g channel. Other Coding Metrics for any a > 0 and b is used, the expression for D(p,J) becomes (2.336) for x # X and where the integral may be a summation when y is a discrete random variable. We now investigate various quantized channels defined by (2.331) and consider examples of coding metrics associated with these equivalent cod- ing channels. 2.7.1 Energy Quantizer Consider the function y=f(e) (2.337) where 3, 30 s Vem m = 1,2,...,M (2.338) This function just quantizes each sampled energy detector output into one of four levels (two-bit quantizer) where the quantization intevel is of length Suppose X is the transmitted symbol. Then define probabilities kv,q)=Pr{kvsVem<(k+1)um x,q} = P1(kv,q)=Pr{kusVem<(k+1)um= = ,}} 0,1,2 (2.339) po(3lv,q)Pr{3usVemm x,q = p1(3/v,q) = Pr{3v < Vemm=x,q} (2.340) Using the probability density for em given by (2.330) i = 0,1;k = 0,1,2 = (2.341) Non-Coherent Frequency-Hopped Systems where (2.342) BqJq, The coding channel conditional probability density is then P1(Yx/V,q) (2.343) where (2.344) is independent of the channel input symbol X. For this channel the maximum-likelihood metric is h(y,xJq)=logp1(yx/v,q) - log Po(yx/v,q (2.345) which will give a coding parameter (2.346) For the uniform channel with no receiver noise where parameters a and B are given by (2.319), q=1,2,...,Q (2.347) the choice of V that minimizes this coding parameter can be found numeri- cally. Figure 2.118 shows the minimizing value of V = v/NJ versus E/N J along with corresponding values of the cutoff rate R for M = 2 defined by (2.334). Instead of the ML metric (2.335) for this coding channel, a simpler met- ric is to approximate the soft decision energy metric with no jammer state information by choosing metric m(y,x) = yx. (2.348) Other Coding Metrics This uses no jammer state information and has coding parameter given by (2.333), which becomes (2.349) This, too, can be minimized over vzo and examined for special cases such as the uniform channel. For the uniform channel with large values of E/N, we expe