Advanced Topics and New Directions - A Rapid Introduction to Adaptive Filtering - page 113

in [ 6 ], local energy relations were found for the APA [ 5 ]. This relations guarantee a
local robust behavior (robust at each time instant).

6.2 Variable Step Size/Regularization Adaptive Algorithms

As we have seen in Chap. 4 , the choice of the step size poses a tradeoff between

speed of convergence and steady state error. To overcome this tradeoff, it seems
natural to look for variable step size strategies. In this way, we expect to have a
large step size at the beginning of the adaptation process (to improve the speed of
convergence) and then decrease it as we get closer to the steady state (to allow for a
smaller final misadjustment).
Instead of using e 2

μ

as a measure of the proximity to the steady state (and
adjust the stepsize accordingly), in [ 7 ] the authors proposed to use e

( n )

.If
the noise is i.i.d., the proposed measure would be a better estimator of the proximity
to the steady state. In addition, several VSS algorithms may not work very reliably
since they depend on several parameters that are not simple to tune in practice. To
overcome this issue, a VSS algorithm is proposed in [ 8 ]. Remember that the NLMS
can be seen as a VSS-LMS algorithm where the step size is chosen to nullify the a
posteriori error. But as we mentioned, this would cause the filter to compensate the
presence of the noise v

( n ) e ( n − 1 )

so that the power of
the a posteriori error is equal to the one of the noise. This idea was further extended
to the APA [ 9 ].
Several of the VSS algorithms proposed in the literature suffer from a drop in per-
formance when the SNR is low. Large noise sample lead to large estimation errors, so
a small value of

( n ) . Then, the idea in [ 8 ] is to find μ( n )

would be required to keep the algorithm stable. However, when the
noise samples present small energy, the small value of

μ

wouldmake the convergence
unnecessarily slow. In this way, the family of robust VSS algorithms introduced in
[ 10 - 13 ], can improve the steady state performance without compromising the speed
of convergence (see Sect. 6.4 for further details).
Finally, it should be noticed that instead of varying the step size

μ

μ , the regular-

ization parameter could be varied. This is particularly suitable for APA, where

the use of regularization can be essential when dealing with large filters and highly
colored nonstationary signals, as in acoustic echo cancelation. The interested reader
can find more information on variable step size algorithms in [ 14 - 16 ] and variable
regularized algorithms in [ 5 , 17 - 19 ].

δ

6.3 Sparse Adaptive Filters

In several system identification applications, the system to be estimated is sparse .
That is, it presents a small number of large coefficients, while the rest of them are
either zero or have small magnitudes. In other words, these systems have the property