Signal Processing - January 2016 - 131
of combination schemes. The combin[table 5] interfilter communication schemes
ation of an arbitrary number of filters
(exemPlified for combinations of two filters).
can be mainly used to:
gradual transfer [47]
simPle coPy [48]
feedback [49]
■ Simplify the selection of a paramfiLteR CoMMuniCation w 2 (n) ! ,w 2 (n) + (1 - ,) w 1 (n)
w 2 (n) ! w 1 (n)
w 1 (n), w 2 (n) ! w (n)
eter. For instance, paying attention
Condition
m (n) $ m 0
m (n) $ m 0 and n = kN 0 n = kN 0
to the tracking scenario depicted in
PaRaMeteRs
0 1 , 1 1; 0 % m 0 1 1
N 0 $ 2; 0 % m 0 1 1
N0 $ 2
Figure 2(a), the robustness of the
scheme would be increased for
inclusion of an additional mechanism (similar to the sigmoidal
Tr {Q} 2 10 -3 if a third filter with step size equal to one were
activation) to make all mixing parameters remain positive and add
incorporated to the combination [50].
up to one. This scheme was proposed by [51] and [19] as an exten■ Alleviate several compromises simultaneously. For
sion to the standard convex combination of two adaptive filters,
instance, regarding the selection of the step size, n, and the
and obtains the combined output as
length of an adaptive filter, M, we can combine four adaptive
filters with settings {n 1, M 1}, {n 2, M 1}, {n 1, M 2}, and
K
{n 2, M 2} . Another example was proposed in [25] and it is
y (n) = / m k (n) y k (n) .
(22)
k=1
included in the section "Acoustic Echo Cancelation," where a
combination of several filters (linear and nonlinear) is
As in the case of combining two filters, instead of adapting
designed to alleviate simultaneously the compromise related
directly the K mixing parameters, K auxiliary parameters
with the step size selection, and with the presence or absence
a k (n) are updated following a gradient descent algorithm. The
of nonlinearities in the filtering scenario.
relation between m k (n) and a k (n) is based on the softmax actiIn the literature, two different approaches for the combination
vation function
of several adaptive filters have been proposed, both for affine and
exp [a k (n)]
convex approaches. These schemes differ in the topology
(23)
m k (n) = K
, for k = 1, ..., K.
/ exp [a j (n)]
employed to perform the combination as described next.
j=1
thE hiErarChiCaL sChEME
This approach combines K adaptive filters employing different
layers, where only combinations of two elements are considered at
a time. For instance, the output of a hierarchical combination of
four filters depicted in Figure 9(a) reads:
y (n) = m 21 (n) {m 11 (n) y 1 (n) + [1 - m 11 (n)] y 2 (n)}
+ [1 - m 21 (n)] {m 12 (n) y 3 (n) + [1 - m 12 (n)] y 4 (n)}, (20)
where m ij (n) refers to the mixing parameter of the jth combination in the ith layer. All mixing parameters are adapted to minimize the power of the local combined error. For their update, we can
follow the same adaptive rules (convex or affine) reviewed previously for the case of a combination of two filters.
thE onE-LayEr sChEME
We can combine an arbitrary number of filters employing an alternative approach based on one-layer combination as depicted in
Figure 9(b). Focusing on affine combinations of M adaptive filters
with outputs y k (n), with k = 1, ..., K, [19] and [50] proposed two
one-layer combination schemes whose output is given by
y (n) =
/ m k (n) y k (n) + =1 - / m k (n)Gy K (n) .
K-1
K-1
k=1
k=1
(21)
Different adaptive rules were proposed in the literature to
update m k (n), with k = 1, ..., K - 1, following, for instance, LMS
or RLS approaches [19], or estimating the K - 1 affine mixing
parameters as the solution of a least-squares problem [50]. The
incorporation of convex combination constraints forces the
This activation function is a natural extension of the sigmoid used
in the binary case to map several real parameters to a probability
distribution [52], as required by a convex combination, where all
parameters must remain positive and sum up to one.
Although both multifilter structures can improve the performance beyond the combination of just two filters, one useful characteristic of hierarchical schemes is its ability to extract more
information about the filtering scenario from the evolution of the
mixing parameters, since each combination usually combines two
adaptive filters that only differ in the value of a setting (step size,
length, etc.), and the combination parameter selects the best of
both competing models.
reduced-cost combinations
Combination schemes require running two or more filters in parallel, which may be a concern in applications in which computational cost is at a premium. In many situations, however, the
additional computational cost of adding one or more filters can be
made just slightly higher than the cost of running a single filter.
In the following sections, we describe a few methods to reduce the
cost of combination schemes.
usE a Low-Cost fiLtEr as CoMpanion to a
high-Cost onE
Although straightforward, several useful results fall in this class.
For example, consider the previously mentioned case of a combination of an RLS and an LMS filter. This structure enhances the
tracking performance of a single RLS filter, and only requires a
modest increase in computational complexity. A lattice implementation of RLS has a computational cost of about 16 M
IEEE SIGNAL PROCESSING MAGAZINE [131] jANuARy 2016
Table of Contents for the Digital Edition of Signal Processing - January 2016
Signal Processing - January 2016 - Cover1
Signal Processing - January 2016 - Cover2
Signal Processing - January 2016 - 1
Signal Processing - January 2016 - 2
Signal Processing - January 2016 - 3
Signal Processing - January 2016 - 4
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Signal Processing - January 2016 - Cover3
Signal Processing - January 2016 - Cover4
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