Digital Signal Processing Reference
In-Depth Information
To ensure that the error will not exceed some given value
, more than
N
quantized
sample values should be averaged. It follows from Equations (12.43) and (12.44)
that
ε
12
q
2
t
2
1
β
ε
2
|
N
≥
σ
|
+
.
(12.45)
x
2
This inequality can be used to calculate number
N
of signal samples that have to
be processed to ensure with probability
β
that the random estimation error will
ε
not exceed the predetermined value
.
To evaluate the efficiency of the estimation method, it can be compared with
others, specifically with those based on extremely rough randomized quantizing
and on multibit deterministic quantizing. The variances of the estimates
m
|
x
|
obtained in those cases are given, respectively, by the following equations:
1
N
Var[
m
|
x
|
r
]
q
=
m
|
x
|
(
q
−
m
|
x
|
)
,
(12.46)
1
N
σ
2
|
Var[
m
|
x
|
id
]
q
=
|
,
(12.47)
x
2
where
m
|
x
|
is the variance of the rectified
signal. The indexes
r
and
id
indicate that the corresponding estimates are obtained
by applying the extremely rough randomized and the extremely fine idealized
deterministic quantizers respectively.
Obtaining of the estimate
m
|
x
|
id
by processing the signal samples, taken without
quantizing errors, is more complicated, because in this case multibit rather than
one-bit numbers have to be averaged. To obtain the estimates
m
|
x
|
r
and
m
|
x
|
id
,it
is necessary to take and to process
N
r
is the absolute mean value and
σ
|
|
x
N
id
signal samples respectively. It follows
from Equations (12.46) and (12.47) that
,
t
2
β
ε
N
r
≥
2
m
|
x
|
(
q
−
m
|
x
|
)
,
(12.48)
t
2
β
ε
2
|
N
id
≥
σ
|
.
(12.49)
x
2
Although the absolute mean values of signals are not measured very frequently,
this signal processing operation can fairly often be found in more complicated
algorithms. The given analysis shows what happens if the signals to be esti-
mated are quantized randomly, pseudo-randomly and deterministically. It can be
seen from the given relationships that application of pseudo-randomized quan-
tizing allows an estimation to be performed in a very simple way and the loss
