Digital Signal Processing Reference
In-Depth Information
function of
X
i
has the form
1
π
S
i
f
X
i
(
X
i
;
β
,S
i
)
=
2
,
−∞
<
X
i
<
∞
,
(5.21)
S
i
+
(
X
i
−
β)
and where
K
√
W
i
>
S
i
=
0
,
i
=
1
,
2
,
...
,N
.
(5.22)
A larger value for the weight
W
i
(smaller scale
S
i
) makes the distribution of
X
i
more concentrated around
, thus increasing the reliability of the sample
X
i
.
Note that the special case when all the weights are equal to unity corresponds
with the sample myriad at the nominal scale factor
K
, with all the scale factors
reducing to
S
i
=
β
K
.
Again, the location estimation problem considered here is closely related
to the problem of filtering a time series
{
X
(
n
)
}
using a sliding window. The
output
Y
,attime
n
, can be interpreted as an estimate of location based
on the input samples
(
n
)
. Further, the aforementioned
model of independent but not identically distributed samples captures the
temporal correlations usually present among the input samples. To see this,
note that the output
Y
{
X
1
(
n
)
,X
2
(
n
)
,
...
,X
N
(
n
)
}
,asanestimate of location, would rely more on
(give more weight to) the sample
X
(
n
)
, when compared with samples that are
further away in time. By assigning varying scale factors in modeling the input
samples, leading to different weights (reliabilities), their temporal correlations
can be effectively accounted for.
The weighted myriad smoother output
ˆ
(
n
)
β
(
)
W
,
X
is defined as the value
that maximizes the likelihood function
i
=
1
K
β
f
X
i
(
X
i
;
β
,S
i
)
. Using Equa-
tion 5.21 for
f
X
i
(
X
i
;
β
,S
i
)
leads to
N
S
i
ˆ
β
K
(
W
,
X
)
=
arg max
β
2
,
S
i
+
(
X
i
−
β)
i
=
1
which is equivalent to
1
2
X
i
N
−
β
S
i
=
ˆ
β
(
W
,
X
)
arg min
β
+
K
i
=
1
N
[
K
2
2
]
=
arg min
β
+
W
i
(
X
i
−
β)
(5.23)
i
=
1
=
(β).
arg min
β
P
(5.24)
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