Page 113 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.27

Problem 3.11. In communications the phase shift induced by propagation be-
tween transmitter and receiver, p , is often modeled as a random variable. A
common model is to have
1
(φ) = − π ≤ φ ≤ π
f p
2π
= 0 elsewhere
This is commonly referred to as a uniformly distributed random variable.
(φ).
(a) Find the CDF, F p
(b) Find the mean and variance of this random phase shift.
(c) It turns out that a communication system will work reasonably well if the
coherent phase reference at the receiver, ˆ p , is within 30 of the true value
◦
of p . If you implement a receiver with ˆ p = 0, what is the probability the
communication system will work.
(d) Assuming you can physically move your system and change the propagation
delay to obtain an independent phase shift. What is the probability that your
system will work at least one out of two times?
(e) How many independent locations would you have to try to ensure a 90%
chance of getting your system to work?
Problem 3.12. You have just started work at Yeskia, which is a company that man-
ufactures FM transmitters for use in commercial broadcast. An FCC rule states
that the carrier frequency of each station must be within 1 part per million of the
assigned center frequency (i.e., if the station is assigned a 100 MHz frequency
then the oscillators deployed must be | f c − 100 MHz| < 100 Hz). The oscillator
designers within Yeskia have told you that the output frequency of their oscilla-
tors is well modeled as a Gaussian random variable with a mean of the desired
2
channel frequency and a variance of σ . Assume the lowest assigned center
f
frequency is 88.1 MHz and the highest assigned center frequency is 107.9 MHz.
(a) Which assigned center frequency has the tightest absolute set accuracy
constraint?
(b) For the frequency obtained in (a) with σ f = 40Hz, what is the probability
that a randomly chosen oscillator will meet the FCC specification?
(c) What value of σ f should you force the designers to achieve to guarantee
that the probability that a randomly chosen oscillator does not meet FCC
−7
rules is less than 10 .
Problem 3.13. X is a Gaussian random variable with a mean of 2 and a variance
of 4.
(a) Plot f X (x).
(b) Plot F X (x).
(c) Plot the function g(x) = P (|X − 2| < x).