.27) where the expectation is over the jammer statistics. 1.21 Chernoff Bound For a given jammer coordinate sequence J fixed consider the Chernoff bound to the bit error probability of (1.26) as follows: P,(J) = Pr(n = VE,JJ (1.28) for any l > 0. Next, following the approach of Kullstam [20], [21], use the inequality cosh x (1.29) to obtain the form Pb(I) < (1.30) The l > 0 that minimizes this Chernoff bound is (1.31) giving the result (1.32) Uncoded Bit Error Probability for Arbitrary Jammer Waveforms In addition, note that a factor of 1/2 can be applied to the above Chernoff bound (see Appendix 4B, Chapter 4, Part 1). The final Chernoff bound is (1.33) This bound applies for all N and J and only assumes the PN sequence {ck} is an i.i.d sequence for binary symbols equally likely to be 1 or -1. 1.2.2 Gaussian Assumption The Central Limit Theorem [22] states that the normalized sum of inde- pendent random variables approaches (in distribution) a Gaussian random variable as the number of terms increases. This basically applies to cases where the variances of the terms are more or less evenly distributed over some bounded range. For fixed J the Gaussian assumption means n is a Gaussian random variable with zero mean since for each k (1.34) and variance given by Var{ney (1.35) The conditional bit error probability is then (1.36) This approximation to the conditional bit error probability applies for large values of N, which is typically the case with direct-sequence spread systems. Using the inequality x = 0 (1.37) we have (1.38) This inequality shows that in applying the Chernoff bound assuming n is a Gaussian random variable, we obtain exactly the same result as the general Chernoff bound divided in the previous section where no Gaussian assump- 3 Although this factor of 1/2 does not always apply in theory, it is a realistic approximation for all practical cases of interest. Coherent Direct-Sequence Systems tions were made. Thus, the Gaussian assumption results in a bit error prob- ability that is appropriate for d