 first compute Mgs from (3.57) using p(E)E) from (3.60) with Yh replaced by (log,M)E,/N, as in (3.79). IDEAL PERFORMANCE (E= O) AVERAGE PERFORMANCE IN PRESENCE OF TIMING ERROR ESTIMATION 8 -ARY FSK 4-ARY FSK 10-4/ INJ, dB Figure 3.23. Average bit error probability performance in the presence of timing error estimation; Nh = 10. Frequency Synchronization of Non-Coherent FH/MFSK Systems Figure 3.23 is an illustration of the average bit error probability perfor- mance of (3.62) for 4-ary and 8-ary FSK with N = 10. In computing these results, (3.78) together with (3.75) and (3.77) were used for the conditional bit error probability. 3.3 FREQUENCY SYNCHRONIZATION OF NON-COHERENT FH MFSK SYSTEMS Frequency synchronization of a conventional (non-spread) non-coherent MFSK receiver has been typically achieved by transmitting a single known MFSK frequency, which is repeated as often as necessary within the true data stream, until the desired degree of frequency synchronization accu- racy is obtained. Spectral estimates in the form of discrete Fourier trans- forms of the receiver signal are formed with those adjacent to the transmitted MFSK frequency used to form the frequency error estimator. For example, Chadwick and Springett [12] used the ratio of the difference of these two adjacent spectral estimates to twice the spectral estimate of the transmitted frequency as an approximation to the maximum likelihood estimator of frequency error. The advantage of this estimator, or, for that matter, any estimator formed from spectral estimates of the standard MFSK tones, is that frequency error estimation could be used during the true data transmission for frequency tracking. Unfortunately, however, using the standard MFSK frequencies forms a very poor estimator since it can be shown (see Appendix 3A) that, for any function of the standard MFSK frequencies, the slope of the conditional mean of the estimator is zero for zero frequency error. A more suitable approach, which, we shall see shortly, also applies in the SS application, i