Digital Signal Processing Reference
In-Depth Information
The above considerations will become clearer, and will be more rigor-
ously revisited, in the sequence of the chapters. Nevertheless, it is worth
remarking these points in this introduction to illustrate the interest in
bringing unsupervised equalization and source separation to a common
theoretical framework.
On the other hand, BSS can become a more challenging problem as the
aforementioned assumptions are discarded. The case of a mixing system
with memory corresponds to the more general problem of convolutive mix-
tures. Such a problem is rather similar to that of MIMO equalization. As a
rule in this topic, we consider convolutive BSS as a more general problem
since, in MIMO channel equalization, we usually suppose that the trans-
mitted signals have the same statistical distributions and belong to a finite
alphabet. This is not at all the case in other typical applications of BSS.
If the hypothesis of linear mixing is discarded, the solution of BSS prob-
lems will require special care, particularly in applying ICA. Such a solution
may involve the use of nonlinear devices in the separating systems, as
done in the so-called post-nonlinear model. It is worth mentioning that
nonlinear channels can also be considered in communication and different
approaches have been proposed for nonlinear equalization, including the
widely known decision feedback equalizer (DFE). Overall, our problem will
certainly become more intricate when nonlinear mappings take place in
H(
·
)
and/or in
, as we will discuss in more detail in Chapter 6.
Furthermore, other scenarios in BSS deserve the attention of researchers,
as those of underdetermined mixtures, i.e., in scenarios in which
M
W(
·
)
<
N
in
Figure 1.1; correlated sources; sparse sources, etc.
1.3 Organization and Contents
We have organized the topic as follows:
Chapter 2 reviews the fundamental concepts concerning the characteri-
zation of signals and systems. The purpose of this chapter is to emphasize
some notions and tools that are necessary to the sequence of the topic. For
the sake of clarity, we first deal with deterministic concepts and then we
introduce statistical characterization tools. Although many readers could be
familiar with these subjects, we provide a synthetic presentation of the fol-
lowing topics: signals and systems definitions and main properties; basic
concepts of discrete-time signal processing, including the sampling theorem;
fundamentals of probability theory, including topics like cumulants, which
are particularly useful in the context of unsupervised processing; a review
on stochastic processes with a specific topic on discrete-time random signals;
and, finally, a section on estimation theory.
In order to establish the foundations of unsupervised signal processing,
we present in Chapter 3 the theory of optimal and adaptive filtering in the
