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20105ee211A_1_homework6 - EE 211A Fall Quarter 2010...

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1 EE 211A Digital Image Processing I Fall Quarter, 2010 Handout 17 Instructor: John Villasenor Homework 6 Due: Tuesday, 16 November 2010 1. For this problem we use a 2x2 matrix U defined as follows: 2 1 1 2 U - - = a) Perform the SVD decomposition on the matrix U . Calculate the matrices 1/ 2 , , and Ψ Λ Φ such that 1/ 2 T U = ΨΛ Φ b) Give the basis images of this SVD transform. Verify the result of a) using these basis images. 2. We have seen that for 1D Markov inputs, the KL transform becomes the DCT as the correlation ρ becomes 1. A critical question, however, concerns how fast this convergence occurs. In other words, for ρ values less than 1, how different are the transform domain energies between the KL and DCT? A zero mean 1 st order stationary Markov sequence can be generated using ( ) ( 1) ( ) u n u n n ρ ε = - + where ( ) n ε is a zero-mean and white random variable. Derive, in terms of ρ , the variance of ( ) n ε so that 2 [ ( )] 1 E u n = .
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