herent FH / MFSK Systems non-coherent detection of the transmitted MFSK symbol and spreading sig- nal energy into adjacent MFSK frequency detectors. In particular, it is clear that in the presence of the residual offset, a signal attentuation degradation occurs that is given by (3.118) and a loss of orthogonality also occurs that is given by (3.119) Thus, if PvB(v/B) denotes the B-conditional pdf of the residual error v(B), then analogoous to (3.93) (3.120) It then follows that conditioned on E, the average bit error probability Pb(e) (3.121) where Pb(e,v) is obtained from (3.78) together with (3.77) and (3.75) with D(,v) and Din(E,v) used in place of D() and Din n(e). Since, as previously noted, OBB is comparatively small, we may make a sim- plifying assumption analogous to (3.95), namely, (3.122) in which case (3.121) simplifies to (3.123) Analogous to Figure 3.23, Figure 3.33 is an illustration of the average bit error probability performance of (3.123) for 4-ary and 8-ary FSK with no timing error (E = 0) and Nh = 10. Comparing these two figures, we observe that the average bit error probability is considerably less degraded by a resid- ual frequency error than by a residual timing error. Time and Frequency Synchronization of Frequency-Hopped Receivers IDEAL PERFORMANCE (B = O) AVERAGE PERFORMANCE IN THE PRESENCE OF FREQUENCY ERROR ESTIMATION (E = O) 4-ARY FSK P6(0) 8-ARY FSK 10-40 Eb/Nj, dB Figure 3.33. Average bit error probability performance in the presence of frequency error estimation; E = 0,Nh = 10. Finally, if both the fine time and fine frequency estimators of (3.53) and (3.106), respectively, are employed, then the overall error probability per- formance in the presence of the combination of the two residual synchro- nization errors can be computed from E-1/2 X PE&(E n/e)dvdn (3.124) References or from the approximate simplified expression (3.125) 3.4 REFERENCES [1] M. K. Simon, A. Polydoros, and G. K. Huth, "Time synchronization of a fre- quency-hopped MFSK communication system," ICC '81 Conference 