e-hop rate product, which produces a relation analogous to (1.81) of Chapter 1. Figures 3.9 and 3.10 are plots of false alarm probability PFA versus normalized dwell time Ta/Th with detection probability PD as a parameter for Yh = -20 dB and either 2-ary or 8-ary FSK, respectively. Clearly, as the number of signalling levels M increases, the required post-detection accumulation increases pro- portionally. When m-diversity is employed, then the appropriate pdf's analogous to (3.9) and (3.10) become exp(- PN(Y) (3.26) 0; otherwise FH Acquisition Techniques PD = 01 M = 2 DWELL TIME/HOP TIME, Nn Figure 3.9. False alarm and detection probability performance of serial search FH/MFSK acquisition system; Yh = -20 dB. Ps+N(Y) = otherwise. (3.27) Similarly the out-of-sync and in-sync pdf's for the largest of the M nonco- herently combined normalized demodulator outputs are given respectively IN(Y) = 0; otherwise (3.28) Time and Frequency Synchronization of Frequency-Hopped Receivers PD = 01 M = 8 10-41 DWELL TIME / HOP TIME, Nn Figure 3.10. False alarm and detection probability performance of serial search FH/MFSK acquisition system; Yh = -20 dB. exp[-(Y₹ + qs+n(Y) = (m-1)/2 X exp[ 0; otherwise (3.29) FH Acquisition Techniques with first two moments m+j+n-1 = 1,2 (3.30) where LX-1)(x) is the k-th generalized Laguerre polynomial of order m - 1 [3] and Ckj are multinomial coefficients which satisfy the recursion rela- tionship (3.31) Thus, recognizing that MS+N and un correspond to (3.30) with n = 1, and fur- ther obtaining S +++ and ON from (3.20) using (3.30) evaluated for n = 2, one uses (3.24) to obtain the system operating characteristic for the m-diversity case. We mention in passing that for acquisition in noise, but in the absence of information modulation (see Figure 3.4 for the appropriate system), one merely sets M = 1 in (3.16)-(3.20), which, when substituted into (3.24), gives the simple result (3.32) Note that if the acquisition system of Figure 3.1 were used in place of that in Figure 3.4, then (1.32) would st