By Gray Robert

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**Additional resources for An Introduction to Statistical Signal Processing last edition**

**Example text**

K − 1} be a collection of members of F. Then a set of the form {{xt ; t ∈ I} : xki ∈ Fki ; i = 0, 1, . . , 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.

Spaces with finite or countably infinite numbers of elements are called discrete spaces. 4] An interval of the real line , for example, Ω = (a, b). We might consider an open interval (a, b), a closed interval [a, b], a half-open interval [a, b) or (a, b], or even the entire real line itself. 4] that are not discrete are said to be continuous. , the space Ω = (1, 2) ∪ {4} consisting of a continuous interval and an isolated point. Such spaces can usually be handled by treating the discrete and continuous components separately.

3], that is, for sample spaces that are discrete. We shall always take our event space as the power set when dealing with a discrete sample space (except possibly for a few perverse homework problems). 5). For example, the power set of the binary sample space Ω = {0, 1} is the collection {{0}, {1}, Ω = {0, 1}, ∅}, a list of all four possible subsets of the space. Unfortunately, the power set is too large to be useful for continuous spaces. To treat the reasons for this is beyond the scope of a book at this level, but we can say that it is not possible in general to construct interesting probability measures on the power set of a continuous space.