ccu- mulated values are then used to form the frequency synchronization esti- mate. Finally, this estimate, which is formed in the receiver, is directly used to update the current frequency position of the local hop generator. Alternately, as for the time synchronization only case, the system in Figure 3.24 can be modified in the same manner that the system in Figure 3.14 was changed to Figure 3.15 to allow for a system whose receiver is a passive processor. 3.3.1 The Case of Full-Band Noise Jamming 3.3.3.1 Signal Model and Spectral Computations The transmitted signal in the j-th hop interval is still characterized by (3.35) with, however, fs replaced by fo. Using similar replacements in (3.36) and (3.37), the received signal y(t) and jammer noise J(t) are still, respectively, given by these equations. The output of the local FH generator in the pres- ence of a timing error T and frequency error l can be expressed as r(t) = 2 sin[2(fo + + fIF)t]; (3.97) Cross-correlating y(t) with r(t), the output x(t) of the band-pass filter gives J'(t) j-1)Th-(j-1)Th (T = 0) jTh StsjTh-T (T<0) cos[2(fIF1)t-j]+J(t) (T = 0) (j -1)Th-TSISIT (T<0) (3.98) where J'(t) is still given by (3.40). Sampling x(t) at the Nyquist rate and tak- ing the sine and cosine discrete Fourier transforms of these samples evalu- ated at the two center standard MFSK frequencies f=fif = +1 1/2Th, one has, Time and Frequency Synchronization of Frequency-Hopped Receivers in the j-th hop interval, (3.99) from which the spectral estimates (3.100) are obtained. Letting t = iTh/Ns 1 (j - 1)Th in (3.98) and substituting into (3.99), we arrive at the following results (for large Ns) for the first two sta- tistical moments of aj+ and bj+, that is, F B), (1 - F B), (1 - e|)] sin Qjt; E S1 E 1 1 (3.101) where t(1/2-B)1-)-2(j-1T (3.102) sinc[x,y] sin(xy)/x, (3.103) and B All Th is the frequency synchronization error normalized by the hop rate (also, the MFSK frequency spacing). Noting again that aj+ and bj+ are conditionally Gaussian random variables, then the rando