HW07 - University of Illinois Spring 2011 ECE 361 Problem...

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University of Illinois Spring 2011 ECE 361: Problem Set 7: Solutions Communication over Band-Limited Channels 1. [The pleasure that you get is less than the pain that you have to endure] For any given fixed positive value of x , let g ( a ) denote Q ( x + a )+ Q ( x - a ) - 2 Q ( x ). Then, g ( a ) = g ( - a ) and g (0) = 0, and so let us assume without loss of generality that a > 0. Now, ∂g ( a ) ∂a = - φ ( x + a ) + φ ( x - a ) = 1 2 π exp - ( x - a ) 2 2 - exp - ( x + a ) 2 2 = 1 2 π · exp - x 2 + a 2 2 · 2 sinh( ax ) 0 , for a 0, and thus we conclude that g ( a ) is a strictly increasing function on [0 , ). Since g (0) = 0, it must be that g ( a ) > 0 for all a > 0. Hence, Q ( x + a ) + Q ( x - a ) > 2 Q ( x ) for a 6 = 0. Note that Q ( x + a ) + Q ( x - a ) 1 as a → ∞ . 2. [Error probability when ISI is ignored] We now have h [ - 1] = 0 . 1 , h [0] = 1 . 0 , h [1] = 0 . 2 , h [2] = - 0 . 1, and conditioned on the values ± E T of X [ m + 1], X [ m - 1], and X [ m - 2], the conditional error probability is P ( E X [ m + 1] , X [ m - 1] , X [ m - 2]) = Q h [0] E T + h [ - 1] X [ m + 1] + h [1] X [ m - 1] + h [2] X [ m - 2] σ = Q E T σ 1 ± 0 . 1 ± 0 . 2 0 . 1 (a) The maximum probability of error occurs when the ISI detracts most from the desired signal, for example, if X [ m ] = E T , then X [ m + 1] = X [ m - 1] = -
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