ays unique. However, it can be shown numerically that the subse- quent optimization over a and S is only unique for K2. In the K = 3 case, for example, Figure 2.58 shows that the Pb bound has a well defined sad- dlepoint at the approximate values S = .28 and a = .537. The existence of a saddlepoint implies that it does not matter whether the optimization is per- formed first over a or 5. Recall that this was not the case for partial-band noise [11], although it is in fact true for band multitone jamming. When 5 = .29 5 = .27 .5757 Figure 2.58. Demonstration that the Pb bound for FH/8-ary FSK signals with diversity in independent multitone jamming has a saddlepoint with respect to the nor- malized diversity S and the jammer power coefficient a. Similar results are obtained whenever K 2. Coding Countermeasures 5 = .29 5 .27 Figure 2.59. Unlike Figure 2.58, there is no saddlepoint when K = 1. Instead, the worst case a that minimizes the coefficient (K1n(1/F) in the Pb bound of (2.121) has an abrupt discontinuity near S = .29. In this case, the order in which the Pb bound is optimized over a and S affects the performance, although not as drastically as in the partial-band noise scenario. K = 1, Figure 2.59 demonstrates that the optimum a has an abrupt dis- continuity at S III .29: to be more precise, (2.122) That is, the Pb bound coefficient (KIn(1/F) has two equal-valued local min- ima at the disjoint values of a given in (2.122) when S = .2911. And it is this value of S, denoted by Sopt, that subsequently determines the optimum diver- sity condition by maximizing (KIn(1/F) for any awc when K = 1: = .3679; K=1 (2.123) provided Eb/N, > = = K = 1. Then the Pb bound of (2.121), jointly optimized over a and S, has the com- pact form (2.124) Non-Coherent Frequency-Hopped Systems Table 2.9 Parameters associated with performance of FH/MFSK signals with optimum diversity in worst case independent multitone jamming, as defined in (2.124). Yo, dB 1.283 or 2.552 .3679 .5495 .5760 .5243 .4379 where from (2.120). In (2.123), the