2) + Td2 Td1 = Td2 > Td, (1.135) which is the same result as (1.134). Clearly, then, for some PFA PEA < 1, say PEAL, exists a corresponding value of Td1 exists, namely, (1.136) which minimizes G(2). Letting G*(2) denote this minimum value, i.e., G*(2)TPAT1 - (1.137) then the ratio T&/G*(2) is a measure of the minimum improvement in acqui- sition time and variance of the double dwell system over the single dwell sys- Figures 1.19 and 1.20 are plots of Td/G*(2) versus PEA with PD as a para- meter and y' = -20 dB and -10 dB respectively. We note from these results that for fixed y' and small PFA, the minimum performance improvement offered by the two-dwell system over the single dwell system improves with increasing detection probability up to a certain point. Beyond that point, Td/G*(2) decreases with increasing PD. In fact, as PD approaches unity, then from (1.125), P also approaches unity, and, from (1.131) and (1.132), Td2 approaches Td. Also from (1.81), when PD approaches unity, then PEA tends to unity for any y' and all NB = BTd Thus, from (1.133), PEA1 also approaches unity, and, from (1.132), Td1 approaches Td2. Finally, using the above facts in (1.130), we see that G(2) = G*(2) approaches Td or Td/G*(2) approaches unity. To generalize the above procedure to arbitrary N, we begin by generaliz- ing (1.119) to become (1.138) i.e., all N unconditional detection probabilities are made equal by appro- priate choice of the N detection thresholds. Next, following steps analo- gous to (1.120)-(1.122), it can be shown that the following recursion The Multiple Dwell Serial PN Acquisition System PD=.01 FALSE ALARM PROBABILITY, PFA Figure 1.19. Acquisition performance improvement factor for two-dwell system over single dwell system versus false alarm probability with detection probability as a parameter; y' = -20 dB. relation holds: (1.139) Since the left-hand side of (1.139) again represents the system detection probability for the N-dwell system, and the product on the right-hand side represents the same probability 