e known, then yc and ys are nonzero mean Gaussian random variables with variance No/2. The conditional probability density function is (2.200) 4We assume all possible carrier frequencies are spaced SO as to result in orthogonal signals. Non-Coherent Frequency-Hopped Systems Averaging this probability density over , which is uniformly distributed over [0,2]] and independent of A, gives [9] PA)/3 (2.201) Averaging this result over the Rayleigh random variable A returns us to the unconditioned probability density function given by (2.197). 2.4.1 Broadband Jamming-No Diversity The performance of the FH/MFSK system against broadband noise jamming is the same as that of conventional non-coherent MFSK systems in white Gaussian noise with single-sided spectral density N1 = J/W ss. Denote ym = (Ymc) Yms) m = 1,2,...,M (2.202) as the cosine and sine components of the received signal at the M possible tones representing an M-ary symbol. For the Rayleigh fading channel described above, the maximum-likelihood (ML) decision rule is based on probability density functions (2.203) where (2.204) and p1(y) and po(y) are given by (2.197) and (2.199), respectively. Without any coding or diversity, the ML decision rule is to choose that symbol m that yileds the maximum value of (2.205) or equivalently the maximum value of em=ymc tyms, (2.206) the energy of the m-th frequency component of the received signal. Thus, the optimum decision rule is the same as that for the non-fading channel when there is no coding or diversity. Since the decision rule is the same whether or not the fading amplitude A is given, consider the conditional bit error probability given A (see Section PA(2.207 Averaging this over the Rayleigh random variable with probability density Slow Fading Uniform Channels function (2.191) results in the bit error probability expression (2.208) (E/N ) ] where E is given by (2.196). In Chapter 4, Section 4.7.1, Part 1, a simple union bound for (2.207) was given by (2.209) which, when averaged over the Rayleigh fading random va