NO DIVERSITY 8.08 dB 15.49 dB Eb/Ns, dB 53.01 dB 14.99 dB Figure 2.50. Same as Figure 2.49, except K = 4. Optimum diversity provides 38 dB improvement relative to m = 1 system, and is only 6.9 dB worse than in broadband noise at Pb = 10-5. 16.22 dB 14.99 dB 13.75 dB 14.95 dB E /N, BB Figure 2.51. Overview of performance of FH/MFSK signals with optimum diver- sity in worst case n = 1 band multitone jamming. Note that the K = 1 curve is for the local minimum at Mopt' and arbitrarily better performance can be achieved for sufficiently large yet practically implementable values of m. Non-Coherent Frequency-Hopped Systems We now consider band multitone jamming for n E [2, M]. To avoid the complexity of an exact performance analysis, we will again employ the union/Chernoff bound approach. Since n 2, if an M-ary bound is jammed, we are guaranteed to have two or more high energy detector outputs. Therefore, from (2.74), Pr(M) (2.106) Again assuming that symbol 1 is sent, we need the statistics of the differ- enced energy detector output e2j - e1j conditioned on H Referring to Table 2.2, which considers the relative phase of the received signal and jammer tones, and incorporating the joint likelihood that either or both symbols 1 and 2 are hit when the M-ary band is jammed, we can write M - 2 B - 1 - 2 00 n(M - n) M(M - 1) B=1-1 (2.107) Since is assumed to be uniformly distributed over [0, 2), (2.108) where Io() is the zeroeth-order modified Bessel function of the first kind. Then, as in (2.82), (2.83), and (2.86): Pos M." (2.109) Coding Countermeasures where - min 120 ame- n(M + (2.110) Because of the Bessel function, we cannot express the minimization over 1 in closed form. Likewise, we must resort to numerical techniques to deter- mine the worst case jamming performance, for which F must be maximized over a E (0, 4). However, it can be demonstrated that this joint extremum is unique, and occurs at a = ao, tabulated below for selected combinations of K and n. Defining the parameter B as in (2.90), for the region > 1, we