By Robert Gray
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Extra resources for An Introduction to Statistical Signal Processing new edition
It is a deep and difficult result of measure theory that the Borel field of the real line is in fact different from the power set of the real line; that is, there exist subsets of the real line that are not in the Borel field. While we will not describe such a subset, we can guarantee that these “unmeasurable” sets have no physical importance, that they are very hard to construct, and that an engineer will never encounter such a subset in practice. It may, however, be necessary to demonstrate that some weird subset is in fact an event in this sigma-field.
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 there that when the one-dimensional sets constraining the coordinates are intervals, then the two-dimensional sets are rectangles. Analogous to the twodimensional example, finite-dimensional events having separate constraints on each coordinate are called rectangles. , in two dimensions we require that x20 ≤ 1 − x21 .
As we progress, however, through experience and the book, we will use the more modern style of not using specialy notation for vectors and we will simply denote a vector as a = (a0 , a1 , . . , ak−1 ). In another notational twist, when doing linear algebra it will be useful to distinguish between column vectors and row vectors. , a row vector will be written as a transpose of a column vector. For the moment, however, it makes no difference. Thus, for example, k is k−dimensional Euclidean space.
An Introduction to Statistical Signal Processing new edition by Robert Gray