Digital Signal Processing Reference
In-Depth Information
is diagonal. In thi
s
w
a
y,
assume that
{
,
}
will be fully characterized by
σ
}
their diagonal entri
es
.
Th
us let
{
σ
,
denote
M
×
1 vectors that collect the
diagonal entries of
{
,
}
, i.e.,
σ
=
σ
=
diag
()
,
diag
(
).
Then from Equation 1.36 we find that
σ
=
F
σ
,
where
F
is the
M
×
M
matrix
F
=
I
−
µ
A,
A
=
2
.
Repeating the arguments that led to Equation 1.22 w
e
can then establish that,
for sufficiently small step sizes, the evolution of E
2
σ
w
i
is described by the
following
M
-dimensional state-space model:
2
2
W
i
= F W
i
−
1
+
µ
σ
v
Y
,
(1.37)
where the
M
×
1 vectors
{W
i
,
Y}
are defined by
E
g
2
[
u
i
]
2
σ
2
E
w
i
u
i
σ
/
E
g
2
[
u
i
]
2
F
2
F
E
w
i
u
i
σ
/
σ
.
.
W
=
,
Y =
,
(1.38)
i
E
g
2
[
u
i
]
E
w
i
2
F
M
−
2
u
i
2
F
M
−
2
σ
/
σ
E
g
2
[
u
i
]
2
F
M
−
1
2
F
M
−
1
σ
/
E
w
i
u
i
σ
and the
M
×
M
coefficient matrix
F
is given by
0
1
0
0
1
0
0
0
1
F =
0
,
(1.39)
0
0
1
−
p
0
−
p
1
−
p
2
...
−
p
M
−
1
{
p
i
}
where
th
e
are the coefficients of the characteristic polynomial of
F
.Ifwe
σ
=
(
)
select
vec
I
, then
2
σ
=
2
U
T
2
2
w
i
w
i
=
w
i
=
w
i
because
U
is orthogonal. In this case, the top entry of
W
i
will describe the
evolution of the filter MSD.
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