t decision detector since the detector output y = (e1, e2, em) is unquan- tized. If the detector is forced to make a decision as if there was no coding the result is the hard decision detector with outputs 1, 2, M where the detector output is the integer i if ei > ej for all j # i. Essentially the hard decision detector makes the usual uncoded non- coherent MFSK decision and results in a symmetric coding channel shown in Figure 4.13 for M = 4 and M = 8. Conventional Non-Coherent MFSK (a) M= 4 (b) M=8 Figure 4.13. Hard decision non-coherent MFSK channels. If the output of the hard decision detector is given as y, then the hard deci- sion coding channel for fixed M is characterized by 1 - Ps, y=x p(y) = (4.79) M - 1 , y x x General Analysis of Anti-Jam Communication Systems where Ps is the symbol error probability given by (4.63). For this hard deci- sion channel the maximum-likelihood metric is given by (4.38) and the para- meter D(A) is (4.80) Thus, (4.81) Note that for M = 2, D becomes identical to (4.40) with E replaced by Ps; i.e., the crossover probability of the BSC. For both hard and soft decision detectors the cutoff rate is given by (4.27). Figure 4.14 shows the cutoff rates for both the hard and soft decision detec- tors with the additive white Gaussian noise channel. The coded bit error bounds for the hard decision detector can be obtained for the convolutional codes discussed above by shifting the bit error bounds shown in Figure 4.12 for the soft decision detectors by the amount of difference between hard and soft decision detectors shown in Figure 4.14 for R versus E/No. This is the basic motivation for evaluating the easy-to-compute cutoff rate R associ- ated with various channels and detectors. M = 8 A: Soft Decision B: Hard Decision Es /No (dB) Figure 4.14. Cutoff rates for MFSK. FH/MFSK and Partial-Band Jamming 4.8 FH/MFSK AND PARTIAL-BAND JAMMING The previous section also applies to FH/MFSK with broadband noise jam- ming where No = N1 and Eb/N, is given by (4.3). Here the case of partial- band n