Digital Signal Processing Reference
In-Depth Information
If the noise sample
v
to
w
T
will be affected
and the performance of the algorithm will be severely degraded. That is, the LMS
is highly sensitive to large perturbations. Situations where large perturbations in
the noise realization could appear with high probability are common.
1
A typical
example is the case of acoustic echo cancellation [
20
], where in the noise realization
v(
(
n
)
is large, the closeness of
w
(
n
−
1
)
there might be a component associated to human speech from a local speaker.
The human speech can have bursty components of high energy, which can be thought
as a realization of impulsive noise. In that application, and with a local speaker (the
double-talk
situation), the performance of algorithms like the LMS or the NLMS
2
could be very poor. Therefore, it is important to obtain
robust
algorithms. Here we
interpret the term robust as “slightly sensitive to large perturbations (outliers)”.
It is known from Chap.
2
that the SEA presents robustness (in the sense defined
above) to impulsive noise. However, we also saw that the SEA can present slow
convergence speed. In order to obtain robustness without compromising the speed
of convergence several approaches have been taken. One of them is the use of robust
statistics [
35
-
38
], where considerations about the statistics of the noise signal
n
)
)
should be taken into account. Other useful approach is the use of
mixed-norm
algo-
rithms, where the algorithms are derived from cost functions that combines
v(
n
2
|
(
)
|
e
n
2
is the cost function used to
|
(
)
|
|
(
)
|
and
e
n
e
n
obtain the LMS algorithm, and
|
e
(
n
)
|
is the cost function used to obtain the SEA. In
2
improves the speed of convergence.
The algorithms obtained are an appropriate weighted combination of the LMS (or
NLMS) and the SEA. Other useful approach is the used of
switched-norm
algorithms
[
10
]. In this way, the algorithm is able to determine if a robust behavior is needed
or if the convergence speed can be improved. This decision is only determined by
the instantaneous environment that the adaptive filter is facing, and without consid-
erations about the perturbation statistics. For further extensions of this approach the
reader can consult [
11
-
13
].
a sense,
|
e
(
n
)
|
provides robustness and
|
e
(
n
)
|
6.5 Distributed Adaptive Filtering
Wireless sensor networks (WSN) are a promising technology in various fields [
41
].
They comprises a large number of sensors which are able to communicate with
each other wirelessly using a radio transceiver. They also have limited sensing and
data processing capabilities. The sensors are deployed in large geographical areas,
without any particular planning, in order to perform a particular task in a collabo-
rative manner. Although each sensor unit has limited sensing and data processing
capabilities, they are able to perform complex tasks. They compensate their limited
individual abilities by collaborating between a large number of them. There exists
1
This is the situation when the probability density of the noise has heavy tails [
35
].
2
Or any algorithm in which the update is linear in the error filtering signal.
