By Robert M. Gray
This quantity describes the basic instruments and methods of statistical sign processing. At each degree, theoretical rules are associated with particular purposes in communications and sign processing. The e-book starts with an summary of uncomplicated chance, random items, expectation, and second-order second concept, by way of a wide selection of examples of the preferred random procedure types and their easy makes use of and houses. particular purposes to the research of random indications and platforms for speaking, estimating, detecting, modulating, and different processing of indications are interspersed in the course of the textual content.
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Extra info for An Introduction to Statistical Signal Processing
25) correspond generally to intuitive notions, at least for the first three properties. The first property requires that a probability be a nonnegative number. In a purely mathematical sense, this is an arbitrary restriction, but it is in accord with the long history of intuitive and combinatorial developments of probability. Probability measures share this property with other measures such as area, volume, weight, and mass. The second defining property corresponds to the notion that the probability that something will happen or that an experiment will produce one of its possible outcomes is one.
Obviously, the sigma-field formed by including countable settheoretic operations does not have this defect. This is why sigma-fields must be used rather than fields. 19) can be related to a condition on limits by defining the notion of a limit of a sequence of sets. This notion will prove useful when interpreting the axioms of probability. Consider a sequence Fn , n = 1, 2, . , of sets with the property that each set contains its predecessor, that is, that Fn−1 ⊂ Fn for all n. Such a sequence of sets is said to be nested and increasing.
We wish to convey an understanding of the structure of such event spaces, but we do not wish to dwell on the technical difficulties that can be encountered. Hence we shall study only one of the possible constructions – the simplest possible definition of a product sigma-field – by making a direct analogy to a product sample space. This definition will suffice for most systems studied herein, but it has shortcomings. At this time we mention one particular weakness: the event space that we shall define may not be big enough when studying the theory of continuous time random processes.