal power is 1mW (in a 1 system).

2
(a) Using Eq. (3.25) compute the average signal power, E[X ], as a function of
σ X .
(b) Compute the probability that a cellular phone will work at a randomly
chosen location on campus.
(c) Assume you physically move and obtain an independent received signal
strength. What is the probability that your cellular phone will work at least
one out of two times?