Signal Processing - January 2016 - 135

LMS update, removing any ability to promote sparsity. This
scheme has shown improved performance with respect to the
standard LMS algorithm when the unknown system is highly
sparse; however, standard LMS outperforms ZA-LMS scheme
when the system is dispersive. For this reason, there exists a tradeoff related to the degree of sparsity of the system, which unfortunately is usually unknown a priori, or can even vary over time.
Here we cover two approaches specially designed to alleviate
this compromise:
■ Scheme A: An adaptive convex combination of the ZANLMS filter, i.e., a sparsity-norm regularized version of the
NLMS scheme and a standard NLMS algorithm [72]. This
approach constitutes the straightforward application of combination schemes to alleviate the compromise regarding the
selection of parameter t in the ZA-NLMS filter.
■ Scheme B: A block-wise biased adaptive filter, depicted in
Figure 12, where the outputs of Q nonoverlapping blocks of
coefficients of an adaptive filter are weighted by different scaling factors to obtain the overall output [73]
y c (n) =

Q

/ m q (n) y q (n),

(25)

q=1

where y q (n) is the partial output of each block, and
m q (n) ! [0, 1] with q = 1, ..., Q are the shrinkage factors that
adapt to minimize the power of e c (n) = d (n) - y c (n) . This
scheme manages the well-known bias versus variance compromise in a block-wise manner [73]: if the MSD in the qth
block is much higher than the energy of the unknown system
in this block, m q (n) will tend to be zero, biasing the output of
the block y q (n) toward zero but reducing the output error.

The shrinkage factors m q (n) therefore act as estimators for the
support (set of nonzero coefficients) of the filter.
Regarding the estimation of these scaling factors m q (n),
with q = 1, ..., Q, it can be shown that their optimal values lie
on the interval (0, 1) [73]. Therefore, proceeding as in [38], we
can reinterpret each element in the sum of (25) as a convex
combination between y q (n) and a virtual filter whose output
remains constant and equal to zero. This is useful, because it
implies that we can employ rules similar to cvx-LMS and cvxPN-LMS reviewed in the section "Estimating the Combination
Parameter" for adjusting parameters m q (n) . It should be noted
that, for this kind of scheme, the ability to model sparse systems
increases with the number of blocks Q, since the length of each
block decreases. However, the computational cost associated
with the adaptation of the mixing parameters also increases
with Q.
To illustrate both approaches, we have carried out a system
identification experiment with an unknown plant with 1,024
taps, whose sparsity degree changes over time considering white
noise as input signal and SNR = 20 dB . We start with a very
sparse system (only 16 active taps), that abruptly changes at
n = 40, 000 to a plant with 128 active coefficients, and finally, at
n = 80, 000, we employ a more dispersive unknown system with
512 active taps. Figure 13 shows the MSD reached by Scheme A
and Scheme B, considering two different possibilities for the
number of blocks Q.
As expected, Scheme A shows a robust behavior with respect
to the sparsity degree of the unknown plant, behaving as well as
its best component filter. Scheme B provides an attractive alternative, whose performance improves when the length of each
block is reduced. Comparing both schemes, the computational

u(n )
y1(n )

w1(n )

y2(n )

w2(n )

λ1(n )
λ2(n )

wQ(n )

+

ec(n )

λQ(n )

yQ(n )

ec(n )

+
Adaptive Filter
w(n )

e(n )

yc(n )

+
+

y (n )
+

+
d (n )

[fig12] a block diagram of scheme b. each block of the adaptive filter w (n) is represented as a complete filter computing its output
using just the indicated coefficients.

IEEE SIGNAL PROCESSING MAGAZINE [135] jANuARy 2016
Table of Contents for the Digital Edition of Signal Processing - January 2016
