Digital Signal Processing Reference
In-Depth Information
FIR
2
( )
q
1
y(n)
x(n)
FIR
2
( )
q
2
FIR
2
( )
q
r
FIGURE 7.6
Parallel-cascade filter.
quadratic term:
r
r
r
i
]
r
i
x
)
=
q
i
[
x
T
q
i
y
i
(
(
)
=
(
)
(
)
y
n
n
n
n
,
(7.9)
i
=
1
i
=
1
where
x
T
r
i
x
y
i
(
n
)
=
(
n
)
r
i
=
(
n
)
(7.10)
is the output of a linear FIR filter with impulse response described by the
coefficients of the vector
r
i
. This implies that if the kernel matrix describing
a second-order Volterra filter has rank equal to
r
, then it can be implemented
by a filter bank with
r
channels. Each channel is formed with a linear filter
followed by a squaring function and a multiplier, as shown in Figure 7.6.
A good approximation of the second-order Volterra filter can be frequently
obtained by using only
m
channels, with
m
r
, and thus allowing significant
computational savings.
30
Tw oproblems arise with the adaptation procedures
applied to this type of filter. The first is that the minimum-error solution is
not unique and, consequently, the filter coefficients can oscillate between the
different solutions. The second problem is the presence of local minima, again
because the output of the filter is not linear with respect to the parameters to
be updated. One remedy to the first problem consists in constraining the first
i
1 coefficients of the
i
th branch to zero and setting the
i
th coefficient to 1.
These coefficients do not need updating. The decomposition obtained in this
way is known as the LDL
T
−
decomposition. It is obtained by expressing the
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