An Introduction to Statistical Signal Processing by Gray R.M., Davisson L.D.

By Gray R.M., Davisson L.D.

This quantity describes the fundamental instruments and strategies of statistical sign processing. At each degree, theoretical rules are associated with particular purposes in communications and sign processing. The e-book starts off with an outline of simple chance, random items, expectation, and second-order second conception, via a wide selection of examples of the preferred random strategy versions and their uncomplicated makes use of and houses. particular purposes to the research of random signs 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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The countably infinite version of DeMorgan’s “laws” of elementary set theory (see appendix A) require that if Fi , i = 1, 2, . . are all members of a sigma-field, then so is ∞ i=1 Fi = ∞ c Fic . i=1 It follows by similar set-theoretic arguments that any countable sequence of any of the set-theoretic operations (union, intersection, complementation, difference, symmetric difference) performed on events must yield other events. Observe, however, that there is no guarantee that uncountable operations on events will produce new events; they may or may not.

An alternative proof follows by observing that since F and G are disjoint, 1F ∪G (r) = 1F (r) + 1G (r) and hence linearity of integration implies that P (F ∪ G) = 1F ∪G (r)f (r) dr = (1F (r) + 1G (r))f (r) dr = 1F (r)f (r) dr + 1G (r)f (r) dr = P (F ) + P (G). This property is often called the additivity property of probability. , that the integral of the sum of two functions is the sum of the two integrals. Repeated application of additivity for two events shows that for any finite collection {Fk ; k = 1, 2, .

20) that is, the whole sample space considered as a set must be in F; that is, it must be an event. 3 Probability Spaces 29 F is in order. If the set F is a subset of Ω, then we write F ⊂ Ω. If the subset F is also in the event space, then we write F ∈ F. Thus we use set inclusion when considering F as a subset of an abstract space, and element inclusion when considering F as a member of the event space and hence as an event. Alternatively, the elements of Ω are points, and a collection of these points is a subset of Ω; but the elements of F are sets — subsets of Ω, — and not points.

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