Page 122 - Fundamentals of Communications Systems

3.36 Chapter Three

X and Y are uncorrelated random variables. So, here we have shown that
if X and Y are uncorrelated random variables this does not imply X and Y
are independent.

(c) The joint PDF of two Gaussian r.v’s X and Y can be written as

2
1 1 x − m X
f XY (x, y) = exp − 2
2 2(1 − ρ ) σ X
2πσ X σ Y 1 − ρ XY
XY
2

(x − m X )(y − m Y ) y − m Y
− 2ρ XY +
σ X σ Y σ Y
If X and Y are uncorrelated, then cov(X, Y ) = 0 ⇒ ρ XY = 0
2
2

1 1 1 x − m X y − m Y
f XY (x, y) = √ √ exp − +
2 σ X σ Y
2πσ X 2πσ Y
2
2
1 1 x − m X
1 1 y − m Y
= √ exp − · √ exp −
2 σ X 2 σ Y
2πσ X 2πσ Y
= f X (x) · f Y (y)
This implies that X and Y are independent.
3.6 Miniprojects
Goal: To give exposure
■ to a small scope engineering design problem in communications.
■ to the dynamics of working with a team.
■ to the importance of engineering communication skills (in this case oral
presentations).

Presentation: The forum will be similar to a design review at a company
(only much shorter). The presentation will be of 5 minutes in length with an
overview of the given problem and solution. The presentation will be followed
by questions from the audience (your classmates and the professor). Each team
member should be prepared to make the presentation on the due date.

3.6.1 Project 1
Project Goals: This problem works you through the generation of realizations
of random variables by computer.
Consider real random continuous variables.
(a) Note that if a continuous random variable has a distribution function,
F X (x), defined on the real line which is one-to-one and onto the interval