(4B.48) and similarly (4B.49) Next, for the special case where (4B.50) General Analysis of Anti-Jam Communication Systems a*=B* = (4B.51) sufficient conditions can be given by (4B.52a) (4B.52b) Note that these conditions are always satisfied if our decision rule is a max- imum-likelihood decision rule where <<f((z) for all 2 0 0 (4B.53a) f2(z) >f((z) for all z < 0. (4B.53b) Assuming conditions (4B.52) C1(a)+C2(B)C1(a*)+C2(a*) > 2PE1 2PE2 (4B.54) <19(a)+1C2(B), (4B.55) which is again a factor of one-half less than the original Chernoff bound on the average error probability (4B.36) for T1 = TT2 = 1/2. Appendix 4B. Factor of One-Half in Error Bounds For the special case where Z happens to be a maximum-likelihood met- (4B.56) then the conditions (4B.52) hold and = B(1 - a) (4B.57) = B(B) (4B.58) where B() is given by (4B.11). Recall that B(1/2) is the Bhattacharyya bound. Applications In most applications of interest, we consider two sequences of length N, X1,X2EXN (4B.59) that can be transmitted over a memoryless channel with input alphabet X and output alphabet Y and conditional probability p(y/x), x E X , yeY. (4B.60) This is shown in the following figure. Memoryless x E X p(ylx) General Analysis of Anti-Jam Communication Systems The receiver obtains a sequence (4B.61) from the channel and must decide between hypotheses H1:X1 is sent H2:X2i is sent (4B.62) which have a priori probabilities 1 = Pr{H1} and TT2 = Pr{H2}. The receiver will typically use a metric m(y,x) xEX, yeY (4B.63) and the decision rule of choosing H if and only if (4B.64) By defining (4B.65) we have the case considered in this chapter. APENDIX 4C. REED-SOLOMON CODE PERFORMANCE A Reed-Solomon (RS) code [15] is a block code with an alplhabet size of Q = 2m. Its block length is N = Q - 1s - symbols which can be extended to N = Q and N = Q + 1. Here, the performance of only the N = Q block length RS codes is presented since adjacent block lengths are only slightly different. If an RS code has r redundant symbols, then the minimum Hamming dis- tance 