Download PDF by Gray Robert: An Introduction to Statistical Signal Processing last

By Gray Robert

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Additional resources for An Introduction to Statistical Signal Processing last edition

Example text

We assume for convenience that the sample times are ordered in increasing fashion. Let {Fki ; i = 0, 1, . . , 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.

3: Upper left: {(x0 , x1 ) : x0 ∈ (1, 3)}, Upper right{(x0 , x1 ) : x1 ∈ (3, 6)}, Lower left{(x0 , x1 ) : x1 ∈ (4, 5) ∪ (−∞, −2)}, Lower right {(x0 , x1 ) : x0 ∈ (1, 3); x1 ∈ (3, 6)}, One- and two-dimensional events in two-dimensional space. one-dimensional sets in 2 , the two-dimensional Euclidean plane. Note, for example, that the unit circle {(x, y) : x2 + y 2 ≤ 1} is not a one-dimensional set since it requires simultaneous constraints on two coordinates. More generally, for a fixed finite k the product sigma-field B( )Zk (or simply B( )k ) of k−dimensional Euclidean space k is the smallest sigmafield containing all one-dimensional events of the form {x = (x0 , x1 , .

In fact, the notations are interchangeable; we could denote waveforms as {x(t); t ∈ } or as {xt ; t ∈ }. The notation using subscripts for sequences and parentheses for waveforms is the most common, and we will usually stick to it. Yet another notation for discrete time signals is x[n], a common notation in the digital signal processing literature. It is worth remembering that vectors, sequences, and waveforms are all just indexed collections of numbers; the only difference is the index set: finite for vectors, countably infinite for sequences, and continuous for waveforms.

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An Introduction to Statistical Signal Processing last edition by Gray Robert

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