d multitone jamming improves with K = logM. Coding Countermeasures [12]. Recognizing this possibility, a jammer may turn to the independent mul- titone jamming format as the best remaining ECM option. Recall from Section 2.2.2.3 that the assumption implicit in the indepen- dent multitone strategy is that each FH tone frequency is independently jammed with a CW signal of power Sla with probability p; with m-diversity, the expression of (2.51) is modified to (2.115) With perfect side information, an error can only occur if at least one of the M - 1 untransmitted symbols is hit by a jamming tone on each diversity hop. The likelihood of this occurring on the j-th hop is Pr{H=1-(1-p)M-1 (2.116) Independent of H, the transmitted symbol on the j-th hop is hit with prob- ability p; however, conditioned on H, a particular untransmitted symbol on that hop is jammed with probability p/e. Again referring to Table 2.2 and assuming that symbol 1 is sent on the j-th hop, we can write Pr{e2j p(1-f) (2.117) P(1-p); B=1-1 As in the band multitone jamming case of (2.107), an error can only occur if B > 0 with a non-zero probability, SO a is restricted to the range (0,4]. Note that from (2.116), -p1- p)1-(1-p)M-2] (2.118) Applying the identity of (2.108), and expressing p as a function of a defined in terms of the parameter S in (2.115), the union/Chernoff bound on Pb is given by (2.109) with 120 mp (2.119) Non-Coherent Frequency-Hopped Systems As in previous jammer cases analyzed in this section on diversity, we want to determine optimum values of a and m for arbitrary but fixed values of K and Eb/N1. From (2.115), we can replace m by the normalized diversity para- meter S: (2.120) SO that (2.109) becomes (2.121) Now we want to minimize the (positive) exponential coefficient (KIn(1/F) over a E (0,4) for given values of K, Eb/N, and S; then we want to maximize the result over S. As always, we must verify that all of the required extrema are unique. From the form of (2.119), it can be easily argued that the minimization over 1 is alw