
5.25. Let
Z (t) = X (t) Y (t) ,
where X (t) and Y (t) are independent random processes. Derive the spectral density function of Z (t)
in terms of the spectral density functions of X (t) and Y (t) .
Solution: The autocorrelation function is given by
R
ZZ
(τ) = E {X (t) Y (t) X (t + τ) Y (t + τ)}
= E {X (t) X (t + τ ) Y (t) Y (t + τ )}
= E {X (t) X (t + τ )}E {Y (t) Y (t + τ)},
where the last simplification uses the property of independent processes. Then,
S
ZZ
(ω) = F (R
XX
(τ) R
Y Y
(τ))
=
1
2π
∞
−∞
R
XX
(τ) R
Y Y
(τ) e
−iωτ
dτ
=
1
2π
∞
−∞
R
XX
(τ)
∞
−∞
S
Y Y
(α) e
iατ
dα
e
−iωτ
dτ
=
1
2π
∞
−∞
∞
−∞
R
XX
(τ) S
Y Y
(α) e
−i(ω−α)τ
dαdτ.
Letting ω −α = β, then
S
ZZ
(ω) =
1
2π
∞
−∞
∞
−∞
R
XX
(τ) S
Y Y
(ω −β) e
−