hw06_web

# hw06_web - Homework 6 These are all practice problems you...

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Homework 6 These are all practice problems – you do not need to submit anything. The solutions will be provided later in the week. SS-1. Problem 5.4 in Stark and Woods ( Stark and Woods problems are at the end of this PDF.) SS-2. Problem 5.8 in Stark and Woods SS-3. Problem 5.9 in Stark and Woods 1. Problem 5.10 in Stark and Woods 2. Problem 5.15 in Stark and Woods SS-4. Problem 6.55 in Leon-Garcia 3. Problem 6.57 in Leon-Garcia 4. Problem 6.61 in Leon-Garcia SS-5. Problem 5.21 in Stark and Woods 5. Problem 6.85 in Leon-Garcia 6. Do Problem 5.22 in Stark and Woods with the following clariﬁcations: After ﬁnding the whitening transform, use the inverse transform to create correlated random variables with the given covariance matrix. On the same graph as the scatter plot of the correlated random variables, plot lines showing the eigenvectors of the covariance matrix. 7. Generate 5000 pairs ( X , Y ) of zero-mean, correlated Gaussian RVs with the covariance matrix given in Problem 5.22. Consider the performance of the following two data compression approaches: (a) Keep all the values of X , but replace all the values of Y by their mean, 0. Thus, the data size is approximately reduced by half. Plot the compressed data on top of the original data, using a different marker and color. Compute the average square-error between the original data and the compressed version. (b) Use the KLT to transform ( X , Y ) into uncorrelated random variables ( W , Z ) such that Var [ W ] is maximized (under the unitary transform of the KLT). Replace all of the values of Z by their mean, 0. Again, the data size is approximately reduced by half. Apply the inverse transform to the compressed data to create ( X 0 , Y 0 ) , which are compressed versions of ( X , Y ) . Plot ( X 0 , Y 0 ) on top of the original data using a different marker and color. Compute the average square-error between ( X 0 , Y 0 ) and the original data ( X , Y ) . SS-6. For each of the following matrices, determine if it is a valid covariance matrix for some random vector. In the case that it is a valid covariance matrix, determine if the random variables with that covariance matrix are linearly independent.

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(a) K 1 = 4 1 . 6 - 1 . 6 1 . 6 1 - 0 . 5 - 1 . 6 - 0 . 5 0 . 69 4 (b) K 2 = 4 1 . 6 - 1 . 6 1 . 6 1 - 0 . 6 - 1 . 6 0 . 6 1 (c) K 3 = 4 1 . 6 - 1 . 6 1 . 6 1 0 . 5 - 1 . 6 0 . 5 1 (d) K 4 = 4 1 . 6 - 1 . 6 1 . 6 1 - 0 . 5 - 1 . 6 - 0 . 5 1 8. Let X be a zero-mean random vector with covariance matrix K = 4 1 - 1 1 2 0 . 5 - 1 0 . 5 1 (a) Specify a vector b with k b k 2 = 1 such that the variance of b T X is minimized , and give the value of that variance. (b) Specify a vector a with k a k 2 = 1 such that the variance of a T X is maximized , and give the value of that variance. (c) Give an equation to whiten X . In other words, give an equation for a random vector Y in terms of X , V , and D such that Y is a vector of uncorrelated random variables with unit variance.
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hw06_web - Homework 6 These are all practice problems you...

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