m variables rja_e Rician-distributed with pdf's (conditioned on E and B with E < and 0; elsewhere Frequency Synchronization of Non-Coherent FH / MFSK Systems where - e|])2 (3.105) and Yh is as defined in (3.46). To produce an estimator with small variance, the spectral estimates must again be accumulated over many hop intervals, say, N before forming the estimate B. In particular, letting R+ be defined identical to (3.52) using, how- ever, rj+ of (3.100), then, analogous to (3.53), the estimator of frequency syn- chronization is given by (3.106) whose first two statistical moments are (3.107) (3.108) where p(B|B) is the B-conditional pdf of 3. Proceeding directly to the results for large Nh, we again apply the central-limit theorem to (3.52) from which Z+ = R2 are Gaussian random variables whose first two statistical moments are given by (3.109) Using the Gaussian conditional pdf's for R+ and R_ together with (3.106), the pdf of B conditioned on B is given by p(B|B) (3.110) elsewhere Time and Frequency Synchronization of Frequency-Hopped Receivers Yh=00 141=017 Yn=5 dB 10 dB 15 dB -0.35 -0.25 -0.15 -0.05 -0.05 Yh = 5 dB 10 dB E = 0.5 -0.15 15 dB -0.30 -0.35 Figure 3.25. Conditional mean MBB versus B with Yh and E as parameters; N = 10. Y = O dB B=0.45 E|=0.5 B=0.25 B=0.45 /E/=O B=0.25 Figure 3.26. Conditional standard deviation OBB versus Nh with Yh, B and E para- meters. Frequency Synchronization of Non-Coherent FH / MFSK Systems where 07-07-19 (3.111) The conditional mean MBB from (3.107) and the conditional standard devi- ation OBB from (3.108) are presented in Figures 3.25 and 3.26, respectively. It is seen from Figure 3.25 that the slope of MBB is non-zero at the origin but decreases significantly as the absolute value of the normalized time syn- chronization error E is increased. Figure 3.26 shows that, for the range of parameter values of interest, OBB is comparatively small SO that the random variable is very close to the mean MBB with high probability. 3.3.2 The Case of Partial-Band Noise Jamming Analogo