Lecture_14 - Advanced digital communication Dr. Saad Qaisar...

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Unformatted text preview: Advanced digital communication Dr. Saad Qaisar Todays Lecture 2011 Saad Qaisar EE830: Advanced Digital Communication 2 Inner Product Projections Orthonormal Bases Grahm-Schmidt Orthonormalization Inner Product 2011 Saad Qaisar EE830: Advanced Digital Communication 3 For the vector space Cn of complex n- tuples , the inner product is defined as: The norm or length of v is then Thus, as far as length is concerned, a complex n-tuple u can be regarded as * 1 , n j j j u v v u = = 2 2 2 [ ( ) ( ) ] j j j j j v v v = + Inner Product Example 2011 Saad Qaisar EE830: Advanced Digital Communication 4 Geometric Example of inner product as angle between two vectors . One-dimensional projections 2011 Saad Qaisar EE830: Advanced Digital Communication 5 An important problem in constructing orthogonal expansions breaking a vector v into two components relative to another vector in the same inner-product space. One component, vu , is to be orthogonal (i.e., perpendicular) to u the other, v|u, is to be collinear with u. Tip: Collinear means lying on same line u One-dimensional projections 2011 Saad Qaisar EE830: Advanced Digital Communication 6 Figure illustrates this decomposition for vectors in R2 One-dimensional projection theorem: 2011 Saad Qaisar EE830: Advanced Digital Communication...
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This note was uploaded on 02/21/2012 for the course ECON 830 taught by Professor Staff during the Spring '08 term at Michigan State University.

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Lecture_14 - Advanced digital communication Dr. Saad Qaisar...

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