Pr{T #1|=1 < Pr{7 = k|l = 1} = P {1 ->k} = (M - 1)Pr{1 2} (2.227) where l = 1 was assumed to be the transmitted symbol and Pr{1 i} the pairwise error bound, which is the same as the M = 2 bit error proba- bility. Thus, using the Chernoff bound (2.228) (E/N ) ) ² Slow Fading Uniform Channels and noting that here E = KEb, we obtain the bit error bound Pb = 1M 4m(m + K(Eb/Nj)) m (2.229) (2m + K(Eb/N3))2 Figures 2.108 to 2.111 show these bounds for various values of K and diver- sity m together with the corresponding exact bit error probabilities given by (2.221) and (2.222). Next choose the value of diversity m that minimizes the bound in (2.228). This is shown in Figure 2.112. Figure 2.113 shows the same bound with m = EXACT BOUND Figure 2.110. Rayleigh fading 16FSK-diversity. Non-Coherent Frequency-Hopped Systems EXACT BOUND m = 4 Figure 2.111. Rayleigh fading 32FSK-diversity. = 2K. For fixed data rate, this case M results = 2, 4, in 8, the 16, bandwidth and 32. chan- of each K where FH/MFSK M chip signal being the as a same coded for M-ary symbol, a coding FH/MFSK nel Regarding (see Figure each 4.1 in Chapter 4, Part 1) with output (2.230) y = (y1, y2, ym) the cosine and sine components of the a sequence channel output of coded signal chips at where yi is is obtained. In a coded system, vectors of these the I-th chip tone the channel outputs are the corresponding uses a metric m(y, 1) which is transmitted and Any coding system each possible input. cosine assigns and a value sine to components. each channel output corresponding to Non-Coherent Frequency-Hopped Systems The ML energy metric we have considered previously is m(y,l) = yic + yis l =1,2,...,M. = (2.231) For the ML metric and simple diversity, the expresion for the bit error probability given by (2.221) and (2.222) is quite complex. Obtaining an exact bit error probability expression is much more difficult with other metrics. Hence we examine bounds on the coded bit error probabilities. Based on the general approach of Chapter 4, Part 1, for an arbitrary 