20111ee114_1_hw5_sol

20111ee114_1_hw5_sol - EE114, Winter 2011 Problem Set 5...

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Unformatted text preview: EE114, Winter 2011 Problem Set 5 Solution ________________________________________________________________________ Problem Set #5 Solution 1) Let ( 29 n m a , and ) , ( n m b be two discrete, aperiodic sequences as specified below. The underscore indicates the location of the origin. Find the convolution ( 29 ( 29 ( 29 n m b n m a n m c , , , = ( 29 ( 29 3 1 2 1 4 1 , 1 1 2 1 1 2 ,-- = +- + =- = n m b n m a ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ---- = --- +--- +- +- = -- + ---- + -- = -- + --- + -- = 3 1 2 3 4 4 4 1 6 8 1 1 2 3 1 2 3 1 1 6 4 2 2 4 1 4 2 8 1 1 2 3 1 2 1 4 1 3 1 2 1 4 1 6 2 4 2 8 2 3 1 2 1 4 1 * 1 3 1 2 1 4 1 1 3 1 2 1 4 1 2 , * , n m b n m a 2) Consider a continuous 2D signal of the form ( 29 [ ] y x y x f 3 4 2 cos ) , ( + = . Suppose you wish to design a sampling/reconstruction system that fails to satisfy the Nyquist requirement, and that gives the aliased signal ( 29 ( 29 [ ] y x y x f 2 2 cos , 2-- = after sampling and reconstruction. What should the sampling frequency of ( 29 y x F F , in this system be? After aliasing, 1 F on the x-dimension is reconstructed as EE114, Winter 2011 Problem Set 5 Solution ________________________________________________________________________ x r kF F F + = 1 1 where k is an integer chosen so that 2 2 1 x r x F F F - . We are given that 4 1 = F , and 1 1- = r F . So we have: k F kF kF kF F F x x x x r 5 5 4 1 1...
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20111ee114_1_hw5_sol - EE114, Winter 2011 Problem Set 5...

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