# notes8 - ECE 562 Fall 2011 September 23, 2011 Optimum...

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ECE 562 Fall 2011 September 23, 2011 Optimum Receiver for Linear Memoryless Modulation and Performance Linear modulated signals are complex one-dimensional signals s m ( t ) = p E m e m g ( t ) m = 1 , 2 , . . . , M, 0 t T s where g ( t ) is the unit energy pulse shaping function, which is the basis for the signal space. Thus the suﬃcient statistic, when symbol m is sent, is given by: R = h r, g i = p E m e m + W = s m + W where W is CN (0 , N 0 ). Thus p m ( r ) ∼ CN ( s m , N 0 ), i.e., p m ( r ) = 1 πN 0 exp ± - | r - s m | 2 N 0 ² Assuming equal priors, ˆ m MPE ( r ) = arg max m p m ( r ) = arg min | r - s m | 2 Thus the optimum detector is a Minimum Distance detector, i.e., we pick the signal s m that is closest in distance to r . We saw in class how to obtain the optimum decision regions in the ( r I , r Q ) space. Let Γ m denote the region where a decision in favor of symbol m is made. Probability of (symbol) error

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notes8 - ECE 562 Fall 2011 September 23, 2011 Optimum...

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