ece4305_L12

# ece4305_L12 - Gram-Schmidt Orthogonalization Procedure...

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Gram-Schmidt Orthogonalization Procedure ECE4305: Software-Deﬁned Radio Systems and Analysis Professor Alexander M. Wyglinski Department of Electrical and Computer Engineering Worcester Polytechnic Institute Lecture 12 Professor Alexander M. Wyglinski ECE4305: Software-Deﬁned Radio Systems and Analysis

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Gram-Schmidt Orthogonalization Procedure Motivation G-S Procedure An Example Orthonormal Basis Functions I Recall from Lecture 10 that { φ j ( t ) } is an orthonormal set of functions on the time interval [0 , T ] such that: T Z 0 φ i ( t ) φ j ( t ) dt = ± 1 i = j 0 otherwise I Furthermore, it is possible to represent a signal waveform s i ( t ) as the weighted sum of these orthonormal basis functions, i.e.: s i ( t ) = N X k =1 s ik φ k ( t ) (1) I How do we determine the set of orthonormal basis functions { φ j ( t ) } ? I Use Gram-Schmidt Orthogonalization Procedure Professor Alexander M. Wyglinski ECE4305: Software-Deﬁned Radio Systems and Analysis
Gram-Schmidt Orthogonalization Procedure Motivation G-S Procedure An Example Orthonormal Basis Functions I A complete orthonormal set of basis functions is needed for a set of M energy signals denoted by s 1 ( t ) ,..., s M ( t ) I Choose s 1 ( t ) and normalize it: φ 1 ( t ) = s 1 ( t ) p E s 1 where E s

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ece4305_L12 - Gram-Schmidt Orthogonalization Procedure...

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