 as long as Xn # Xn. Thus, the pair- wise error probability is bounded by (4.19) where w(x, X) is the number of places where x # x n = 1,2,...,N.T = This is sometimes called the Hamming distance [1]. Since (4.19) applies for all 1 0, define D = min D(A) = min E{exp(Xm(y,xz)-m( (4.20) x x x which gives the final form of the pairwise error probability Chernoff bound, (4.21) Note that the parameter D depends only on the coding channel and the choice of decoder metric. In deriving the above results, we assumed a metric m(y, X; z) which was arbitrary. For the special case of the maximum-likelihood metric of (4.11), (4.20) has the special form (4.22a) where the expectation is over the jammer state random variable Z. When there is no jammer state information and the maximum-likelihood metric (4.12) is used, then (4.20) has the form (4.22b) When (4.22) is used in the pairwise error bound (4.1), this is referred to as the Bhattacharyya bound [2]. In order to analyze realistic cases where the channel conditional probabilities are not known and, thus, receivers are based on the maximum-likelihood metric (mismatched receivers), the gen- eral form of D is given in (4.20). The pairwise error probability is the basis of general bit error bounds for coded communication systems. This is based on the union bound where the bit error probability is upper bounded by the sum of the probabilities of all General Analysis of Anti-Jam Communication Systems ways a bit error can occur. For any two coded sequences X and X let a(x, X) denote the number of bit errors occurring when X is transmitted and X is chosen by the decoder. If p(x) is the probability of transmitting sequence X then the coded bit error bound has the form (4.23) where C is the set of all coded sequences. Thus, we have the general form Pb = G(D) (4.24) where G() is a function determined solely by the specific code whereas the parameter D depends only on the coding channel and the decoder metric. The above coded bit error bound is based on two inequalities: the Chernoff boun