general results here. Also for the hard deci- sion ML metric, the broadband jammer is the worst case noise jammer. In section 2.5.2 we had shown that for the energy metric without jammer state information, the broadband jammer was not the worst case for the uniform channel. We shall examine the hard decision channel with no jammer state information after deriving the general coding parameter expression for arbitrary metrics. For more general metrics the general Chernoff bound can be used to com- pute the coding parameter D. To examine metrics that assume no knowledge of J, consider the jammer state information or metrics with various quanti- zations on the chip energy out of each detector. Here assuming the most gen- eral metric form, m(y,x/q), the usual pairwise error Chernoff bound is (2.320) Worst Noise Jammer Distribution-Slow Fading Non-Uniform Channel where E{ is the expectation over all channel statistics including jammer noise when x is sent and q is the hopping sequence. Averaging this over the hopping sequence according to probability p and minimizing over the Chernoff bound parameter, 1, gives the final pairwise error bound (2.321) where D(ppm(m( (2.322) when x # X. The minimax bound is (2.323) where D* = min max D(p, I). = (2.324) The parameter D* can be computed for non-fading channels as well as fad- ing channels with any fading statistics. These can all be included in the gen- eral analysis discussed in Chapter 4, Part 1. As an example consider the hard decision metric m(y,x) = -w(y,x) (2.325) which assumes no knowledge of the jammer state J. Here when x # X E{exp Am(y,R)-m(y,x)].x, - = - (2.326) = min (2.327) where (2.328) Non-Coherent Frequency-Hopped Systems Note that for the hard decision metric (2.325) where no jammer state information is used, the minimax point (p*,J*) for D(p,. J) is also the mini- max point for E given by (2.328). But E is exactly the uncoded symbol error probability with the same minimax point as specified by (2.302), (2.304), and (2.306). For the uniform channel specified