s always proportional to the logarithm of the channel conditional probability which in this case has the form given in (4.60). Without coding where only symbol-by-symbol deci- sions are made, the maximum-likelihood metric can be any monotone func- tion of the channel conditional probability such as (4.62). For high signal-to-noise ratios the optimum metric can be approximated by (4.62). Although this is not the maximum-likelihood metric, it is the most commonly used metric for coded non-coherent MFSK systems. Volume II, Chapter 2 shows that it is the maximum-likelihood metric when there is Rayleigh fading in the channel. The choice of this metric results in the para- meter D given by (4.70) with E = REb where R is the code rate in bits per coded M-ary symbol. For the binary alphabet (M = 2), = the binary convolutional codes found by Odenwalder are given by (4.42) for rate R = 1/2 and by (4.43) for rate R = 1/3. Thus, for non-coherent BFSK with the metric given by (4.62), we would merely use D given by (4.70) with E = REB. For M = 4, Trumpis [12] has found an optimum constraint length K = 7 convolutional code of rate R = 1 bit per coded 4-ary symbol whose performance is given by the bound 352D10 + 1187D¹¹ + ]. (4.77) Trumpis also found an optimum convolutional code for alphabet M = 8 and constraint length K = 7 with rate R = 1 bit per 8-ary symbol whose perfor- mance is given by the bound 8D9 + 49D10 + 92D11 + (4.78) General Analysis of Anti-Jam Communication Systems A: K = 7, R = 2 BFSK B: K = 7, R = 3 1/m C K = 7, R = I D: K = 7, R = Eb INO (dB) Figure 4.12. Convolutional code bit error bounds-MFSK. Figure 4.12 shows all these codes for the non-coherent MFSK modulation in the conventional white Gaussian channel. The metric given by (4.62) results in a non-negative real number. Since this metric is not quantized, it is referred to as a soft decision metric analo- gous to the soft decision metric for coherent BPSK modulation discussed earlier in Section 4.4. Also, the detector of Figure 4.9 is referred to as a sof