ssumed to have a flat spec- tral density N1 and band-pass expansion about the sum of fs and the j-th hop Time Synchronization of Non-Coherent FH / MFSK Systems frequency given by - fi)t. (3.37) Letting T denote the time synchronization error between the received sig- nal and the local frequency hop generator, then in its normal (not delayed or advanced) synchronization position, the output of this generator can be expressed as r(t) = 2 sin[2m(fs+fi+1 (j - -1)Th-I<jTh- - T (3.38) where fIF is the IF center frequency of the band-pass filter. Cross-correlating y(t) with r(t) and assuming, as previously mentioned, that the hop frequency difference is outside the bandwidth of the IF filter, then the output x(t) of this filter is given by -1)Th-ts(j-1)Tn(0) J'(t); (j - (t = 0) 2S or (j - 0) (3.39) where J'(t) - V2J,,((t)sin[2nf1pt]. (3.40) If the band-pass filter output is now sampled at the Nyquist rate, then there will be Ns = 2B1FTh samples in each hop interval where B 1F denotes the IF noise bandwidth of the band-pass filter. Letting Xij = x(iThINs + (j - 1)Th) denote the i-th sample (i = 0,1, , N s - 1), in the j-th hop interval, then these samples are statistically independent Gaussian random variables with variance o2 = NBIF = Nj(Ns/2Th). Taking the sine and cosine discrete Fourier transforms of these samples and evaluating them at f = fIF, one has, in the j-th hop interval, (3.41) Time and Frequency Synchronization of Frequency-Hopped Receivers from which the spectral estimate (3.42) is obtained. Letting t = iThINs + (j - 1)Th in (3.39) and substituting in (3.41), we arrive at the following results (for large Ns) for the first two statistical moments of aj and bj, i.e., (3.43) where E Ill T/T is the time synchronization error normalized to the hop interval and Qj = - 2(j 1)fifTh. Also, aj and bj are Gaussian random variables with conditional pdf's (for 1) (3.44) where we have introduced the notation (3.45) Further, letting Yh denote the ratio of signal energy per hop-to-jamming noise spectral density, i.e., (3.