/channel use. Since Mopt = 1 at n = n*, Ropt in (2.187) is the overall sys- tem code rate. If we prefer to express this parameter in M-ary information symbols/channel use, as in (2.177), we have (2.188) Finally, from (2.184) and (2.186) and (2.187), we find that In(2Pb) (2.189) Note that r opt depends only on K, n* depends only on K and Pb, and neither parameter depends on the type of jamming. From (2.183), the correspond- ing SNR is (2.190) which does not depend on the jamming environment through (2.164), as well as K, but not explicitly on Pb These parameters are shown in Table 2.22 as a function of K for worst case noise and tone jamming when Pb = 10-5. The ran- dom block coding results perpetuate a pattern that we observed for almost Table 2.22 Random (n,k) block coding with FH/MFSK signals in worst case jamming. If n > n*, the performance cannot be improved by adding diversity; at n*, ropt = k/n* minimizes the required E/N, = 10-5 Noise 10.13 127.1 210.5 10.97 382.1 Non-Coherent Frequency-Hopped Systems all of the specific codes considered in this chapter. The performance improves monotonically with K in worst case partial-band noise, but it peaks at K = 2 for worst case multitone jamming. If we must be capable of dealing with either threat, the best performance for a coded FH/MFSK system is usually achieved at K = 2. Recall that of all the coded FH/MFSK systems considered, the small- est EIN that could guarantee Pb < 10-5 in any non-adaptive jamming envi- ronment was 8.11 dB, realized for the (1023, 959) RS outer code combined with the (7,1) Trumpis 4-ary convolutional inner code and optimum diver- sity. Table 2.22 says that block codes exist that can perform at least .5 dB bet- ter than that system, presumably because they are better matched to the jamming channel; the incentive to find these codes is evident. 2.4 SLOW FADING UNIFORM CHANNELS We assume a slow fading channel where each hopped FH/MFSK signal expe- riences an independent fade with the same statistics. In particular, assume that during each ho