By Gray Robert
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Additional resources for An Introduction to Statistical Signal Processing last edition
K − 1} is an example of a finite-dimensional set. Note that it collects all sequences or waveforms such that a finite number of coordinates are constrained to lie in one-dimensional events. 3(d). Observe that when the one-dimensional sets constraining the coordinates are intervals, then the two-dimensional sets are rectangles. Analogous to the two-dimensional example, finite-dimensional events having separate constraints on each coordinate are called rectangles. , in two dimensions for a circle of radius one we require that x20 ≤ 1 − x21 .
It may, however, be necessary to demonstrate that some weird subset is in fact an event in this sigma-field. This is typically accomplished by showing that it is the limit of simple Borel sets. In some cases we wish to deal not with a sample space that is the entire real line, but one that is some subset of the real line. In this case we define the Borel field as the Borel field of the real line “cut down” to the smaller space. Given that the sample space, Ω, is a Borel subset of the real line , the Borel field of Ω, denoted B(Ω), is defined as the collection of all sets of the form F ∩ Ω, for F ∈ B( ); that is, the intersection of Ω with all of the Borel sets of forms the class of Borel sets of Ω.
Stochastic systems theory is the combination of systems theory with probability theory. The essence of stochastic systems theory is the connection of a system to a probability space. Thus a precise formulation and a good understanding of probability spaces are prerequisites to a precise formulation and correct development of examples of random processes and stochastic systems. Before proceeding to a careful development, several of the basic ideas are illustrated informally with simple examples.
An Introduction to Statistical Signal Processing last edition by Gray Robert