onal error bounds (4B.23) PE B(a). (4B.24) Appendix 4B. Factor of One-Half in Error Bounds The average error probability is simply (4B.25) The choice a = = 1/2 which often minimizes this bound yields the usual Bhattacharyya bound PEB1. (4B.26) Again using the inequality (4B.17) a tighter bound is as follows: = max{,() (4B.27) for Osas1.For the case and a = 1/2, we again have a reduction of the bound by a factor of one-half when compared with (4B.26). Most cases of interest have equal a priori prob- abilities where the MAP and ML decision rules are the same. Maximum-Likelihood Metric and Chernoff Bounds Assume that Z is some sort of metric used to make the decision such that when Z = Z, choose H <0 choose H. (4B.28) The decision function is then (4B.29) and the conditional errors are (4B.30) (4B.31) For any a > 0 and any B > 0 1 - u(z) <e-az (4B.32) u(z) < eBz (4B.33) General Analysis of Anti-Jam Communication Systems resulting in the Chernoff bounds (4B.34) (4B.35) Thus, the averaged error probability is PE() 2(B) (4B.36) where (4B.37) (4B.38) Jacobs [8] considered the conditions f1(-z) = f1(z) all (4B.39a) f2(-z) = f2(z) all z  . (4B.39b) Then using the inequality 1 all w (4B.40) and a change of variables of integration, he derived the following inequali- ties: G(a))= (4B.41) (4B.42) and similarly (4B.43) Appendix 4B. Factor of One-Half in Error Bounds Thus, the often satisfied conditions given by Jacobs in (4B.39) result in a fac- tor of one-half in the usual Chernoff bounds. Less restrictive but more difficult to prove conditions are that (4B.44a) (4B.44b) where a* minimizes C1(a) and B* minimizes C2(B). Note that for the special case of a* = 0 (4B.45) which is always satisfied when (4B.46) Indeed, conditions (4B.44) are also true for some non-negative range of a values. We assume it is true for the minimizing choices of the Chernoff bound parameters. Note that conditions (4B.44) are less restrictive than those of (4B.39) since (4B.39) implies (4B.44). Now consider the inequalities, =2/`fi(z)dz = 2PE1 (4B.47) 