Digital Signal Processing Reference
In-Depth Information
of length
N
20. From the figure it is clear that the greater separation between
the later exponential samples is captured in the fuzzy SR matrix.
=
2.3.3.4 Element Invariant Property
The fuzzy samples and the fuzzy order statistics constitute the same set and the SR
relation is invariant to the fuzzy ordering.
Analytically, this property means if
x
i
is the
j
th order statistic, i.e.,
x
i
=
x
,
(
j
)
.
9
This property is a direct consequence of the definitions of the
fuzzy samples and the fuzzy order statistics. As a result, a one-to-one mapping
can be established between
then
x
i
=
x
(
)
j
X
={
x
1
, x
2
,
...
, x
N
}={
x
, x
,
...
, x
)
}
and
(
1
)
(
2
)
(
N
X
, its image
in
X
is the weighted average of all the elements of
X
, where the weight of
each element is its membership function value with respect to
x
. Moreover,
the element and its image have the same spatial and rank indexes in each set.
X
={
x
1
,x
2
,
...
,x
N
}={
x
,x
,
...
,x
)
}
. For each element
x
∈
(
1
)
(
2
)
(
N
2.3.3.5 Order Invariant Property
The fuzzy order statistics have the same rank orders as their crisp counterparts if
the membership function obeys certain conditions, which are satisfied by Gaussian,
triangular, and uniform membership functions.
This property guarantees that the rank of a fuzzy order statistic's value is
consistent with its rank index, i.e.,
x
(
1
)
≤
x
(
N
)
.
9
With this property,
the fuzzy samples and the crisp samples have the same internal rank order
relations. This property holds if and only if the membership function
x
(
2
)
≤
...
µ
(
·
,
·
)
R
is such that
)
=
µ
(
x, t
+
t
)
R
C
(
x, t,
t
µ
(
x, t
)
R
is a monotonically nondecreasing function of
x
0.
This constraint controls the decay of the membership function and is satisfied
by many common membership functions, such as the Gaussian, triangular,
and uniform functions.
∈
, for
∀
t
,
t
∈
,
t
≥
2.3.3.6 Distribution of Fuzzy Order Statistics
Since the fuzzy order statistics are formed as weighted averages of crisp sam-
ples, the distributions of fuzzy order statistics contain hybrid characteristics
of both the crisp order statistics and the local means. The contribution of each
component is jointly controlled by the membership function spread and the
local distribution of the crisp samples. Thus, the distribution of a fuzzy order
statistic can be approximated by a linear combination of the distribution of
the corresponding crisp order statistic and the local mean,
10
))
λ
f
x
(
j
)
(
f
x
(
j
)
(
x
)
=
(
1
−
p
(
x
x
)
+
p
(
x
)
f
(
x
)
,
(2.31)
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