meaning for integer values. Since the minima in Figure 2.30 are quite broad, there is essentially no loss in performance when the continuous value of Mopt, defined below in (2.70), is truncated to the nearest integer.) The loss in performance for m > Mopt due to non-coherent combining is clearly evi- dent in Figure 2.30. m=MOPT Eb/N,,BB Figure 2.28. Same as Figure 2.27, except FH/MFSK signals with K = 3 or M = 8, and m diversity chips/8-ary symbol. Non-Coherent Frequency-Hopped Systems m MOPT Eb/NJ, dB Figure 2.29. Same as Figures 2.27 and 2.28, except that K = 5 or M = 32 and m diversity chips/32-ary symbol. MOPT = 8 MOPT = 12 DIVERSITY m CHIPS/M-ARY SYMBOL Figure 2.30. Variation in performance of FH/MFSK signals in worst case partial- band noise with amount of diversity for EbIN1 = 12 dB. The Pb upperbound is min- imized at m = Mopt and the performance degrades for larger values of m due to non-coherent combining of the diversity chip energies. Coding Countermeasures Referring again to Figures 2.27-2.29, the performance in worst case jam- ming with optimized diversity is the lower envelope of the set of curves with parameter m, shown as a dotted curve in each graph. Analytically, we are looking for the solution (2.69) which implies that we want the smaller of the minima of (2.63) and (2.64) over m 1. We find that the desired solution derives from (2.64), provided Eb/N1 is large enough: provided (2.70) Pwc = 3/ The constraint in (2.70) simply guarantees that Mopt 1; for smaller values of E/N, Mopt = 1 and (2.25) yields the exact Ph (which is obviously discon- tinuous from the upperbound of (2.70) at its lower limit on E/N). The result of (2.70) was first documented in [2], where the extrema Mopt and Pwc were termed "quasi-optimum" since they are based on an upper- bound. It is plotted in Figure 2.31 for 1 < K < 5, which shows that the per- formance with optimum diversity in worst case partial-band noise improves Eb/Nj,dB Figure 2.31. Performance of FH/MFSK signals with optimum diversity in worst case partial-ban