g now to (1.91), we recognize that the restrction NACQ < q is equivalent to considering only the i = 0 term in the summtation on i, since the index i represents the number of times that the entire group of cells has been previously examined. Thus, i = 0 implies that the q cells are being examined for the first time. Making this simplification in (1.91) produces The Single Dwell Serial PN Acquisition System the result (1.95) Now making the equivalence between the coefficients of zi in (1.95) and (1.94) produces, after much simplification, the desired result, namely [6], [7] (1.96) where (1.97) and the notation a represents the largest integer less than or equal to a. Furthermore, the term corresponding to h=h+1 clearly has meaning only if j - 1 - -(h+1)K > 0; otherwise its contribution is assumed equal to zero. The cumulative distribution of NAco, namely, (1.98) PFAS.OOL PFA - 01 PFA.1 Figure 1.11. Normalized cumulative distribution function for q = 102, K = 10 and various values of PEA (reprinted from [6]). Pseudonoise Code Acquisition in Direct-Sequence Receivers PFAB.01 PFA = 2 Figure 1.12. Normalized cumulative distribution for q = 10 ², K = 102 and various values of PEA (reprinted from [6]). represents the probability of acquisition in j or fewer dwells. Although a closed form expression for PACQ(j) using pj of (1.96) appears impossible, one can readily obtain numerical results for moderate values of q using digital computation. Figures 1.11-1.14 illustrate the normalized acquisi- tion probability of PACO(j)/PD as a function of j/q for various values of q, K, and PFA. In all cases, as PFA goes to zero, we obtain the optimum per- PFA2.2 Figure 1.13. Normalized cumulative distribution function for q = 5 x 10 , K = 102 and various values of PEA (reprinted from [6]). The Single Dwell Serial PN Acquisition System Figure 1.14. Normalized cumulative distribution function for q = 5 10 superscript(3), K = 103 and various values of PEA (reprinted from [6]). formance corresponding to PACO(j)/PD = j/q. For a given q, inc