), (2.140), and (2.141), we have (2.142) For the special case of n = 1 band multitone jamming with K = 1, we could use (see (2.77)) (2.143) although we saw in Figure 2.36 that we could achieve arbitrarily small error rates by using large enough values of m. Here, we combine (2.130), (2.133), (2.138), (2.140), and (2.143) to yield the performance bound (2.144) For n = 1 band multitone jamming with K 2, we want to use (see (2.104)) (2.145) where 8 is specified in Table 2.7 for 2<K<5. Non-Coherent Frequency-Hopped Systems Perhaps the most commonly used convolutional codes for binary (K = 1) channels are the constraint length 7, rate 1/2 and 1/3 structures determined by Odenwalder [17]; assuming soft decision Viterbi decoding, the performance is given by G(D) = 36D10 211D12 + 1404D14 + 11,633D16 + 77,433D18 + 502,690D20 + 3,322,763D2 + 21,292,910D + 134,365,911D ²6 + and R = ;(7,1) code (2.146) G(D) = + and R=1;(7,3) code. (2.147) ((2.147) is not accurate for D VN 10-1.) For larger channel alphabets, Trumpis found the optimum constraint length 7, rate 1/2 and 1/3 (over these larger alphabets) convolutional codes for 4-ary and 8-ary orthogonal signal sets, respectively [18]: G(D) = =7D7 +39D8 104D9 + 352D10 + 1187D11 (7,1) code (2.148) G(D) = D7 + 4D 8 + 8D9 + 49D10 + 92D11+ (7,3) code. (2.149) Since each information bit produces an M-ary code symbol above, R = 1 for both cases. For K > 2, the M-ary orthogonal convolutional codes [19] have and R=1. (2.150) Finally, for all values of K, we have the powerful class of dual-K convo- lutional codes with code rate 1/v over GF(2) [2], [20]. That is, for every M- ary (K-bit) input word, V M-ary code symbols are generated, corresponding to R = Klv bits/chip. The performance is defined by GD)11) (2.151) We now want to investigate how effective these convolutional codes are at combatting worst case partial-band noise and n = 1 band multitone jam- ming. The performance of these codes with FH/MFSK signals is illustrated in Figures 2.66-2.75 for 1 < K < 5 and both kinds of jamming.