hw4 - ECE 562 Fall 2011 October 10, 2011 HOMEWORK...

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Unformatted text preview: ECE 562 Fall 2011 October 10, 2011 HOMEWORK ASSIGNMENT 4 Due Date: October 18, 2011 (in class) 1. Semi-Orthogonal Signal Set. Consider the signal set with M = 2 N signals given by: s m ( t ) = E g m ( t ) m = 1 ,...,N j E g m- N ( t ) m = N + 1 ,...,M. where { g k ( t ) } N k =1 are real-valued orthonormal functions. Clearly this signal set satis- fies: Re [ k,` ] = 0, for k 6 = ` . (a) Argue that R k = h r ( t ) ,g k ( t ) i ,k = 1 ,...,N , form sufficient statistics for optimal decision making at the receiver for an AWGN channel. (b) Now define the M real-valued statistics y m = r m,I m = 1 ,...,N r ( m- N ) ,Q m = N + 1 ,...,M. Show that the MPE decision rule is given by m MPE = arg max m y m (c) Find an expression for P e for the MPE decision rule 2. In this problem you will show the following result for M-ary orthogonal modulation lim M P c = 1 b > ln 2 b < ln 2 where P c is the probability of correct decision making. Recall that we showed in class that P c = Z - h 1- Q ( x + p 2 b log 2 M ) i M- 1 1 2 e- x 2 / 2 dx (a) Show that for any...
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hw4 - ECE 562 Fall 2011 October 10, 2011 HOMEWORK...

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