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Hw6Soln_1 - ECE 5670 Digital Communications ECE department...

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ECE 5670 : Digital Communications ECE department, Cornell University, Spring 2011 Homework 6 Solutions Instructor: Salman Avestimehr Office 325 Rhodes Hall 1. (a) We know from Lecture 14 that the mean of the squared error in the estimate of the th channel coefficient is E [ | h | 2 ] 1 + E [ | h | 2 ] SNR . (1) For large enough SNR, this means that the mean of the squared error decreases as 1 SNR . (2) So, if we increase the SNR by a factor of two, the mean of the squared error decreases by a factor of a half. (b) i. We simply use of a pulse of amplitude E at every L time instances and nothing else: x [ iL ] = E, x [ iL + j ] = 0 , 0 i k - 1 and 1 j L - 1 (3) ii. Since this is just a repetition code, the optimal estimator for h l just is just ˆ h l = c l k - 1 i =0 y [ iL + l ] k , l = 0 , . . . , L - 1 (4) where c l is chosen to minimize the estimation error E [( h l - ˆ h l ) 2 ] . Note that we can write E [( h l - ˆ h l ) 2 ] = E [( h l - c l k k - 1 X i =0 ( Eh l + n [ iL + l ])) 2 ] (5) = E [( h l - c l Eh l - c l k k - 1 X i =0 n [ iL + l ]) 2 ] (6) = (1 - c l E ) E [ h 2 l ] + c 2 l k 2 k - 1 X i =0 var ( n [ iL + l ]) (7) = (1 - c l E ) 2 E [ h 2 l ] + c 2 l σ 2 k (8)
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