k E [1,m]; for larger m, there is a computational gap for 3 VI k < m/3. We could extend (2.97) to smaller values of k, but the result would be even more complex. As it is, compare (2.97) with the simplicity of the Pb bound of (2.90). And we still have to maximize these exact expressions over a < 1 to establish the worst case jamming condition. If we treat k rather than a as the maximiz- ing variable, it is clear from (2.97)-(2.99) that P is maximized over a for a given k by selecting ma = k. (2.100) Non-Coherent Frequency-Hopped Systems Then why did we not replace ma by k in (2.97)-(2.99) and be done with it? The reason is that we must also satisfy the constraint u < 1, which implies ma s MN, (2.101) In the singular cases for which KEMN is precisely integer, k < KEMN and (2.100) is valid for all k in this range. In general, when KEM1 is non- integer, we have (2.102) The worst case jamming condition is now established by maximizing the expressions for P in (2.97)-(2.99) over the range 1 V k < min(n KE/MN3); this operation must be performed using a computer for each combination of K, Eb/N1, and m, further emphasizing the practical utility of the Pb upperbound. Note that since (2.97) can be evaluated only if k = ma > m/3, (2.101) limits us to the range (2.103) These exact results are first used to assess the accuracy of the upper- bounds plotted in Figures 2.39-2.41. As we observed in the partial-band noise case of Figures 2.24-2.26, for worst case n = 1 band multitone jam- ming, Figures 2.42 and 2.43 demonstrate that the bounds are pessimistic by several dB for small m ~ 2, but are accurate to within about 1/2 dB of the exact performance for moderate m VN 10. Actually, the asymptotic accu- racy is about 30% better for the band multitone case relative to the par- tial-band noise curves. Note that for the values of m selected, we could not plot exact Pb's for SNR's that were too small to satisfy the constraint of (2.103). Figure 2.44 gives a different perspective on the accuracy of the Pb bounds. In the K = 2 case, w