
9.5. Determine the probability density function f
˙
R
( ˙r) of the time derivative of the envelope process for
a narrow-band Gaussian process with zero mean.
Solution: From Section 9.2.4, it is found that
f
R
˙
R
(r, ˙r) =
r
√
2πσ
2
X
*
σ
2
˙
X
−ω
2
m
σ
2
X
exp
−
1
2
1
r
2
σ
2
X
+
˙r
2
σ
2
˙
X
−ω
2
m
σ
2
X
2
r ≥ 0, −∞ < ˙r < ∞
The marginal density can be obtained by
f
˙
R
( ˙r) =
∞
0
f
R
˙
R
(r, ˙r) dr
=
1
√
2π
*
σ
2
˙
X
−ω
2
m
σ
2
X
exp
−
1
2
˙r
2
σ
2
˙
X
−ω
2
m
σ
2
X
for −∞ < ˙r < ∞,
which is a Gaussian process.
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