) since it is always true from Figure 1.23 that H(1) + HM(1) = 1. Hence, the case H(1) = 1 effectively corresponds to the existence of only one absorb- ing state, namely ACQ. A Unified Approach to Serial Search Acquisition with Fixed Dwell Times Table 1.1 Branch gains for single and multiple dwell systems. Single Dwell Multiple (N) Dwell Hp(z) HM(z) (1 - PD)z HFA(z) HNFA(Z) (1 - PFA)Z + (1 - PFA1/0)Z As simple examples of the application of the unified approach just described, we cite the single and multiple dwell serial search acquisition sys- tems for which the branch gains are shown in Table 1.1.²6 Taking the deriv- ative(s) of these gains as required in (1.56a) and evaluating them with their argument(s) equated to unity produces results identical to (1.3) and (1.112) with V = q (i.e., H contains only one cell). Alternately, for the single dwell system with worst case a priori probability distribution, we get the additional result (1.159) (worst case) where again we have set v = q. As previously mentioned, V will almost always exceed q in accordance with either (1.152a) or (1.152b). Thus, we now enter into a brief discussion of how to modify the flow graph to account for the fact that the H region actually contains 2q/Nu [or (2q/Nu) - 1] cells rather than just the single (v-th) cell shown in Figure 1.23. For example, consider a single dwell system with half- chip search update (q/Nu = 2) and let Q1, Q2, Q3, Q4 denote the four (pos- sibly only three) cells corresponding to the four possible values of relative code misalignment with magnitude less than a chip (i.e., within the triangu- For multiple dwell case, the gains H(z) should actually be written as H(z) where Z is the vector [21,Z2, ZN]. Also, the prime notation in (1.156) now refers to an N-dimensional par- tial differentiation with respect to the N compliments of Z with each component set equal to unity. Pseudonoise Code Acquisition in Direct-Sequence Receivers [1-P0101)] [1-PD(Q2)] [1-PD(Q3)] [1-PD1Q41]z H,REGION Figure 1.24. Expansion of the H region f