-5 13.62 30.05 42.91 4.74 X 10-5 12.12 30.79 42.56 4.84 X 10-5 11.11 31.45 42.42 4.57 X 10-5 10.36 32.06 Non-Coherent Frequency-Hopped Systems In particular, Table 2.6 demonstrates that optimum diversity recovers more than 29 dB at K = 1 (even more for larger K) at Pb = 10-5. Furthermore, comparing the upperbounds of (2.20c) and (2.70), we see that the perfor- mance with optimum diversity in worst case partial-band noise is only about 3 dB worse than with no diversity in broadband noise. (2.70) shows that the desired amount of diversity Mopt can become large for sufficiently low Pb's, especially for larger alphabet sizes M, under worst case partial-band noise jamming conditions (e.g., Mopt = 20 at Pb = 10-5 when K = 5). This is an indication that reliable communication over such chan- nels demands a substantial amount of coding redundancy. Since time diver- sity corresponds to a primitive repetition code, it is conceivable that more powerful codes may achieve the required redundancy with improved effi- ciency; this will be examined in Section 2.3.2. A surprising result is that with optimum diversity, Pwc = 3/3 independent of K and EbIN1, provided that the constraint in (2.70) is satisfied. This invari- ance can probably be attributed to the form of the P upperbound in (2.69); an exact joint determination of Mopt and Pwc would in all likelihood exhibit some complex variation with K and Eb/N Table 2.6 reminds us that in the absence of diversity, at low Pb's, an effective partial-band noise jammer con- centrates its available power over a small fraction of Wss (e.g., Pwc ~ 5 x 10-5 at Pb = 10-5 for 1 < K < 5). Optimum diversity forces the jammer to retreat back towards the broadband (p = 1) strategy, with the resultant loss of most of its prior advantage (see Figures 2.32-2.34). It should be noted that the solution to the joint optimization of the Pb bound in (2.69) over m and p is not a saddlepoint; that is, if we reverse the order of that optimization and consider instead the expression (2.71) the result would dif