Digital Signal Processing Reference
In-Depth Information
However, the obtained estimation time will be optimal, in the given sense, only
for
(1)
opt
(1)
opt
Θ
2
will be optimal,
i.e. the optimal estimation will be performed under the condition that the signal
samples are taken at the minimum time intervals for which these samples are still
uncorrelated.
In the case under discussion this is
T
Θ
∈
[
Θ
0
,Θ
2
].For
Θ
≥
Θ
2
, the observance time
=
t
0
.
Processing of uncorrelated sam-
ples is optimal for
A
≤
2. For
A
>
2, the optimal sampling interval is equal
to
T
(1)
.
opt
Example 12.1
The input signal is characterized by
X
10
−
6
s. Therefore the signal
/σ
x
=
10
,
t
0
=
bandwidth is
f
0
=
0
.
5
/
t
0
=
500 kHz. The mean value
μ
x
of the signal should be
10
−
3
. The confidence probability
estimated with the relative random error
ε
≤
β
=
0
.
95(
t
β
=
2). The signal is quantized roughly and only two threshold levels
are used.
Under the given conditions,
10
−
6
Θ
=
0
.
04 s
,
A
=
3
.
083
,
α
=
25
×
,
Θ
2
=
0
.
12 s
,
N
2
=
123 320
,
N
2
Θ
2
=
14 798 s
,
N
(
1
)
(
1
)
opt
N
2
Θ
2
Θ
opt
(1)
opt
N
(1)
Θ
=
0
.
08 s
,
=
166 640
,
=
0
.
9
.
opt
The sampling frequency corresponding to the optimal estimation conditions is
f
s
=
10
3
Hz
2
.
083
×
.
12.2.2 Simplifying Hardware
The complexity of ADCs and the hardware used for processing digitized signals
first of all depends on the number of bits of the corresponding quantizer. It is
therefore of considerable practical interest to determine the minimum number of
quantizer threshold levels at which it is still possible to solve the given estimation
task with the required accuracy.
Criterion Nz
Minimizing this criterion allows the best conditions to be determined for estimat-
ing the parameter
x
by processing a relatively small number of few-bit signal
samples. Assume that the signal autocorrelation function is given by Equation
(12.20) and that sampling is performed in such a way that the signal samples
taken are uncorrelated. The variable
z
is a positive integer. The conditions are
μ
