Digital Signal Processing Reference
In-Depth Information
2
where the notation
u
i
denotes the squared weighted Euclidean norm of
u
i
, specifically,
2
=
u
i
u
i
u
i
.
e
p
(
,
e
a
(
Relation 1.8 can be used to express
e
(
i
)/
g
[
u
i
] in terms of
{
i
)
i
)
}
and to
eliminate this term from Equation 1.7. Doing so leads to the equality
2
u
i
e
a
(
2
u
i
e
p
(
u
i
·
w
i
+
i
)
=
u
i
·
w
i
−
1
+
i
).
(1.9)
By equating the weighted Euclidean norms of both sides of this equation, we
arrive, after a straightforward calculation, at the relation:
+
e
a
(
)
2
+
e
p
(
)
2
2
2
2
2
u
i
·
w
i
i
=
u
i
·
w
i
−
1
i
.
(1.10)
This energy relation is an exact result that shows how the energies of the
weight-error vectors at two successive time instants are related to the energies
of the
a priori
and
a posteriori
estimation errors.* In addition, it follows from
e
(
i
)
=
u
i
w
i
−
1
+
v
(
i
)
, and from Equation 1.7, that the weight-error vector
satisfies
I
w
i
−
1
u
i
u
i
g
[
u
i
]
u
i
g
[
u
i
]
v
w
i
=
−
µ
−
µ
(
i
).
(1.11)
1.4
Weighted Variance Relation
The result (Equation 1.10) with
I
was developed in Reference 5 and
subsequently used in a series of works to study the robustness of adaptive
filters (e.g., References 6 through 9). It was later used in References 10 through
12 to study the steady-state and tracking performance of adaptive filters. The
incorporation of a weighting matrix
=
in References 13 and 14 turns out to
be useful for transient (convergence and stability) analysis.
In transient analysis we are interested in characterizing the time evolution
of the quantity E
2
w
i
, for some
of interest (usually,
=
I
or
=
R
u
). To
arrive at this evolution, we use Equation 1.8 to replace
e
p
(
i
)
in Equation 1.10
* Later in Section 1.13 we provide an interpretation of the energy relation (Equation 1.10) in terms
of Snell's law for light propagation.
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