) Substituting (3.9) and (3.10) into (3.13) and the results of these integral evaluations into (3.12) gives for the in-sync condition (M-1)exp(-Y|)[1-exp(-Y)]- - - M-2 qs+n(Y). (3.14) 0; otherwise. The corresponding result for the out-of-sync condition is obtained by letting Yh = 0 in (3.14),i.e., M(315 As previously mentioned, we need to determine the first two central moments of Y in order to evaluate the operating characteristic of the acqui- sition system. From (3.14), we can determine the mean of Y * for the per- fectly in-sync condition as = (M M-2 (3.16) FH Acquisition Techniques and for the out-of-sync condition, (3.17) Note that the leading terms of (3.16) and (3.17) correspond to the means of y* as determined from Ps+N(y) and PN(y*) in (3.10) and (3.9), respectively (see (1.75) and (1.76) of Chapter 1). Similarly, the mean-squared value of Y * under the two sync conditions is determined as (3.18) (3.19) Thus, the in-sync and out-of-sync variances of Y* are given by = - Ms+N (3.20) where once again the leading terms will correspond to the variances of y* (see (1.75) and (1.76) of Chapter 1). Post-detection accumulation of Y * produces the approximately Gaussian (Nh large) random variable Z* = Y (3.21) which when compared with the normalized threshold n* gives rise to a false alarm probability (exceeding the threshold when in the out-of-sync condition) (3.22) Time and Frequency Synchronization of Frequency-Hopped Receivers and a detection probability (exceeding the threshold when in the in-sync condition) (3.23) os+n/on where, as in previous chapters, Q(x) is the Gaussian probability integral. Eliminating B between (3.22) and (3.23) produces the desired system oper- ating characteristic (3.24) os+N/ON where MS+N, MN, os+N, and ON are determined from (3.16)-(3.20) and are all functions of the signalling alphabet size M and hop signal-to-noise ratio Yh. Alternately, since the dwell time Td of the system is related to the accu- mulation size Nh by (3.25) then (3.24) can be expressed in terms of the dwell tim