aximizes P1m = + NJ exp { E/NJ + NJ Whe k=1 + yks) } (2.218) Pom(Y1) or equivalently maximizes (2.219) e1 = k=1 elk where elk = ykc + yiks (2.220) is the energy of the 1-th tone in the k-th chip interval. E is the average energy per MFSK chip. - EXACT BOUND E /N (dB) Figure 2.107. Rayleigh fading BFSK-diversity. Non-Coherent Frequency-Hopped Systems here the ML decision rule is based on the sum of the m for chip the Note that for each of the M tones. The energy metric is thus optimum broadband energies fading channel with equivalent white Gaussian noise or white Rayleigh This was not the case with coding over a non-fading used since noise Gaussian jamming. noise channel. There, however, the energy metric was it is convenient. For m diversity, the bit error probability is given by [24], = and + 1(m-1) (E/N) (2.221) (E/N1) EXACT BOUND Eb/N, (dB) Figure 2.108. Rayleigh fading 4FSK-diversity. Slow Fading Uniform Channels where E = (K/m)Eb, M = 2K, and Bkl satisfies (2.222) For the binary case where M = 2, the bit error probability is simply Pb m-1 m k 8)k (2.223) where (2.224) (Eb/N ) Figure 2.107 shows this bit error probability versus Eb/N J for various values of diversity. Note that, for each value of Eb/N J , there is an optimum EXACT BOUND m = 4 Figure 2.109. Rayleigh fading 8FSK-diversity. Non-Coherent Frequency-Hopped Systems diversity. Excessive diversity results in non-coherent combining losses that begin to cancel the beneficial effects of having independent observations. Next, a Chernoff bound will be derived for the bit error probability when M = 2. Assuming the ML metric, the optimized Chernoff bound is the Bhattacharyya bound given by (2.225) But F(Y, Y2) is the joint Gaussian density function where all random vari- ables are i.i.d with zero mean and variance Nj/2. Thus (2.226) (Eb/N ) ) ² Figure 2.107 shows this bound (dotted lines) together with the exact bit error probability. For M > 2 the union bound can be combined with the above Chernoff bound. The symbol error probability is union bounded by P,=