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5_homework5 - EE 211A Fall Quarter, 2011 Instructor: John...

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1 EE 211A Digital Image Processing I Fall Quarter, 2011 Handout 15 Instructor: John Villasenor Homework 5 Due: Thursday, 3 November 2011 Reading: Textbook pp. 146 – 180. 1. Consider a “random vector” u consisting of two vectors below that occur with equal probability. Ο Ο Π Ξ Μ Μ Ν Λ = 1 2 0 u and Ο Ο Π Ξ Μ Μ Ν Λ− = 0 2 1 u (a) Determine the autocorrelation matrix u R (not the covariance matrix) of . u (b) Find the orthonormal eigenvectors and associated eigenvalues of . u R (c) Give the matrix , * t φ which is used to obtain the forward KL transform of . u (d) Obtain the KL transforms n t n u v * = for . 1 , 0 = n Verify that 0 v represents an expansion of 0 u in terms of the KL transform basis functions, i.e. that , ) 1 ( ) 0 ( 1 0 0 0 0 v v u + = where n are columns of , or equivalently, conjugate of the rows of . * t (e) Find , v R the autocorrelation matrix of the vectors , n v and confirm that ), ( k v Diag R λ = where k are the eigenvalues of . u R (f) Now find 0 w and , 1 w where 0 w and 1 w are obtained by taking the unitary DFT of 0 u and 1 u respectively.
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This note was uploaded on 12/27/2011 for the course EE211A 211A taught by Professor Villasenor during the Spring '11 term at UCLA.

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5_homework5 - EE 211A Fall Quarter, 2011 Instructor: John...

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