g. (See Appendix 4B.) The bit error probability is then given by Pb = M-IP 1M < 1 MD. (4.73) Figure 4.10 compares (4.65) and (4.73) for M = 2, 4, 8, 16, and 32. There is approximately a 1 dB difference between the exact bit error probability EXACT UNION - CHERNOFF BOUND M = 16 Eb/No (dB) Figure 4.10. Exact versus union-Chernoff bound for uncoded MFSK bit error prob- ability. General Analysis of Anti-Jam Communication Systems and the general coded bit error rate bound applied to this special case of no coding. Recall that the bound in (4.73) is based on the union bound (4.25) and the Chernoff bound (4.16). The 1 dB difference between the exact bit error probabilities and the bounds is due primarily to the Chernoff bound. For the uncoded case, we can derive the exact pairwise error probability which is given by (4.74) for j # 1. This is just the two signal case equivalent to the BFSK bit error probability. The symbol error union bound is, thus, = (M - 1)1e-Es/2No - (4.75) EXACT UNION BOUND Eb/No (dB) Figure 4.11. Exact versus union bound for uncoded MFSK bit error probability. Conventional Non-Coherent MFSK and the corresponding uncoded bit error bound using only the union bound (4.76) Figure 4.11 shows this bound and the exact bit error probability for M = 2, 4, 8, 16, and 32. For the special case of no coding the Chernoff bound was not required since an exact expression for the pairwise error probability is avaialable. For the coded cases, however, the Chernoff bound for the pairwise error prob- abilities is needed since easy-to-evaluate exact pairwise error probability expressions are difficult to obtain. 4.7.2 Uncoded With the use of coding, each coded MFSK symbol has an associated M-ary signal of energy denoted E and of duration Tc- With no coding these para- meters become Es and Ts as in the previous section. The maximum-likelihood decision rule with coding requires a metric that includes a zero-th order Bessel function. This is because in order to have the additivity property, the maximum-likelihood metric i