SI) with metric given by (2.317) and the result- ing coding parameter given by (2.318). Without JSI the decoder metric becomes (2.325) with the resulting coding parameter given by (2.327) and (2.328). Suppose next we want a binary input channel with two-bit outputs. Here the hard decision with 1-bit quality measure suggested by Viterbi [27] and discussed in Section 2.7.2 can be used. For M = 8 there are 16 outputs here of the form (m, 1) and (m, 0) for M = 1, 2, We can now convert these into two-bit binary channel outputs by taking, for example, the output (4,1) and converting it into three two-bit binary outputs of the form (4,1) (0, 1) (1, 1)(1,1) (2.393) where recall m = 4 corresponds to the 011 bits. Similarly, if (4,0) is the out- Other Coding Metrics M=4,L=2 M = 4, L = 1 E/N , (dB) Figure 2.133. List metric cutoff rates for worst case tone jammer. put of the channel in Section 2.7.2, then we have the conversion (4,0) (0,0),(1,0),(1,0). (2.394) With interleaving and deinterleaving the result is a binary input channel with two-bit outputs as shown in Figure 2.136 where the transition probabilities can be obtained using the type of analysis given in Section 2.7.2 (see Viterbi [27]). This approach can be extended to three-bit outputs for each input bit by using three threshold levels 1 > Y2 > Y3 > 1 where (2.351) is generalized as (m, 11), em-zymaxek Ymaxek>emY2maxe (2.395) ,01), 2maxek>emy3maxek (m, 00), Non-Coherent Frequency-Hopped Systems M = 8, L = 1 M=8,L=2 = Figure 2.134. List metric cutoff rates for worst case tone jammer. Again for the M = 8 example we have the conversion (2.396) (4,01) (001), (101), (101) decision bits are followed by two "quality" for each bits. binary With inter- input where leaving the the and channel hard deinterleaving which can there be are used three-bit in the outputs usual three-bit soft decision decoder for binary codes. to obtaining a binary input channel each transmitted with a quantized bit given Another to consider approach the conditional probability let b1, of b2, bk b