(3.90a) are the worst case partial-band fraction and optimum diversity for the per- fectly synchronized system (see Chapter 2, Part 2), and analogous to (3.78), Yh is related to Eb/N, by (3.90b) Actually, the quantities in (3.90a) are derived from a minimax solution of a Chernoff bound on the error probability. Nevertheless, it is convenient to use them in the exact expression for error probability of (3.89). 3.2.3.3 Average Error Probability Performance in the Presence of Time Synchronization Error Estimation If the estimator CO of (3.53) is used for FH time synchronization of the non- coherent receiver, then a residual time offset n(e) I E - (CO arises which affects system performance in the same manner as just discussed for an uncompensated time error E. In particular, it is clear that in the presence of the residual offset, a signal attenuation degradation occurs that is given by D(n)1--)21-n(n( = (00 (3.91) Likewise, a loss of orthogonality degradation analogous to (3.71) occurs that is given by (3.92) Thus, if PnE (ne) denotes the E-conditional probability density function of the residual offset n(e), and is given by (3.93) where the right-hand side of (3.93) is given by either (3.55), (3.60), or (3.65), it then follows that the average bit error probability Pb is (3.94) In (3.94), P(h) is the n-conditional bit error probability obtained from (3.78) together with (3.77) and (3.75) or (3.83) with E replaced by N. Substitution of (3.78) and (3.93) in (3.94) requires evaluation of a double integral to obtain numerical results for . To somewhat simplify matters, we observe from Figure 3.17 that, for the range of values of interest, OEE is comparatively small SO that (00 is very close to its conditional mean Me s with high probability. Equivalently, P n E (ne) may be approximated by a delta Time and Frequency Synchronization of Frequency-Hopped Receivers function located at E - or from (3.93) (3.95) Substituting (3.95) into (3.94) gives the much simplified result (3.96) To evaluate (3.96) (assuming large Nh), we