riable, becomes (2.210) 2(E/N ) In Figure 2.106 we plot Pb and its bound versus Eb/NJ where Eb = E/K is the average bit energy and M = 2K with K = 1, 2, 3, 4, and 5. Note that EXACT BOUND Eb/N, (dB) Figure 2.106. Rayleigh fading-uncoded. - Non-Coherent Frequency-Hopped Systems the impact of fading is similar to the worst case partial-band and multitone jammer. Also notice that although the union bound is very tight for the non- fading case (see Figure 4.11 in Chapter 4, Part 1), this bound is quite weak in this uncoded Rayleigh fading case. Furthermore, the bound increases as K increases, whereas the true bit error probability decreases with increas- ing K. With no fading the union bound results in a sum of terms that decrease exponentially with Eb/N, whereas in the Rayleigh case these terms decrease at a much slower rate, namely inversely with Eb/N . We saw earlier that coding and/or diversity can improve the performance of worst case partial-band jamming. These techniques similarly improve the performance in Rayleigh fading channels. 2.4.2 Broadband Jamming-Diversity and Coding Consider diversity of order m where each MFSK signal of duration T sec- onds is transmitted as m MFSK chip tones each of duration T = Tlm sec- onds. Each chip is independently hopped over the spread-spectrum frequency band. Also assume that each chip tone has independent Rayleigh fading of the same statistics. This is a reasonable assumption for FH/MFSK signals that hop over a wide frequency band Wss where Wss>>Afo (2.211) where Afc is the "coherence bandwidth" of the fading channel [24]. If we denote (2.212) as the cosine and sine components of the received signal at the 1-th frequency during the k-th chip interval and (2.213) as the sequence of such components for the 1-th tone, then the conditional probability of the entire set of mM chip cosine and sine components given the 1-th tone was transmitted as p(Y1, sent) (2.214) where YM) = II Pom(Y), (2.215) (2.216) (2.217) Slow Fading Uniform Channels The ML decision rule chooses l that m