Optimal probability-of-error thresholds and performance for two versions of the sign detector

Expressions are obtained for specifying the optimal error probability (minimum P/sub e/) thresholds lambda /sub 01/ and lambda /sub 02/ for the traditional and modified sign detectors, respectively. These thresholds are shown to depend on the parameters p, P/sub 1/, and M where: M is the number of observations z/sub i/ used in the test statistic; P/sub 1/=P(H/sub 1/) is the prior probability for hypothesis H/sub 1/ that signal s/sub 1/ is present and 1-P/sub 1/=P(H/sub 0/) corresponds to the hypothesis H/sub 0/ that signal s/sub 0/ is present; and p=Pr(z/sub i/>or=0 mod H/sub 1/) with s/sub 0/=0 for the traditional sign detector and p=Pr(z/sub i/>or= lambda mod H/sub 1/)=Pr(z/sub i/< lambda mod H/sub 0/) with lambda =(s/sub 0/+s/sub 1/)/2 for the modified sign detector. The expressions for lambda /sub 01/ and lambda /sub 02/, are given explicitly, and shown to be independent of P/sub 1/ for sufficiently large M. Optimal P/sub e/ versus M performance curves, corresponding to both versions of the sign detector, are obtained for a representative range of values for p and P/sub 1/.<
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