 constraint on a ensures that p VI 1 in (2.115), while the limit on S (which translates into the restriction that EIN, > yo) guarantees that Mopt 1. Interestingly, e-1 = .3679, but we have not been able to prove explicitly that 8 = e-1 for K = 1. Since we do not have a sad- dlepoint for K = 1, it is illuminating to reverse the order of optimization in (2.123): we find that Sopt = .2911 again, but III .370 and awc K=1. (2.125) The performance improvement is only 8'18 = .025 dB for K = 1, which is insignificant SO that the absence of a saddlepoint here is only of academic interest, unlike the partial-band noise case. (2.123) and (2.124) are valid for any K. The computed values of a wo Sopt 8, and Yo are shown in Table 2.9 for various K. The effectiveness of diversity in combatting independent multitone jamming is illustrated in Figure 2.60 for 8-ary FSK. As a benchmark, at Pb = 10-5, the improvement with opti- mum diversity relative to no diversity is at least 37.8 dB, and is closer than 4.4 dB to the broadband noise performance, since the Mopt curve is an upper- bound. The best asymptotic performance for small Pb with optimum diver- sity is achieved at K = 3, which has the largest exponential coefficient 8; this is underscored graphically in Figure 2.61. (Recall that K = 3 was also asymptotically optimum for n = 2, 3, and 4 band multitone jamming.) 2.3.1.4 Time Diversity Overview At this stage, we are in a position to assess the relative effectiveness of the different kinds of jammers we have analyzed against FH/MFSK systems with simple repetition coding. For sufficiently large EIN, (how large depends on Coding Countermeasures (BER UPPERBOUND) WC INDEPENDENT MULTITONE OPTIMUM DIVERSITY NO DIVERSITY m=1 BROADBAND NOISE P = I 9.09 dB ED/NJ, dB 51.25 dB 13.50 dB Figure 2.60. Effectiveness of diversity against worst case independent multitone jamming for FH/8-ary FSK signals. Improvement with optimum diversity relative to no diversity exceeds 37.8 dB, and performance is degraded less than 4.4 dB relative to broadband 