en error probabilities are difficult to evaluate and SO easily computed bounds are sometimes used. In the following Bhattacharyya and Chernoff bounds for various decision rules are examined. Maximum A Posteriori (MAP) The decision rule that minimizes PE is the MAP rule, (4B.6) 22(z) which satisfies the inequalities (4B.7) 1 -u(s)= - (4B.8) Appendix 4B. Factor of One-Half in Error Bounds for any a > 0, , B > 0. These inequalities are typically used to derive the bounds (4B.9) (4B.10) Next, define the function (4B.11) Then an upper bound on the average error probability is PEB(1-B)1-B(a - (4B.12) for any a > 0, B > 0 where in general a and B are chosen to minimize these bounds. The special case where a=B=1 results in the Bhattacharyya bound [2] PE<2V (4B.13) since (4B.14) In most cases of interest such as when f1(z) = f2(-z) for all Z (4B.15) a = 1/2 minimizes the function B(a). When f1(z) and f2(z) are conditional probabilities derived from a communication channel model, this is usually the case. Let us now re-examine the general form for the average error probability General Analysis of Anti-Jam Communication Systems using the MAP decision rule. Note that (4B.16) Following Hellman and Raviv [9], for any a > 0,b > 0 and 0 <a<1 (4B.17) This yields the upper bound on the average error probability = -(B(a). (4B.18) If the minimizing choice of a is in the unit interval [0, 1] then this bound is always a factor of one-half smaller than the bound given in (4B.12). In par- ticular, for the Bhattacharyya bound where a = 1/2, this bound (Hellman and Raviv) shows that there is always a factor of one-half, (4B.19) Thus, the commonly used Bhattacharyya bound (often used in transfer function bit error bounds for convolutional codes) can be tightened by a fac- tor of one-half. Maximum Likelihood (ML) The ML decision rule, (4B.20) tends to keep both conditional probabilities closer in value but only mini- mizes PE when T1 = TT2 = 1/2, the equal a priori probability case. In general, inequalities (4B.21) (4B.22) result in conditi