 diversity of order m = 5. The diversity can be viewed as a special case of a repeat 5 times inner code with the bit error probability which has the bound (see (4.109)) (4.117) where (see (4.105)) D(5) = D5 (4.118) with D given by (4.92) or (4.93). Here the data rate is bits/coding channel 8-ary symbol. (4.119) Suppose the outer code is the optimum rate R = 1/2, K = 7 binary convo- lutional code. Then the final bit error bound is (see (4.42)) (4.120) where (4.121) with bounded by (4.117). Here the concatenation of codes gives an over- all code rate of 13 = 10 3/13 bits/coded channel 8-ary symbol (4.122) (4.123) where E is the energy per coded 8-ary FSK symbol in the coding channel. General Analysis of Anti-Jam Communication Systems 4.10.2 M-ary Super Channel In the previous example, with the inner code being the diversity or repeat 5 times code, the inputs to the inner encoder are 8-ary symbols. Each 8-ary symbol was taken to be equivalent to three bits each in deriving the bound on the BSC super channel crossover probability given by (4.117). The inputs and outputs of the inner encoder and decoder respectively could also be taken to be 8-ary symbols with the symbol error probability bounded by (4.108). In general, we can have a super channel with M-ary input and output sym- bols where in all cases of interest these M-ary channels are symmetric with the super channel probabilities of (4.79) where y and x are now M-ary out- puts and inputs respectively of the super channel and Ps is the symbol error probability of the inner code. In this case, parameter Ds is given by (4.81) Rs =log2M-log2[1+(M-1)Ds] - (4.124) Thus, for M = 4 or M = 8 we can have outer codes found by Trumpis with bit error bounds given by (4.77) and (4.78) respectively where the super channel is characterized by Ds and Rs given by (4.81) and (4.124). Another class of outer codes to consider are the Reed-Solomon codes which are discussed in the next section. 4.10.3 Reed-Solomon Outer Codes Figure 4.22 shows the most popular form of concatenation of