Digital Signal Processing Reference
In-Depth Information
Statistical Characterization of Signals
and System
s
The statistical characterization of signals and systems provides an impor-
tant framework of concepts and mathematical tools that are fundamental to
the modern theory of filtering and signal processing. In signal theory, we
denote by statistical signal processing the field of study that treats signals as
stochastic processes. The word stochastic is etymologically associated with
the notion of randomness. Even though such notion gives rise to different
interpretations, in our field of study, randomness is related to the concept of
uncertainty. Uncertainty on its turn is present in the essence of information
signals in their different forms as well as in the several types of disturbances
that can affect a system.
The subject of statistical characterization of signals and systems is really
extensive and has been built along more than two centuries, as a result
of classical works on statistical inference, linear filtering, and information
theory. Nevertheless, the purpose of this chapter is rather objective and,
in a way, unpretentious: to present the basic foundations and to empha-
size some concepts and tools that are necessary to the understanding of the
next chapters. With this aim in mind we have chosen five main topics to
discuss:
•
the sake of systemizing such theory, we first consider signals that do
not have randomness in their nature.
•
most methods to be presented in the topic tend to be implemented
using this approach.
•
to introduce the suitable tools to deal with random signals. The main
definitions and properties are exposed.
•
with some useful properties. An appendix on the correlation matrix
properties complements the subject.
•
a major area of statistical signal processing with strong connections
with that of optimal filtering, which is the subject of the following
chapter.
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