Digital Signal Processing Reference
In-Depth Information
Fig. 4.4
MSE behavior of
the SEA and SDA for the
equalization problem
0
Sign Error Algorithm (SEA)
Sign Data Algorithm (SDA)
−5
SEA
−10
−15
SDA
−20
−25
0
500
1000
1500
2000
Number of iterations
making it difficult to measure its proximity to the Wiener solution without using
probability tools. Although it could be desirable to have results about almost sure
convergence (that is, usual convergence for almost any realization of the stochastic
processes involved [
23
]) to the Wiener solution, this will not be possible for adaptive
algorithms with fixed step size and noisy observations.
Moreover, the issue of final performance in terms of proximity to the Wiener
solution is not the only one to be considered in an adaptive filter. This is because
adaptive filters are recursive estimators in nature, which leads to the issue of their
stability [
24
-
26
]. This is very important, because the results about the asymptotic
behavior of an adaptive filter will be valid as long as the adaptive filter is stable.
The study of this issue is, in general, very difficult, basically because the adaptive
filters are not only stochastic systems but also nonlinear ones. However, with certain
assumptions about the input vectors some useful results can be derived. Sometimes
the assumptions are very mild and reasonable [
27
-
29
] while other times they are
strong and more restrictive [
30
,
31
]. Although results with milder assumptions might
be more satisfying, they are by far much more difficult to obtain. In fact, in general,
during their derivation some important intuitive ideas about the functioning of the
adaptive filter are obscured. On the other hand, results with stronger assumptions,
although of more restrictive application, are easier to derive and highlight several
important properties of the adaptive filter. In this topic we will use, in general, this
latter approach to show the properties of the adaptive filters.
Finally, for an adaptive filter it is important to analyze, besides its stability and
steady state performance, its transient behavior. That is, how the adaptive filter
evolves before it reaches its final state. As in the SD procedure, it is important how
fast this is done (i.e., the speed of convergence). And even more importantly, how
is the tradeoff between this speed of convergence and the steady state performance.
In general, the requirements of good final performance and fast adaptation are con-
flicting ones. As the transient behavior analysis of adaptive filters is mathematically
demanding (needing stronger assumptions), we shall not cover it in detail. However,
we will provide some useful remarks about it.
