e relation among false alarm probability PFA, detection probability P, and dwell time Td.To determine this relation, we shall assume that the post-detection accumulation size N (actually NFFH or NSFH) is large SO that the statistics at the input to the threshold comparison in Figures 3.5 and 3.7 may be assumed Gaussian. A similar assumption was made in Chapter 1 in connection with our discussion of serial search PN code acqui- sition. Because of this simplifying assumption, it is sufficient to find only the mean and variance of the signal at the post-detection accumulator input in both the in-sync and out-of-sync conditions. To illustrate the procedure, we consider first the case of no diversity and equal symbol and hop rates. Thus, with m = 1 and Ts = Th, Figures 3.5 and 3.7 are identical. Assuming first the out-of-sync condition then the mixer output is noise only and the M non-coherent demodulator (normal- ized) outputs all have the same probability density function (pdf) given by (see (1.73) of Chapter 1) otherwise k ==1,2,...,M. (3.9) On the other hand, for the "ideal" in-sync condition (T = of = 0) and orthog- onal MFSK tone spacing (i.e., an integer multiple of the symbol rate), then M - 1 of the demodulator outputs will be characterized by (3.9), while the remaining output corresponding to the transmitted tone (say fs1) has the pdf (see (1.72) of Chapter 1) youo otherwise (3.10) where gh III STh/No is the hop signal-to-noise ratio, or, in this case, also the symbol signal-to-noise ratio. Letting Y * denote the random variable corresponding to the largest of the M non-coherent demodulator normalized outputs at the i-th sampling (once per hop) instant, then the probability density function of Y * is given (3.11) Time and Frequency Synchronization of Frequency-Hopped Receivers for the out-of-sync condition and + (M - (3.12) for the in-sync condition where PN(Y) and PS1N(Y) are, respectively, the probability distribution functions corresponding to the pdf's PN(y*) and Ps+N(y*) of (3.9) and (3.10), i.e., (3.13