Principles of
Communication
Systems
Assignment 5
DR. Sabbaghian
Question 1
Which of the following functions can be valid autocorrelation functions?
Justify your answer using the fundamental properties of autocorrelation functions.
1. R4 (τ ) = sin(2πf0 τ )
2. R5 (τ ) = τ 2
3.R1 (τ ) = 2 cos(10πτ
) + cos(30πτ )
4.R3 (τ ) = Λ τ2 + 2 Π(τ )
Question 2
Show that for a Rayleigh random variable X with probability density function
x2
fX (x) = xσ 2 e− 2σ2 , x > 0
the mean and variance are given by
r
π
π 2
E[X] = σ
,
Var(X) = 2 −
σ .
2
2
Question 3
Obtain the autocorrelation function, the corresponding power spectral density
(PSD), and the total average power of the following signals:
1:g1 (t) = A cos(20πf0 t + θ0 )
2: g2 (t) = 3 + 6 cos(20πt) + 3 sin(20πt)
Question 4
Give an example of two processes X(t) and Y(t) for which RXY (t + τ, t) is a
function of τ but X(t) and Y(t) are not stationary
1