Digital Signal Processing Reference
In-Depth Information
In particular, the filter excess mean-square error, defined by
E
e
a
(
EMSE
=
lim
i
)
→∞
i
corresponds to the choice
S
=
R
u
since, by virtue of the independence
Assumption 1.15, E
e
a
(
2
i
)
=
E
w
i
−
1
R
u
. In other words, we should select
σ
as
)
−
1
vec
σ
=
(
I
−
F
(
R
u
).
emse
On the other hand, the filter mean-square deviation, defined as
2
MSD
=
lim
i
E
w
i
→∞
=
is obtained by setting
S
I
, i.e.,
)
−
1
vec
σ
msd
=
(
I
−
F
(
I
).
Let
{
emse
,
msd
}
denote the weighting matrices that correspond to the vectors
{
σ
emse
,
σ
msd
}
, i.e.,
vec
−
1
vec
−
1
=
(σ
)
,
=
(σ
).
emse
emse
msd
msd
Then we are led to the following expressions for the filter performance:
2
2
v
EMSE
=
µ
σ
Tr
(
Q
)
,
emse
(1.30)
2
2
v
MSD
=
µ
σ
Tr
(
Q
).
msd
Alternatively, we can also write
2
2
v
vec
T
2
2
v
vec
T
)
−
1
vec
EMSE
=
µ
σ
(
Q
)σ
emse
=
µ
σ
(
Q
)(
I
−
F
(
R
u
)
,
(1.31)
2
2
v
vec
T
2
2
v
vec
T
)
−
1
vec
MSD
=
µ
σ
(
Q
)σ
msd
=
µ
σ
(
Q
)(
I
−
F
(
I
).
While these steady-state results are obtained here as a consequence of vari-
ance Relation 1.21, which relies on independence Assumption 1.15, it turns
out that steady-state results can also be deduced in an alternative manner that
does not rely on using the independence condition. This alternative deriva-
tion starts from Equation 1.10 and uses the fact that E
2
in
steady state to derive expressions for the filter EMSE; the details are spelled
out in References 11 and 12.
2
w
i
=
E
w
i
−
1
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