or of the term a cos -1(Va/2) over the range 0 < a V 4. As shown in Figure 2.14, it has a single maximum at a = 2.52, and decreases monotonically on either side of that peak. The other term that must be maximized in (2.45), au_1(1 - a), increases monotonically with a over its range (0,1). Depending on the range of Eb/N, and n/M, the worst case performance is specified by (2.46a) Because the function being maximized in (2.45) is discontinuous at a = 1, it is easier to maximize it separately over the ranges O<a<1 and 1 < VI (nK/M)(E,IN)), and then to determine the larger of these two maxima for the given ratio n/M. Since peaks at .525 when a = 2.52, (2.45) reduces to (2.46b) where .525; Defining (2.46c) Worst Case Jamming MAXIMUM @ =2.52 Figure 2.14. The behavior of one of the terms involved in the maximization of Ph in (2.45). we must consider two ranges of n/M for the given range of Eb/N}: (a) )nsyM awc=1_=u= (2.46d) 1)Eb/NJ (b) M<n<M (2.46e) 1)Eb/NJ One of the most interesting conclusions presented by (2.46a-e) is that there exist conditions for which it is in fact advantageous to allocate less power to each jamming tone than the received signal power. This occurs only for sufficiently large values of Eb/N, and n ~ M. Non-Coherent Frequency-Hopped Systems In particular, consider the special case of n = M. Here we know with cer- tainty that all M symbols of a jammed band will be hit, SO that the only error mechanism is the random jamming phase effect. This is reinforced by (2.44), which reduces to (2.47) We have already seen in Figure 2.14 that this function of a peaks at a = 2.52. So we are not surprised that awc can exceed 1, nor that it saturates at 2.52 for Eb/N, 2.52/K as shown in Figure 2.15. The impact of allowing awc to exceed 1 is to increase the likelihood that a band is jammed (see Figure 2.16), even though the larger value of a decreases the range of jamming phase for which an error can occur (refer to (2.35)). Figure 2.17 shows that the resulting uncoded FH/MFSK performance is 2 dB worst in the inver