Linear Optimal and Adaptive Filtering - Unsupervised Signal Processing - page 100

i.e., the prediction coefficients are given by (3.105). This is an alternative way
to build the PEF and highlights that the PEF design can be seen as a problem
of linear filtering without a reference signal.
The above results open some perspectives that deserve to be further
exploited: in fact, in the problem of channel equalization, it is usual to
assume that the transmitted signal is a sequence of independent and iden-
tically distributed (i.i.d.) random variables. Therefore, the transmitted signal
is uncorrelated, which means that the equalizer works as a whitening filter.
This requirement naturally leads to the idea of considering the use of an FEP
in the equalization process, when a training sequence is not available.
Equalization without training sequence, i.e., unsupervised or blind
equalization is a central subject of this topic, which will be studied from
now on.
3.9 Concluding Remarks
In this chapter, we presented the fundamental concepts of optimal and
adaptive filtering, which constitute the theoretical basis of the studies on
unsupervised signal processing.
After a brief presentation on the principles and motivations of supervised
filtering, we exposed the classical Wiener theory in the context of the search
of optimal parameters of an FIR linear filter and derived the Wiener-Hopf
equations. As an illustration, we applied the results in some representative
examples, like system identification and channel equalization.
Instead of using the Wiener-Hopf equations, the optimal parameters may
also be obtained via iterative procedures. We developed the steepest-descent
technique, a well-established way to reach the domain of truly adaptive algo-
rithms. Based on the Robbin-Monro principle of stochastic approximation,
we derived the LMS algorithm, and briefly discussed some convergence
issues of this celebrated technique, including some recent results provided
in [136].
Afterward, we considered another family of optimal and adaptive filter-
ing methods, based on the LS criterion, in which we work with a given set of
available data and not with statistic averages. We derived the normal equa-
tions that provide the optimal LS solution for the filter parameters. By using
an appropriate update of these averages as new data is available, we derived
the recursive LS algorithm.
Although the focus of this chapter is on linear FIR filters, we provided
a brief discussion about alternative structures, including IIR and nonlinear
filters. A more in-depth presentation on nonlinear filtering will be given in
Chapter 7.