ility, it is convenient to order the spectral estimates as r1 r(2) , r(M) where r(1) corresponds to the low- est frequency MFSK tone and p(M) corresponds to the highest frequency Time Synchronization of Non-Coherent FH / MFSK Systems MFSK tone. Then, if, in the j-th hop interval, the 1-th tone in the set is trans- mitted, the conditional probability of symbol error for that transmission, denoted by Ps(le) is described by the probability (3.74) Since p(1)E) is given by (3.47) and equals p(rine) of (3.73), then after some simplification (3.74) becomes (3.75) n=1-1 n # 0 where Q(a,B) is Marcum's Q-function [14] and as such (3.76) Finally, the average e-conditional symbol error probability is given by (3.77) and the corresponding e-conditional bit error probability Pb(e) is related to (3.77) by (3.78) Also, since for no diversity the hop signal-to-noise ratio Yh is equal to the MFSK symbol energy-to-jammer-noise spectral density ratio EI, then the bit energy-to-jammer noise spectral density ratio EbIN, is simply given by (3.79) Using (3.77)-(3.79) together with (3.71) and (3.75), Figure 3.18 is an illus- tration of Pb(e) versus Eb/N, in dB with E as a parameter for 4-ary FSK and the minimum orthogonal tone spacing, i.e., M = 4 and k = 1. The perfor- mance degradation, namely, the additional Eb/N, required at a given value of E relative to that required at E = 0, is plotted in Figure 3.19 versus tim- ing error for Pb(e) = 10-2. Also shown is the composition of the performance degradation in terms of its signal attenuation and loss of orthogonality com- ponents. We note that for small timing errors signal attenuation is the dom- Time and Frequency Synchronization of Frequency-Hopped Receivers E = 0.35 SIGNAL - TO-NOISE RATIO, EDINJ, dB Figure 3.18. 4-ary FSK with timing error (minimum orthogonal tone spacing) (reprinted from [13]). 4-ARY FSK Pb = 10-2 MINIMUM ORTHOGONAL TONE SPACING NORMALIZED TIMING ERROR, E Figure 3.19. Performance degradation due to timing error (reprinted from [13]). Time Synchronization of Non-Coh