ining the parameterized probabilities M 2 2 otherwise (2.358) where, recall from (2.291) the hard decision symbol error probability is (2.359) Non-Coherent Frequency-Hopped Systems we have (2.360) Next note that = r{e1maxemx1 -Pre1ymaxemx = 1} (2.361) Similarly when occurs occur p((2,1)|1)=Pr{e2ymaxemx1} = (2.362) where now = (1-e-2/2)1-e-/2o-2 - (2.363) and thus p((2,1)|1) (2.364) Other Coding Metrics By similar steps we have = = = 1} (2.365) M 2 2 and by symmetry for y = (m,b) m=x m x x b=0,1. (2.366) Assuming an ML metric where m(y, xJ) = ln p(yx), then the coding para- meter is (2.367) where = 2V(p((1,1)1)p((2,1)|1)+2Vp((1,0)1)p((2,0)|1) + (M 2)p((2,1)|1) + (M - 2)p((2,0)|1). (2.368) For a suboptimum metric of the general form ==(x,1) = (x,0) (2.369) y (x,1), x x me,0, y (x,0), x x Non-Coherent Frequency-Hopping the coding parameter is given by (2.367) with Dq(J) = min = min + + (M - 2)p((2,1)|1) + (M - 2)p((2,0)|1)}. (2.370) 10-3 5 10 15 20 25 Eb/NJ 30 (dB) 35 40 45 50 Figure 2.120. D for M =2. Other Coding Metrics the special case of the uniform channel with broadband noise jam- is ming, in Figures 2.120 to 2.123. Shown in Figure 2.124 are the decision correspond- exam- For the coding parameter D with various values of the parameter g shown cutoff rates for M = 2. These figures also show the hard bit be seen. ing ples (2 = 1) SO that the advantage of one additional quality can 2.7.3 List Metric useful metric is based on a channel output that provides an detector ordered Another list of the M energy detector output samples. For the M energy y = 32 Eb/N, (dB) Figure 2.121. D for M = 4. Non-Coherent Frequency-Hopped Systems Y = 32 30 35 40 10-4 5 10 15 Eb/Nj 20 (dB) Figure 2.122. D for M = 8. output samples e = (e1, C2, ..., em), the demodulator-to-decoder interface function is the list (2.371) = , ym) detec- = k if is the k-th largest energy = term among > e2 > the e1>e4 M energy then y = (3, where yi e1 example, for M 4 if e3 sample, y2=2 tor output 4). Here samples. 1 = 3 For indicates e1 was the third larg