spread bandwidth Rb = data rate in bits per second. For conventional direct sequence (DS) coherent BPSK signals, Es is the energy per coded symbol while for frequency-hopped (FH) non-coherent MFSK, Es is the energy of each coded M-ary signal. For any specific code the bound on the coded bit error probability will have the form Pb = B(R0) (4.5) which is only a function of the cutoff rate R0- Since the function B(R0) is unique for each code and the cutoff rate parameter, R0, is independent of the code used, we are able to decouple the coding from the rest of the com- munication system. Thus, to evaluate various anti-jam communication sys- items, first compare them using the cutoff rate parameter. Codes can then be evaluated separately. This decoupling of the codes and the coding channels (shown enclosed by dotted lines in Figure 4.1) is possible at least for antipo- dal and orthogonal signals such as those commonly used in anti-jam com- munication systems. The basic model for this analysis is shown in Figure 4.2. Here a coded sym- bol sequence of length N is denoted (4.6) with corresponding channel output sequence (4.7) In addition, a corresponding side information sequence is denoted Z = = (z1, 22, ...,ZN). (4.8) System Model MEMORYLESS CODING ENCODER CHANNEL DECODER p(y/x,2), q(z) Figure 4.2. General memoryless channel. With adequate interleaving and deinterleaving, assume that the coding channel is memoryless. That is, assume that the channel probabilities satisfy (4.9) (4.10) For any coded communication, the decoding process uses a metric of the form m(y, x; z) if side information is available and m(y, x) if it is not avail- able. To simplify the decoding process, the metric is required to have an addi- tivity property where, for sequences of symbols, the total metric of the sequences is the sum of the metrics for each channel input and output pair of the sequence. Thus, for X, y and Z given by (4.6), (4.7), and (4.8) the total metric is m(y1,xj; Z1) + (m(y2, x2; Z2) + + or m(y1,X1) + m(y2,X2) m(yn,XN) dependin