em, we have that (1.49) Note from (1.49) that the mean search update is a linear function of the search cell. Because of this dependence on n, we cannot directly replace q by Nu/u in (1.4) to arrive at a formula for mean acquisition in the presence of code Doppler and Doppler rate. Rather, we should first find the average mean search update obtained by averaging Mn+1 of (1.49) over all q search cells, i.e., (1.50) and then make the above suggested replacement in (1.4). Thus, from (1.49) 13We continue to assume, as before, that the code phase derivatives are positive when they are in such a direction as to aid the search (reduce the acquisition time). Pseudonoise Code Acquisition in Direct-Sequence Receivers and (1.50), we have that KPFA)2 + K2PFA(1 - PFA)}. (1.51) Since the total mean search update must correspond to the total number of chips searched, we have that14 qu= = (1.52) Thus, substituting Nu/u for q in (1.51) results in a quadratic equation in , namely, (1.53) Letting (1.54) (1.55) Finally, substituting N/u for q in (1.4), with u defined by (1.55), gives the resulting expression for mean acquisition time in the presence of code Doppler and code Doppler rate, namely, (1.56) 14Here we have made the assumption that lAfd and Af are small enough such that Un+1 > 0 for all n = 0,1, = q - 1 and thus the search proceeds only in one direction, namely, that dictated by the local PN generator code phase update. The Single Dwell Serial PN Acquisition System Also, the acquisition time variance in the presence of code Doppler and code Doppler rate becomes (1.57) Note again that, when Afc = Afc=0, = we have that B = N/q' = N/q, C = 0, whereby (1.56) and (1.57) reduce to (1.4) and (1.7), respectively. 1.2.4 Evaluation of Detection Probability PD and False Alarm Probability PFA in Terms of PN Acquisition System Parameters The formulas for mean acquisition time and acquisition time variance devel- oped in the previous section are all functions of the detection probability PD, false alarm probability PFA, and dwell time T