Digital Signal Processing Reference
In-Depth Information
autocorrelation function:
1
|
|
≤
−|
t
/
t
0
|
for
t
t
0
,
p
x
(
t
)
=
(12.20)
|
|
>
0
for
t
t
0
.
The corresponding spectral density function is given by
x
sin
2
4
σ
ω
t
0
G
xx
(
ω
)
=
,
(12.21)
t
0
ω
2
σ
2
where
x
is the variance of the signal. Substituting Equation (12.20) into Equation
(12.19) (after omitting manipulations) yields
⎧
⎨
2
0
3 (
A
−
1)
Θ
ω
for
Θ
∈
[
Θ
,Θ
2
]
,
0
t
0
+
3
Θ
2
α
−
3
Θ
t
0
N
=
(12.22)
⎩
A
/α
for
Θ
≥
Θ
,
2
where
+
√
1
(1
−
4
α/
3)
t
0
Θ
=
(12.23)
0
2
α
and
τ
At
0
α
is the estimation time, corresponding to the estimation of
A
c
,
max
Θ
2
=
=
α
μ
x
by processing un-
correlated signal samples.
Equation (12.22) can be used to define the optimal values of
N
and
Θ
with
regard to the criterion
N
Θ.
Denote the respective optimal values of
N
and
Θ
by
N
(1)
(1)
opt
. Then
opt
and
Θ
+
√
1
(1
−
α
)
t
0
(1)
opt
=
Θ
=
2
Θ
0
,
α
(12.24)
+
√
1
)
2
3(
A
−
1) (1
−
α
2 (
A
−
1)
)
=
N
(1)
opt
=
3
√
1
.
α
α
(3
−
2
α
+
−
α
It follows from Equation (12.24) that, under the conditions in question, the optimal
sampling interval is given by
(1)
opt
N
(1)
=
Θ
t
0
=
T
(1)
1
.
(12.25)
opt
A
−
opt
