for all values of K, SO we should not be surprised that the K = 2 performance in Figure 2.41 improves for larger values of m. (This effect would be evident for other values of K in Figure 2.41 if we had gone beyond m = 30.) Mathematically, for large values of m in the saturation region of (2.90), awc << 1 Ps=M(M-2)" M M - (2.91) Eb/Nu,dB Figure 2.39. Performance upperbounds for FH/4-ary FSK signals with diversity m chips per 4-ary symbol in worst case n = 1 band multitone jamming. Coding Countermeasures Eb/N,,BB Figure 2.40. Same as Figure 2.39, except K = 4. Eb/Nj = 14 dB 10-11 DIVERSITY m, CHIPS/SYMBOL Figure 2.41. Behavior of performance upperbounds for FH/MFSK signals in worst case n = 1 band multitone jamming as a function of diversity m. Note that perfor- mance improves for large m in the K = 2 curve, and the performance degrades as K increases. Non-Coherent Frequency-Hopped Systems Another way of seeing this is to recall that under the perfect side informa- tion assumption, an M-ary symbol error requires that all m diversity chips be jammed, denoted by the event H in (2.81). In the saturation region, Pr[H] (2.92) and the error rate must fall accordingly for large amounts of diversity. However, unlike the K = 1 case where it was possible to achieve Ph's less than the first local minimum at m = Mopt (see Figure 2.36) with moderately larger values of m, such is not the case for K 2. Figures 2.41 clearly implies that for practical implementations with non-binary alphabets, the best design tradeoff is to use m = Mopt Finally, Figure 2.41 demonstrates that for K > 2, the performance with diversity degrades as K increases in worst case n = 1 band multitone jamming, in contrast to the behavior in worst case partial- band noise; this echoes our earlier observations for m = 1 signalling (e.g., refer to Figure 2.12). With diversity transmission, where received data decisions involve multi- ple channel uses, it is generally difficult and cumbersome to compute the exact performance. We circumvent this problem by evalua