minimization subject to a given desired probability of acquiring by the end of these Nsw sweeps. Letting PNSW denote this probability, i.e., the probability of acquiring in Nsw or fewer sweeps, then since k represents the sweep (search) at which the sys- tem is first acquired, clearly we would have (1.205) where p(k) is the pdf of k as given in (1.203). Substituting (1.203) into (1.205) Pseudonoise Code Acquisition in Direct-Sequence Receivers and simplifying yields (1.206) Analogous to (1.10), the acquisition time for the first Nsw sweeps is28 (1.207) Thus, in mathematical terms, the optimization problem can be stated as: For a given PD and (x), choose the search lengths L1, L2,..., , LN,w SO as to minimize LT with an acquisition probability equal to P N su This type of problem is most easily solved by the method of LaGrange multipliers. In particular, the set L1, L2, LN,10 corresponds to the sta- tionary points of the function (1.208) where 1 is the LaGrange multiplier (as yet unknown). The stationary points of F correspond to the locations where (1.209) Thus, substituting (1.206) and (1.207) into (1.208) and performing the required partial differentations gives (1.210) or in view of (1.204) P(1-P)2); = 1, 2, , Nsw. SW (1.211) For a given (x), one can for each i implicitly solve for Li as a function of 1, say fi(A). Then 1 can, in principle, be eliminated by satisfying the constraint in (1.206), i.e., (1.212) Typically, for a given P, and form of pdf (x), the solution of (1.212) for l (and thus Li: i = 1, 2, , Nsw) will exist only for certain val- ues of Nsw. To show this, lower and upper bounds on Nsw can be obtained as follows. Since P(Lk) < 1 for all k = 1, 2, ,Now, then from (1.212) we 28We assume here, for simplicity, a zero false alarm probability or, equivalently, that the penalty time associated with the occurrence of a false alarm is zero. In the next section, we shall include this effect as part of the more general formulation of serial search systems with non-uniform search strategies following the u