0 and considering the maximization of (2.246) with respect to I1,J2...,Io where all q. (2.247) Worst Noise Jammer Distribution-Slow Fading Uniform Channel Suppose J* = (J,J,... , ) maximizes C(J). Then for values of l where (2.248) we have the necessary condition aC(J) (2.249) (2.250) For those values of l where Jt = 0 (2.251) we have the necessary condition IC(J) (2.252) (2.253) These necessary conditions become both necessary and sufficient if Pb(1/z) is a concave function of Z. (See the Kuhn-Tucker Theorem [19]). In this case, the minimizing jammer noise power distribution is unique. For the Rayleigh fading FH/BFSK case, Pb(1/z) is concave in Z SO the nec- essary and sufficient conditions become EbWss (2.254) with equality when Jt > 0. The unique worst jammer noise power distri- bution is 1=1,2,...,Q (2.255) which is the broadband jammer with uniform power distribution. This gen- eralizes to arbitrary uncoded FH/MFSK signals including background white Gaussian noise together with the noise jammer. With no fading, the worst distribution for uncoded FH/MFSK is the usual two-level partial-band noise jammer considered previously in this chapter. To show this for the binary case, suppose that the distinct values Non-Coherent Frequency-Hopped Systems of Jq, q' q=1,2,...,Q are given by A1, A2,...,AL where L is the number of these values of the jammer power levels in the sub-bands. Let F - be the number of sub-bands that have jammer power D. Then (2.256) is the fraction of sub-bands with jammer power D for l = 1,2,...,L. The bit error probability thus has the form (2.257) Using the bound (2.258) which is easily obtained by direct differentiation, EbWss (2.259) Eb/N1 using the relationship (2.260) The bound in (2.259) can be achieved with two levels (L = 2) where 1 = 0 2 = J/p (2.261) Worst Noise Jammer Distribution-Slow Fading Uniform Channel P1=1-p (2.262) provided that Eb/N  2. (2.263) If this condition is not satisfied, then the broadband jammer is the worst noise jammer. 2.5.2 Diversity and Coding Next assume that