 definition of Z in terms of Z, we have N - 1 Z* = Yy (1.74) Since, by previous assumption, the yk S are independent random variables, then for large N, Z is approximately Gaussian distributed with mean Z* = NBY* and variance O2 = NBO2 The means and variance of the pdf's in (1.72) and (1.73) are well known [35] to be = signal present (1.75) y*=1;02  1;signal absent. = = (1.76) Thus, Z* = N(1 + y); = N(1 + 2y); signal present (1.77) Z* = N; O2 = N; signal absent. (1.78) Using the Gaussian assumption, the false alarm probability is (1.79) where Q(x) is the Gaussian probability integral. Thus, if PFA is specified, The Single Dwell Serial PN Acquisition System can be determined. The corresponding detection probability under the same assumption is (1.80) Combining (1.79) and (1.80) and reidentifying NB and Y in terms of the sys- tem parameters gives the final relation (1.81) /1+2(A) Thus, given PD, PFA, A2/No and B, the dwell time Td is determined. Before we use (1.81) and the dwell time determined from it in the formulas derived in the previous section for mean acquisition time and acquisition time variance, several modifications based upon practical con- siderations must be made. 1.2.5 Effective Probability of Detection and Timing Misalignment The calculation of detection probability as in (1.71) or (1.80) implicitly assumed that only one cell in the entire search satisfies the "signal present" hypothesis. In actuality, since the PN correlation curve exists over an inter- val of 1 chip around the peak, a system which updates the locally gener- ated code phase, for example, in half-chip increments would yield several cells for which signal could be considered present. Typically, the system is designed on the basis of the worst case correla- tion, which for the half-chip update case would correspond to the pair of correlation points one-quarter chip away from the correlation peak. Since the normalized correlation value at these points is 0.75 (relative to a peak of 1), then the single signal point calculation of detec