Ri-1+1kr + Ri-1+1 (1.241) PN Sync Search Procedures and Sweep Strategies v-k+l (FALSE ALARM) -Ri+1 Rj- I -Rj+1 Rj +1 Figure 1.40a. Portion of state diagram corresponding to Rj, expanding window search, 1jjk - 1 (reprinted from [38]). where HH(1-i-1) (1.242a) F2(i,j) = H2R) (1.242b) F3(k,j) = (1.242c) S(m, n) = (1.242d) if m>n. In deriving the above, the simplifying assumptions were made that the Pseudonoise Code Acquisition in Direct-Sequence Receivers R k-l 2-4+1 (FALSE ALARM) Figure 1.40b. Portion of state diagram corresponding to Rj, expanding window search, j jk (reprinted from [38]). a priori distribution is symmetric (TTK = -), v is large, and R A Nsw;i = 1, , Nsw, all of which are met in practical systems. Also, the dependence of the gains on Z has been dropped. As a particular case of interest, let us consider the "equiexpanding" win- dow search, whereby the radii R increase by the same amount of code chips, i.e., R = (v/2Nsw)i; i=1,...,Nsw Arbitrarily assuming case A, the moment generating function of (1.240) becomes H ri(i-1)v/Nsw - U(z) = Nsw-i H11Hi (1.243) Ri-1+ +1<k<R PN Sync Search Procedures and Sweep Strategies and, for the single dwell detector, the corresponding mean acquisition time is given by [a(k, Nsw; PD) + B(k, Nsw; PD) + y(k, Nsw; PD) (1.244) where a(k, Nsw; PD) = + - k)(1 - (1 - P)2 (1.245a) 1 - PD)2 k(k - 1)(1 - P)2k (1.245b) + (1 - (1.245c) k = 1,2,...,New (1.246a) Ill l't;; k = 1, 2, Nsw. (1.246b) Pseudonoise Code Acquisition in Direct-Sequence Receivers For large v, it is convenient to replace the discrete a priori probability dis- tribution TTK by a continuous distribution corresponding to its envelope (x) and evaluate the sums in (1.246) as integrals, namely, (1.247a) -(v/2)(1-(k-1)/N31)) (1.247b) For the symmetric triangular distribution of (1.227), we have (1.248) P0=04 1.0 2.0 3.0 4.0 5.0 6.0 70 8.0 9.0 10.0 11.0 Figure 1.41. Normalized mean acquisition time versus number of partial windows for expanding window search strategy; single dwell system with triangular a priori distr