he Single Dwell System A more compllete statistical characterization of the acquisition time per- formance of a single dwell PN acquisition system can be had by consider- ing the probability of acquisition in k or fewer dwells. Computation of this cumulative probability requires first obtaining an expression for the proba- bility density function of the number of dwells to obtain successful syn- chronization [6]. Once again in order to readily gain immediate and exact results, one resorts to the signal flow graph approach which provides the system gener- Pseudonoise Code Acquisition in Direct-Sequence Receivers ating function as in (1.1). Starting with this expression, we first rewrite it in the form a power series in Z, namely, PPE1(190 Applying the binomial theorem to the factor involving PFA, the generating function can be rewritten as the triple sum X (1 - Pp)izig+1+hk (1.91) In order to proceed further, we must relate the generating function U(z) to the probability of successful synchronization. Recalling (1.31), then (1.92) is the integer valued random variable which represents the total number of cells that have been examined when successful synchronization (acquisition) occurs. Letting pj denote the probability that the system acquires on the j- th tested, or, in terms of NACQ, Pj=Pr{NAcQ=j};j=1,2,3. (1.93) then Z has the moment generating function U(2)= (1.94) Thus, equating the coefficients of z'; j = 1,2,3, in (1.91) to pj produces, at least in principle, the desired result. Determination of these coefficients in (1.91) is possible [7] but quite tedious. To make matters more tractable, but still meaningful, we shall make the assumption that the system acquires within a single search of q cells, or equivalently we impose the restriction NAce<q. Indeed, if this were not the case in practice, then the serial search synchronization system would give way to the maximum-likelihood system discussed earlier since if all the cells were to be examined there would be no need for a threshold test on each. Returnin