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Unformatted text preview: 6.450 Introduction to Digital Communication October 9, 2002 MIT, Fall 2002 Lecture 10: Waveforms as vectors in signal-space The set of L 2 functions, viewed as a vector space, is usually called signal-space . The signal-space viewpoint is one of the foundations of modern digital communication and the credit for popularizing this viewpoint is often given to the classic text of Wozencraft and Jacobs (1965). We start by giving the axioms of a vector space. This might appear to be somewhat more formal and mathematical than necessary, but our objective is to be able to use common geometric insights about two and three dimensional Euclidean space to reason about signal space. In order to do this, we need to show that signal space (with the appropriate operations) is in fact a vector space. We also need to know which of our insights about Euclidean space actually come from the vector space axioms. 1 Vector spaces A vector space is a set V of elements defined over a field. The elements of the field are called scalars. The only two fields of interest here are the familiar real number field R and complex number field C . A vector space with real scalars is called a real vector space and one with complex scalars is called a complex vector space . In either case, the scalars can be added, multiplied, etc. according to the well known rules of R or C . Neither C nor R include . The most familiar example of a real vector space is R n . Here the vectors are n-tuples of real numbers. R 2 is represented geometrically by a plane where the vectors are points within the plane and R 3 is represented geometrically by 3 dimensional space. The most familiar example of a complex vector space is C n , the set of n-tuples of complex numbers. The axioms below apply to real vector spaces, complex vector spaces, and vector spaces over all other fields. In reading the axioms, keep R 2 , viewed as points on a plane, in mind and observe that the axioms are saying very natural and obvious statements about addition and scaling of such points. Then reread the axioms thinking about C 2 . If you read the axioms very carefully, you will observe that they say nothing about the important geometric ideas of length or angle. We remedy that shortly. The axioms of a vector space V are listed below: (a) Addition: For each v V and u V , there is a vector v + u V called the sum of v and u satisfying (i) Commutativity: v + u = u + v , (ii) Associativity: v + ( u + w ) = ( v + u ) + w for each v , u , w V , 1 (iii) There is a unique V such that v + = v for all v V , (iv) For each v V , there is a unique- v such that v + (- v ) = . (b) Scalar multiplication: For each scalar and each v V there is a vector v V called the product of and v satisfying (i) Scalar associativity: ( v ) = ( ) v for all scalars, , , and all v V , (ii) Unit multiplication: for the unit scalar 1, 1 v = v for all v V ....
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This note was uploaded on 10/07/2009 for the course ENSC 5210 taught by Professor Daniellee during the Spring '08 term at Simon Fraser.
- Spring '08