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

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Unformatted text preview: 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 · 2sinh( ax ) ≥ , 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 . ,h [1] = 0 . 2 ,h [2] =- . 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 ± . 1 ± . 2 ∓ . 1 (a) The maximum probability of error occurs when the ISI detracts most from the desired signal, for...
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HW07 - University of Illinois Spring 2011 ECE 361: Problem...

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