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Unformatted text preview: 6.079/6.975 S. Boyd & P. Parrilo October 2930, 2009. Midterm exam This is a 24 hour takehome 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 1. 2D filter design. A symmetric convolution kernel with support { ( N 1) , . . . , N 1 } 2 is characterized by N 2 coecients h kl , k, l = 1 , . . . , N. These coecients will be our variables. The corresponding 2D frequency response (Fourier transform) H : R 2 R is given by H ( 1 , 2 ) = h kl cos(( k 1) 1 ) cos(( l 1) 2 ) , k,l =1 ,...,N where 1 and 2 are the frequency variables. Evidently we only need to specify H over the region [0 , ] 2 , although it is often plotted over the region [ , ] 2 . (It wont matter in this problem, but we should mention that the coecients h kl above are not exactly the same as the impulse response coecients of the filter.) We will design a 2D filter ( i.e. , find the coecients h kl ) to satisfy H (0 , 0) = 1 and to minimize the maximum response R in the rejection region rej [0 , ] 2 , R = sup  H ( 1 , 2 )  . ( 1 , 2 ) rej (a) Explain why this 2D filter design problem is convex. (b) Find the optimal filter for the specific case with N = 5 and rej = { ( 1 , 2 ) [0 , ] 2  1 2 + 2 2 W 2 } , with W = / 4. You can approximate R by sampling on a grid of frequency values. Define ( p ) = ( p 1) /M, p = 1 , . . . , M. (You can use M = 25.) We then replace the exact expression for R above with R = max { H ( ( p ) , ( q ) )   p, q = 1 , . . . , M, ( ( p ) , ( q ) ) rej } ....
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This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.
 Fall '06
 boyde

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