 to define the "optimum" diversity Mopt> as well as a monot- onically decreasing Pb for sufficiently large values of m. In the uncoded case, we argued that the values of m needed to provide better performance than at Mopt were too large for practical implementations; for example, in Figure 2.86, Eb/N, = 14.2 dB will provide the uncoded system with Pb = 10-5 at m = Mopt II 12, whereas lower bit error rates require diversities in excess of 40. However, the (7,1) Trumpis convolutional code has Mopt = 2 with extremely low Pb's for diversities of the order of 10. We will continue to use our pre- vious definitions of Mopt (refer to (2.79) and (2.104)); however, we should Non-Coherent Frequency-Hopped Systems remember that some of the more capable codes can achieve arbitrarily good performance in worst case n = 1 band multitone jamming for moderate amounts of diversity. We will now mathematically define the performance of coded FH/MFSK systems concatenated with optimum diversity. For arbitrary diversity m, Pb for the concatenated system is proportional to G(Dm), where D itself depends on m through (2.160). Since G() is monotonic for small Pb's, the optimum diversity is the value of m that minimizes Dm. Recall from (2.158) that the performance of uncoded FH/MFSK signals with diversity is pro- portional to Dm, , and we computed Mopt for such systems in Section 2.3.1. The only difference here is that R' is arbitrary, whereas R' = K in the previous analysis. Generalizing the earlier results for arbitrary outer code rates, we conclude that for values of Eb/N, large enough to yield Mopt > 1, (2.162) as in (2.70), (2.77), and (2.104), where, for worst case noise or tone jamming, the optimum diversity has the form (2.163) UNCODED WC n= I BAND MULTITONE 4-ARY ORTHOGONAL CONVOLUTIONAL CODE DUAL-2, RATE 1/2 CONVOLUTIONAL CODE Eb/N = = 14.2 E6/N =11.3 db (7, 1/2) 4-ARY CONVOLUTIONAL CODE Eb/N = 9.4 db DIVERSITY m PER M-ARY SYMBOL Figure 2.86. Same as Figure 2.85, but for worst case n = 1 band multitone jamming. Note that although w