e estimates are used can be seen. Using the Chernoff bounds for these two cases and assuming repeat codes with diversity m, we have the results shown in Figure 1.16. For smaller values of signal-to-noise ratio, there is approximately a 1.25 dB difference between these two metrics. At higher sig- nal-to-jammer noise ratios, however, there is a much greater difference. Exact Chernoff Bound E N (dB) Figure 1.15. Repeat code-soft decision with ideal fade estimate. Coherent Direct-Sequence Systems No Fading Estimates Ideal Fade Estimates m =10 Eb/N (dB) Figure 1.16. Repeat code Chernoff bounds-soft decision. For the hard decision channel with ideal fade estimates the conditional probability of y given x and A is PN(y)x, A) = II(1 - 1 = N w(yk,xk) n=1 N - E(An)) (1.148) Slow Frequency Non-Selective Fading Channels where E(A) is given by (1.121). Here the maximum-likelihood metric is (1.149) Assuming this optimum metric, the Chernoff bound for the conditional pair- wise error probability is the Bhattacharyya bound (1.150) where, without loss of generality, only the first d places in x and X are assumed to differ. The pairwise error probability is thus (1.151) (1.152) For the suboptimum hard decision metric with no fade estimates, D is given by (1.133) where IC is given by (1.134). Since the function f(x) = V4x(1-x) (1.153) is concave for 0 1xxx 1, using Jensen's inequality [22], E{f(X)} = f(E{X}). (1.154) for the random variable X, (1.155) as expected. 1.6.4 Pulse Jammer with No Coding Now consider the case of a fading channel together with a pulse jammer with average power J and jamming with power Jlp for a fraction p of the time. Again assuming that when there is no jammer pulse the bit error probabil- ity is negligible, then the average uncoded bit error probability is (1.156) Coherent Direct-Sequence Systems For the case of Rayleigh fading, this has the form (1.157) The choice of jammer pulse parameter p that maximizes the bit error prob- ability for Rayleigh fading channels is p = 1 for all E/NJ. Thus the con- stan