# Download An Introduction to Statistical Signal Processing new edition by Robert Gray PDF By Robert Gray

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Example text

We next consider what other events must be in F I by virtue of its being an event space. 3. PROBABILITY SPACES 41 coordinate of the vector or sequence or waveform, the next most general kinds of events are finite-dimensional sets that separately pin down the values of a finite number of coordinates. Let K be a finite collection of members of I and hence K ⊂ I. Say that K has K members, which we shall denote as {ki ; i = 0, 1, . . , K − 1}. 3} for a waveform. We assume for convenience that the sample times are ordered in increasing fashion.

PROBABILITY 42 of rectangles and hence is in the event space generated by the rectangles. ) The moral of this discussion is that the product sigma-field for spaces of sequences and waveforms must contain (but not consist exclusively of) all sets that are described by requiring that the outputs of coordinates for a finite number of events lie in sets in the one-dimensional event space F. We shall further explore such product event spaces when considering random processes, but the key points remain 1.

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.