
5.15. Let X (t) = A sin (ω
0
t + Φ) , where ω
0
is a constant and A and Φ are random variables with
marginal probability density functions given by
f
A
(a) = W
0
/2 for −W
0
≤ a ≤ W
0
f
Φ
(φ) = 1/2π for 0 < φ ≤ 2π.
Find the autocorrelation function and the spectral density, R
XX
(τ) and S
XX
(ω) .
Solution: The autocorrelation function is given by
R
XX
(τ) = E
$
A
2
sin (ω
0
t + Φ) sin (ω
0
(t + τ) + Φ)
%
.
Assuming that A and Φ are independent,
R
XX
(τ) =
W
0
−W
0
2π
0
a
2
sin (ω
0
t + φ) sin (ω
0
(t + τ) + φ)
W
0
2
1
2π
dφda
=
W
4
0
6
cos ω
0
τ.
Then
S
XX
(ω) =
1
2π
∞
−∞
R
XX
(τ) exp (−iωτ) dτ
=
1
2π
∞
−∞
W
4
0
6
cos ω
0
τ · exp (−iωτ) dτ
=
W
4
0
12
[δ(ω −ω
0
) + δ(ω + ω
0
)].
This integral was evaluated by replacing the cos ω
0
τ by