e the K coded output the M that energy is result detector in an M-ary outputs. symbol In general m with corresponding energy detector [33]) bits output lm. Using Bayes rule Pr(m)) (see Lee (2.397) Other Coding Metrics M 1=16, L = I M = 16, L = 2 E/N (dB) Figure 2.135. List metric cutoff rates for worst case tone jammer. Table 2.23 Bits to symbol conversion, K = 3, M = 8. 8-ary Input Energy Detector Output b1b2b3 Non-Coherent Frequency-Hopped Systems (0,1) (0,0) (1,1) (1,0) Figure 2.136. Binary input 2-bit output channel. Next observing that there are several possible M-ary symbols having the l- th bit, b1, (2.398) Defining M(b1) = ={m: 1-th bit i is b1}, (2.399) by symmetry (2.400) m EM(b1). Recall - II p(ex|m,q) (2.401) where p(e)m, q) is given by (2.330) for the general slowly fading non-uni- form channel with jammer power distribution J. Defining No + BqJq aqEek (2.402) the final form is (2.403) Other Coding Metrics where (2.404) Pr{e} But Pr{b|e} is the sum of exponential functions of em, m E M (b1) and for sig- nal-to-noise ratios of interest such a sum is dominated by the largest term. Thus, using the approximation Pr{b1e}=G(e)maxr(em) (2.405) me.(((M) gives an approximate ML metric (taking logarithm) given by m(e,b1)=max_m = (2.406) me.M(b1) For the K = 3 (M = 8) example of Table 2.23 M(b1 1) = 8} M(b2 = 0) = { 1, 2,5,6} = M(b2 = 1) ={{,4,7,8} = M(b3 0) = {1,3,5,7} = (2.407) and the metrics for each of the three coded binary inputs, m(e, b1 = 0) = max{e1,e2, e4} m(e, b1 = 1) = max{es,e6,e7, = eg} m(e, b2 = 0) = max{e1,e2,e5,e6} m(e, b2 = 1) = max{e3, e4, e7, eg} m(e, b3 = 0) = max{e1,e e7} m(e, b3=1) = max{e2, e4, e6 eg} (2.408) The above metrics are approximate ML metrics with no quantizations. They result in a binary input channel with real valued outputs. Various quan- tized outputs can be obtained for each coded input bit in the form = = - = = - max (2.409) me.(((M(b)=1) where f() is typically a uniform quantizer function. Here the difference max em em (2.410) me.(((M(b)=1) Non-Coherent Frequency-Hoppe