bility that the central (entrance) cell is not the v-th (H1) but the j-th; in other words, II stands for the probability that H is actu- ally V - j positions to the right (if j (v + 1)/2) or j positions to the left (if - 1)/2). CONTINUOUS/CENTER Z SERIAL SEARCH, U ODD, I SA< U-1 START NUMBER = POSITIONS WHERE H1 IS MET BY SEARCH SEARCH STRATEGY UNCERTAINTY REGION Figure 1.34. Cell numbering for the continuous/center Z serial search with 1 Sk<(v-1)/2. Pseudonoise Code Acquisition in Direct-Sequence Receivers START (FALSE ALARM) utl -k u-k I Figure 1.35. Equivalent circular state diagram for the continuous/center Z serial search, 1k<(v- - 1)/2. Translating the Z motion of the search into an equivalent motion along an equivalent circular path leads to the circular state diagram of Figure 1.35, which for purposes of deriving the transfer function Un-k,Acq(2) from state V - k to state ACQ can be consolidated into Figure 1.36, which contains two forward paths and one feedback loop. Applying Mason's formula [32] to Figure 1.36 trivially provides the result (for 0 < k < (v - 1)/2) H(z)H ()[1 + HM(Z) Hu-2-2k(z)] (1.225) Finally, averaging Uv-k,AcQ(2) over the a priori probability distribution of the ENTER 1.36. Flow graph and corresponding path gains for the continuous/center Z serial search entering at node V - k,0<k<(v-1)/2. Figure Pseudonoise Code Acquisition in Direct-Sequence Receivers code phase uncertainty, and taking certain symmetries into account, gives the desired result for the acquisition process generating function U(z), namely, (v-3)/2 (1.226) Equation (1.226) can be combined with any a priori distribution to provide specific results. For example, for the symmetric triangular distribution (1.227) (2-1) j = V which in [22] is used as an approximation to a truncated Gaussian distribu- tion, U(z) of (1.226) becomes Ho(z) H0(z)) + 1 + (1.228) From the generating function U(z) of (1.228), we can obtain the mean acquisition time TACO by a relation identical to that in (1.3). For example, for a single dwell system