 A, A2, ,AN. That is, dur- ing any coded symbol transmission, complete knowledge of the random fade amplitude is assumed. In this case the maximum-likelihood decision rule obtained by comparing A) with Pn(r/x, A) results in metric (1.139) Without loss of generality take the first d places of x and X to be different and obtain the conditional pairwise error bound (1.140) This is reasonable for slowly varying fading channels where these estimates must also be deinterleaved. Slow Frequency Non-Selective Fading Channels Averaging this bound over the common fading density function PA() gives (1.141) where (1.142) This can be compared with (1.114) for the case with no fading estimate. When the fades are i.i.d. Rayleigh random variables (1.143) This result can be compared with (1.119) where there is no fading estimate. In Figure 1.14, the cutoff rates of these two cases using R given by (1.92) are shown. Note that for small values of E/N, there is about a 1.25 dB dif- ference for the same value of R0. An exact expression can be obtained for the pairwise error probability for the Rayleigh fading case where (1.144) is a chi-square random variable with 2d degrees of freedom having the prob- ability density function z = 0. (1.145) Averaging the exact expression in (1.140) over this probability density func- tion results in the closed form solution (1.146) where (1.147) In Figure 1.15, the exact bit error probability is compared with the Chernoff bound for repeat codes having diversity m = 1,2, and 4. There is about a 1.5 dB to 3 dB difference between the exact bit error probability and the Chernoff bound. As in earlier examples, the Chernoff bound is looser for Ideal Fade Estimates No Fade Estimates Es/Nj (dB) Figure 1.14. Cutoff rates-soft decision. Slow Frequency Non-Selective Fading Channels cases when the rror probabilities fall off more slowly with signal-to-noise ratio. In Figure 1.15 the impact of having ideal fade estimates and using the opti- mum metric (1.139) compared to the suboptimum metric (1.120) where no fad