Digital Signal Processing Reference
In-Depth Information
respect to
u
i
. Relation 1.87 then amounts to saying that the projections of
these vectors along the horizontal direction should have equal norms.
More generally, when a positive-definite weighting matrix
is present in
{
θ
θ
}
Equation 1.10, we let
i
,
denote acute angles whose squared cosines are
i
−
1
given by
e
a
(
)
2
e
p
(
)
2
i
i
=
=
cos
2
cos
2
(θ
)
,
(θ
)
.
(1.88)
i
−
1
i
2
2
2
2
w
i
−
1
·
u
i
w
i
·
u
i
The subscript
is used in
computing it. With this notation, it is straightforward to verify that energy
relation 1.57 becomes
in cos
(
·
)
indicates that a weighting matrix
2
sin
2
2
sin
2
w
i
−
1
(θ
i
−
1
)
=
w
i
(θ
i
)
,
(1.89)
which is a natural extension of Equation 1.87.
1.14
Concluding Remarks
This chapter describes an energy-conservation approach to studying the per-
formance of adaptive filters. By studying the energy balance at each iteration,
the dynamic behavior of an adaptive filter can be characterized in terms of
a variance relation (e.g., Equations 1.16, 1.64, and 1.65) and, subsequently, in
terms of a state-space model (e.g., Equations 1.22 and 1.73). The approach does
not restrict the input data to Gaussian or white distributions. In addition to
providing information about the stability and convergence behavior of the fil-
ter, the energy-conservation arguments also help characterize the steady-state
performance of the filter. Although the analysis in this chapter has relied on
independence Assumption 1.15, steady-state results can be obtained without
relying on this assumption (see, e.g., References 11 and 12).
Acknowledgments
This work was supported in part by the National Science Foundation un-
der grants ECS-9820765 and CCR-0208573. The work of T.Y. Al-Naffouri was
also supported by a fellowship from King Fahd University of Petroleum &
Minerals, Dhahran, Saudi Arabia.
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