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MIT6_079F09_midterm

MIT6_079F09_midterm - 6.079/6.975 October 2930 2009 S Boyd...

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6.079/6.975 S. Boyd & P. Parrilo October 29–30, 2009. Midterm exam This is a 24 hour take-home midterm exam. Please turn it in to Professor Pablo Parrilo, (Stata Center), on Friday October 30, at 5PM (or before). You may use any books, notes, or computer programs ( e.g. , Matlab, CVX), but you may not discuss the exam with anyone until October 31, after everyone has taken the exam. The only exception is that you can ask us for clarification, via email. Please address your emails to both professors and the TA . Please make a copy of your exam before handing it in. When a problem involves computation you must give all of the following: a clear discussion and justification of exactly what you did, the Matlab source code that produces the result, and the final numerical results or plots. Matlab files containing problem data are available on Stellar. All problems have equal weight. Some are easier than they might appear at first glance. Be sure to check your email and the course web site on Stellar often during the exam, just in case we need to send out an important announcement. And one technical comment. For problems that require you to work out a numerical solution, you are welcome to use a solution method that involves solving more than just a single convex optimization problem. (Of course, only when this is necessary.) 1
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1. 2D filter design. A symmetric convolution kernel with support {− ( N 1) , . . . , N 1 } 2 is characterized by N 2 coefficients h kl , k, l = 1 , . . . , N. These coefficients will be our variables. The corresponding 2D frequency response (Fourier transform) H : R 2 R is given by H ( ω 1 , ω 2 ) = h kl cos(( k 1) ω 1 ) cos(( l 1) ω 2 ) , k,l =1 ,...,N where ω 1 and ω 2 are the frequency variables. Evidently we only need to specify H over the region [0 , π ] 2 , although it is often plotted over the region [ π, π ] 2 . (It won’t matter in this problem, but we should mention that the coefficients h kl above are not exactly the same as the impulse response coefficients of the filter.) We will design a 2D filter ( i.e. , find the coefficients h kl ) to satisfy H (0 , 0) = 1 and to minimize the maximum response R in the rejection region Ω rej [0 , π ] 2 , R = sup | H ( ω 1 , ω 2 ) | . ( ω 1 2 ) Ω rej (a) Explain why this 2D filter design problem is convex. (b) Find the optimal filter for the specific case with N = 5 and Ω rej = { ( ω 1 , ω 2 ) [0 , π ] 2 | ω 1 2 + ω 2 2 W 2 } , with W = π/ 4. You can approximate R by sampling on a grid of frequency values. Define ω ( p ) = π ( p 1) /M, p = 1 , . . . , M. (You can use M = 25.) We then replace the exact expression for R above with R ˆ = max {| H ( ω ( p ) , ω ( q ) ) | | p, q = 1 , . . . , M, ( ω ( p ) , ω ( q ) ) Ω rej } . Give the optimal value of R ˆ . Plot the optimal frequency response using plot_2D_filt(h) , available on the course web site, where h is the matrix containing the coefficients h kl . 2
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2. Gini coefficient of inequality. Let x 1 , . . . , x n be a set of nonnegative numbers with positive sum, which typically represent the wealth or income of n individuals in some group. The Lorentz curve is a plot of the fraction f i of total wealth held by the i poorest individuals, i f i = (1 / 1 T x ) x ( j ) , i = 0 , . . . , n,
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