46) then rj is a Rician-distributed random variable with pdf (conditioned on E with 18<1) p(rj|e) elsewhere. (3.47) Time Synchronization of Non-Coherent FH / MFSK Systems If the transmitter hop frequency generator is now advanced and delayed by AT(/) from its nominal synchronization position and the corresponding spectral estimates denoted by ri+ and rj-, then an appropri- ate estimator of time synchronization is (3.48) where KA is a constant whose value is chosen relative to that of the nor- malized advance-delay fraction A. In the absence of jammer noise, we have (3.49) Thus, the maximum region over which (00 is a linear function of E would occur for A = 1/2 in which case KA would be chosen equal to 2, SO that over this interval (CO = E. In what follows, we shall assume these values for A and KA, or equivalently from (3.48), (3.50) Furthermore, since the advance and delay of the transmitter hop generator does not affect the variance of the discrete Fourier transform components, then from (3.43) and (3.47), we have for E < A = 1/2, p(ritle) 0; elsewhere (3.51) If it wasn't for the presence of the additive jamming noise, (3.50) would be a perfect estimator of time synchronization. However, in the presence of noise, (c) of (3.50), which is computed on the basis of spectral estimates from a single hop interval, would possess a large variance. Thus, to produce an esti- mator with small variance, we must first accumulate the spectral estimates over many hop intervals, say N before forming our estimate of E in the man- ner of (3.50). In particular, letting (3.52) Time and Frequency Synchronization of Frequency-Hopped Receivers then we define our estimator of time synchronization by (3.53) In order to calculate the variance of the estimator E, we must first com- pute its pdf (conditioned on e). From (3.51) and (3.52), one can show that (Nh-1)/2 p(R+|E) (3.54) 0; elsewhere. Then, by a straightforward transformation of variables and the fact that R+ and R_ are statistically independent (conditioned on E), we obtain the d