By Robert M. Gray

ISBN-10: 0521838606

ISBN-13: 9780521838603

This quantity describes the fundamental instruments and strategies of statistical sign processing. At each level, theoretical principles are associated with particular functions in communications and sign processing. The ebook starts off with an summary of uncomplicated chance, random items, expectation, and second-order second idea, by means of a large choice of examples of the most well-liked random strategy versions and their simple makes use of and homes. particular functions to the research of random signs and platforms for speaking, estimating, detecting, modulating, and different processing of signs are interspersed in the course of the textual content.

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For example, the infinite two-sided space for a given A is t∈ℜ At = { all waveforms {x(t); t ∈ (−∞, ∞)}; x(t) ∈ A, all t}, with a similar definition for one-sided spaces and for time functions on a finite time interval. Note that we indexed sequences (discrete time signals) using subscripts, as in xn , and we indexed waveforms (continuous time signals) using parentheses, as in x(t). In fact, the notations are interchangeable; we could denote waveforms as {x(t); t ∈ ℜ} or as {xt ; t ∈ ℜ}. The notation using subscripts for sequences and parentheses for waveforms is the most common, and we will usually stick to it.

This is true since any countable sequence of set-theoretic operations on subsets of Ω must yield another subset of Ω and hence must be in the collection of all possible subsets. The collection of all subsets of a space is called the power set of the space. 3 Probability spaces 35 is the largest possible event space for the given sample space, because it contains every possible subset of the sample space. 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).

Although 20 Probability introduced as an example of simple signal processing, the usual name for a real-valued function defined on the sample space of a probability space is a random variable. We shall see in the next chapter that there is an extra technical condition on functions to merit this name, but that is a detail that can be postponed. The output space Ωq can be considered as a new sample space, the space corresponding to the possible values seen by an observer of the output of the quantizer (an observer who might not have access to the original space).

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