Digital Signal Processing Reference
In-Depth Information
1.10
Fourth-Order Moment Approximation
Instead of the small-step-size approximation of Section 1.8, we can choose to
approximate the fourth-order moment that appears in the expression for
in Equation 1.17 as
E
u
i
E
u
i
u
i
g
[
u
i
]
E
2
2
g
[
u
i
]
u
i
u
i
g
2
[
u
i
]
u
i
≈
·
=
P
Tr
(
P
)
,
would become
where
P
=
E
(
u
i
u
i
/
g
[
u
i
]
)
. In this way, Expression 1.17 for
=
−
µ
2
P
Tr
P
−
µ
P
+
µ
(
P
)
,
(1.50)
which is fully characterized in terms of the single moment
P
.Ifwenow
let
P
U
T
=
U
denote the eigen-decomposition of
P
>
0, and introduce the
transformed quantities:
U
T
U
T
=
=
=
.
w
i
w
i
,
u
i
u
i
U,
U
Then variance Relations 1.16 and 1.50 can be equivalently rewritten as
E
,
2
g
2
[
u
i
]
u
i
2
2
2
2
v
E
w
i
=
E
w
i
−
1
+
µ
σ
(1.51)
2
=
−
µ
−
µ
+
µ
Tr
().
shows that it will be diagonal as long as
The expression for
is diagonal.
Thus let again
σ
=
σ
=
diag
()
,
diag
(
).
Then from Equation 1.51 we find that
σ
=
F
σ
,
where
F
is
M
×
M
and given by
2
, B,
2
T
,
F
=
I
−
µ
A
+
µ
A
=
2
,
B
=
µ
δδ
where
. Repeating the arguments that led to Equation 1.22 we
can establish that,
un
der the assumed fourth-order moment approximation,
the evolution of E
δ
=
diag
()
2
σ
w
i
is described by an
M
-dimensional state-space model
similar to Equation 1.37.
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