 Pwc = (2.42) Eb/N, < 9.597 Thus, the average bit error probability performance of FH/MDPSK (M = 16) in the presence of the worst case tone jammer is given by 0.25 < Eb/NJ < 9.597 B=1/2VE,/NJ 1.457 Pbmay = Eb/N J ; Eb/NJ 9.597 Eb/Nj < 0.25. (2.43) Q 2ne Figure 2.4. Individual signal point error probability components as a function of square root of jamming (per tone)-to-signal power ratio. ED # Pb (16) 0.1614 0.18 0.20 0.22 0.24 0.26 0.28 0.30 Bit error probability performance O O 0.1 of 0.12 FH/MDPSK 0.14 (M = 16) B as a function of square root of jamming (per tone)-to-signal power Figure ratio. Differentially Coherent Modulation Techniques Table 2.1. Asymptotic performance of FH/MDPSK for worst case partial-band multitone jamming. Worst case partial-band fraction Maximum average bit error probability Process 0.5220 1.835 0.2593 0.2760 4.376 0.5280 0.1614 9.597 1.457 Similar results can be obtained2 for FH/MDPSK with M = 2, 4, and 8. The asymptotic behavior of these results (i.e., p and Pb inversely related to E/N is given in Table 2.1. Using the results in Table 2.1 and the fact that, for any Eb/NJ K (2.44) with Pb(M) given by (2.20), Figure 2.6 is an illustration of the average bit error probability performance of FH/MDPSK for worst case partial-band multitone jamming. Before concluding this section, we wish to alert the reader to a point of pathological behavior that is directly attributable to the assumption of an on-tune tone jammer and is perhaps not obvious from the analytical or graphical results given. In particular, we observe from Figure 2.4 that QT118 has a jump discontinuity at B = 1 and thus Phmax of (2.43) will have a similar jump discontinuity at Eb/N, = 0.25 (-6 dB). In fact, since from (2.39), = 0.625 as B approaches one from below, then at EIN = -6 dB, Phmax 2It should be noted here that the results in [1] for the performance of FH/MDPSK (M = 4) in the presence of the worst case tone jammer are partially incorrect. In particular, Houston finds = 0.52 as the maximizing value. However, since the