book_5 - Chapter 5 Vector spaces and signal space In the...

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Unformatted text preview: Chapter 5 Vector spaces and signal space In the previous chapter, we showed that any L 2 function u ( t ) can be expanded in various orthog- onal expansions, using such sets of orthogonal functions as the T-spaced truncated sinusoids or the sinc-weighted sinusoids. Thus u ( t ) may be specified (up to L 2 equivalence) by a countably infinite sequence such as { u k,m ; < k, m < } of coecients in such an expansion. In engineering, n-tuples of numbers are often referred to as vectors , and the use of vector notation is very helpful in manipulating these n-tuples. The collection of n-tuples of real numbers is called R n and that of complex numbers C n . It turns out that the most important properties of these n-tuples also apply to countably infinite sequences of real or complex numbers. It should not be surprising, after the results of the previous sections, that these properties also apply to L 2 waveforms. A vector space is essentially a collection of objects (such as the collection of real n-tuples) along with a set of rules for manipulating those objects. There is a set of axioms describing precisely how these objects and rules work. Any properties that follow from those axioms must then apply to any vector space, i.e. , any set of objects satisfying those axioms. R n and C n satisfy these axioms, and we will see that countable sequences and L 2 waveforms also satisfy them. Fortunately, it is just as easy to develop the general properties of vector spaces from these axioms as it is to develop specific properties for the special case of R n or C n (although we will constantly use R n and C n as examples). Fortunately also, we can use the example of R n (and particularly R 2 ) to develop geometric insight about general vector spaces. The collection of L 2 functions, viewed as a vector space, will be called signal space . The signal- space viewpoint has been one of the foundations of modern digital communication theory since its popularization in the classic text of Wozencraft and Jacobs[35]. The signal-space viewpoint has the following merits: Many insights about waveforms (signals) and signal sets do not depend on time and fre- quency (as does the development up until now), but depend only on vector relationships. Orthogonal expansions are best viewed in vector space terms. Questions of limits and approximation are often easily treated in vector space terms. It is for this reason that many of the results in Chapter 4 are proved here. 141 Cite as: Robert Gallager, course materials for 6.450 Principles of Digital Communications I, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 142 CHAPTER 5. VECTOR SPACES AND SIGNAL SPACE 5.1 The axioms and basic properties of vector spaces A vector space V is a set of elements, v V , called vectors , along with a set of rules for operating on both these vectors and a set of ancillary elements called scalars . For the treatment here, the ....
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book_5 - Chapter 5 Vector spaces and signal space In the...

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