Signal Processing - January 2016 - 138

VF (µ1 and µq)

VF (µ2 and µq)

LF (µ1)

LF (µ2)

Combination (CK)

ERLE (dB)

25
20
15
10
5

Mixing Parameters

Only Linear Distortion
LNLR = ∞ dB

Low Nonlinear Distortion
LNLR = 25 dB
(a)

High Nonlinear Distortion
LNLR = 0 dB

1
λ1(n)
λ2(n)

0.5

60
Time (s) →
(b)

120

[fig16] the performance of the proposed scheme as a function of the linear-to-nonlinear ratio (lnlr) of echo powers, including an
abrupt change in the room impulse response at t = 60 s. (a) the echo return loss enhancement (erle) achieved by the cks scheme and
of all possible filters (linear and vfs) using its constituent kernels. (b) time evolution of the mixing parameters. in this experiment, we
employ laplacian speech-like colored noise as input signal; experiments with real speech can be found in [25].

linear and nonlinear filters. In addition, the algorithm shows a
suitable reconvergence when the room impulse response abruptly
changes, thanks to the combination of two linear kernels with different step sizes, as it can be seen after t = 60 s.
conclusions and oPen Problems
Combinations of adaptive filters constitute a powerful approach to
improve the performance of adaptive filters. In this article, we
have reviewed some of the most popular combination schemes,
highlighting their theoretical properties and limits. Practical algorithms to combine adaptive filters need to implement estimation
methods to adjust the combination layer, taking into account the
possibly time-varying conditions in which the filter operates.
We have reviewed several of the methods that have been proposed in the literature, paying special attention to gradient methods. Power-normalized algorithms are particularly interesting,
since they simplify the selection of parameters and result in a more
robust behavior when the statistics of the filtering scenario are
(partly) unknown, which is frequently the case. The versatility of
the approach has been demonstrated through several examples in a
variety of applications. We have seen that, in all studied scenarios,
combination schemes offer competitive performance when compared to state-of-the-art methods for each application. This fact,
together with the inherent simplicity of the approach, make combination structures attractive for demanding applications requiring

enhanced performance, as illustrated by examples and references
given in the article.
In our opinion, some of the most interesting open problems to
be addressed are:
■ the selection and design of component filters with reduced
cross-EMSE, with the goal to minimize the overall EMSE
■ providing and exploiting strategies to reduce the cost of
combined schemes, trying to develop new structures whose
complexity is close to that of an individual filter
■ extensions to other application domains where different kinds
of compromises and performance tradeoffs may be present.
acknowledgments
The work of Jerónimo Arenas-García and Luis Azpicueta-Ruiz was
partially supported by the Spanish Ministry of Economy and Competitiveness (under projects TEC2011-22480 and PRIPIBIN-2011-1266. The work of Magno M.T. Silva was partially
supported by CNPq under Grant 304275/2014-0 and by FAPESP
under Grant 2012/24835-1. The work of Vítor H. Nascimento was
partially supported by CNPq under grant 306268/2014-0 and
FAPESP under grant 2014/04256-2. The work of Ali Sayed was
supported in part by NSF grants CCF-1011918 and ECCS1407712. We are grateful to the colleagues with whom we have
shared discussions and coauthorship of papers along this research
line, especially Prof. Aníbal R. Figueiras-Vidal.

IEEE SIGNAL PROCESSING MAGAZINE [138] jANuARy 2016

Table of Contents for the Digital Edition of Signal Processing - January 2016