.160) in the expressions for D, where R' is the outer code rate in bits per M-ary symbol. This explicit dependence on m will be useful in determining the opti- mum diversity later. As a check on (2.158)-(2.160), consider an uncoded FH/MFSK system with diversity m in worst case partial-band noise. Using (2.134) and (2.160), with R' = K for uncoded M-ary signalling, we compute (2.161) This combines with (2.158) and (2.159) to produce the same Pb expression derived earlier for this system (i.e., (2.164)). These m-diversity formulas can now be used to determine the variation in performance with m of any uncoded or coded FH/MFSK system in worst case jamming. As an example, consider the uncoded 4-ary system with diversity: Table 2.12 says that we require EIN, = 14.2 dB to guarantee Pb < 10-5 in any non-adaptive jamming environment if we use the optimum Non-Coherent Frequency-Hopped Systems WC PARTIAL-BAND NOISE MOPT=12.2 MOPT=13.2 WC BAND MULTITONE DIVERSITY m PER M ARY SYMBOL Figure 2.84. Variation in performance with diversity of uncoded FH/4-ary FSK sig- nals in worst case jamming, for EIN, = 14.2 dB; this SNR was selected to guaran- tee that Pb < 10-5 at m = Mopt = 12.2 for any non-adaptive jamming. amount of diversity. For this value of EIN Figure 2.84 illustrates the depen- dence of the system performance on diversity for worst case partial-band noise and n = 1 band multitone jamming. (We have seen similar curves in Figures 2.30 and 2.41). The multitone jamming curve reminds us that although there is a local minimum at m = 12.2, we can force Pb to be arbi- trarily small for large enough m. However, as we previously argued in Section 2.3.1.2, for practical implementation considerations, we want to keep m small; consequently, we will continue to refer to the diversity at the local minimum Pb for worst case n = 1 band multitone jamming as Mopt, defined in (2.79) and (2.104). The variation in performance of several coded systems in worst case noise and multitone jamming as a function of diversity is shown in Figures 