1.4 A UNIFIED APPROACH TO SERIAL SEARCH ACQUISITION WITH FIXED DWELL TIMES
1.4.1 The Flow Graph Technique
Consider a generalized serial search system of the type mentioned above whereH1denotes the hypothesis that the received and local codes are mis- aligned by less than a code chip (previously designated as a “hit”) and H0 the alternate hypothesis, i.e., the relative alignment is greater than or equal to a chip. The discrete Markovian nature of the underlying process of such a system allows it to be represented by a n⫹2-state flow graph, where n⫺ 1 of the states belong to the cells corresponding to H0, and one state (the n- th) corresponds to H1. Since, depending on the actual value of the relative offset (misalignment) between the codes, up to either 2q/Nuor (2q/NU)⫺1 cells correspond to H1(recall that Nu/qcorresponds to the fractional search update in chips), and the total number of cells for H0andH1combined is q, thennis determined from either
(1.152a) or
(1.152b) These nstates are indexed in a circular arrangement (see Figure 1.23) with thei-th state,i⫽1, 2, . . . ,q⫺1 corresponding to the i-th offset code phase position to the right of the “true” sync position (H1). The remaining two states are the false alarm(FA)stateand the correct acquisition(ACQ)state.
n⫺1⫹ 2q Nu
⫺1⫽q. n⫺1⫹ 2q
Nu⫽q
25This excludes schemes based on sequential detection methods, which will be discussed by themselves later on in the chapter.
These are also indicated in Figure 1.23 where it is observed that the acqui- sition state can be directly reached only from the n-th (H1) state, whereas the false alarm state can be directly reached from any of the n⫺ 1 states corresponding to the offset cells (H0).
A flow graph model such as that illustrated in Figure 1.23 is an exact rep- resentation of the state transition diagram of serial search acquisition sys- tems with no absolute limit on acquisition time. Although we are primarily interested in this class of systems (since their underlying process is indeed Markovian), we shall briefly mentioned at the conclusion of this section how the above model can be used in the presence of a finite limit on acquisition time.
The first way in which the flow graph model of Figure 1.23 is a general- ization of those previously considered is that the a prioriprobability distri- Figure 1.23. A flow graph representation of a generalized serial search acquisition system—the circular state diagram.
bution {pj;j⫽1, 2, . . . ,n) assigned to the nstates at which the search process can be entered is arbitrary.In the absence of any a prioriinformation con- cerning the initial relative position of the codes, the system designer would assign a uniform distribution to the model, i.e., {pj⫽ 1/n;j⫽ 1, 2, . . . ,n}.
This is the case which we have discussed thus far. Another special case of the more general formulation might be a worst case distribution (p1⫽1,pj
⫽0;j⫽1) corresponding to an initial relative code position farthest from the correct sync position.
The second way in which Figure 1.23 is a generalization of the previous notions is that each branch of the flow graph is assigned a generalized gain H(z) which characterizes all possible ways by which the process can move along that branch. The significance of the subscripts on these branches is as follows:
HD(z)⫽gain for verification of detection HM(z)⫽gain for missed verification of detection HFA(z)⫽gain for false alarm occurrence
HNFA(z)⫽gain for no false alarm occurrence
Hp(z)⫽gain for penalty after false alarm occurrence
Proper combination of these gains then allows computation of the gain asso- ciated with the “generalized” branch between any pair of nodes, e.g.,
(1.153) represents the gain in going from node ito node i⫹1 for i⫽1, 2, . . . ,n⫺1.
As before, the flow graph representation of the system is used to compute the moment generating function U(z) of the underlying acquisition process.
Using standard flow graph reduction techniques, it can be shown [14] that (1.154) For the two special cases of uniform and worst case a prioristate probabil- ity distributions, (1.154) reduces to
(1.155a) and
(1.155b) The complete statistical description of the acquisition process is obtained by substituting for the various H(z)’s in (1.154) the expressions appropriate to the particular configuration at hand and then, as before, expanding the result in a power series in z(see (1.94)). In general, obtaining the complete
U1z2 ⫽ HD1z2H0n⫺11z2
1⫺HM1z2H0n⫺11z2 , 1worst case2. U1z2⫽ 1
n
HD1z2 11⫺H0n1z2 2
11⫺HM1z2H0n⫺11z2 2 11⫺H01z2 2,1uniform2 U1z2⫽ HD1z2
1⫺HM1z2H0n⫺11z2 a
n
i⫽1piH0n⫺i1z2.
H01z2 ⫽HNFA1z2 ⫹HFA1z2Hp1z2
set of coefficients of this series is a difficult task and approximations must be made in the manner discussed in [7]. Alternately, if one is pleased with the first few moments of the acquisition time, then the relations of (1.3) and (1.6) as applied to U(z) of (1.154) or its special cases in (1.155) are appro- priate. In particular, for the mean acquisition time , we obtain from (1.3) and (1.155) the results
(1.156a) and
(1.156b) where the primed quantities denote differentiation with respect to z, i.e., H⬘(1)⫽d/dzH(z)0z⫽1, and
(1.157) i.e., the probability of acquiring after any number of dwells (see (1.98)).
Furthermore, note that in the case H0(1)⫽1, (1.154) and (1.157) combine to yield
(1.158)
since it is always true from Figure 1.23 that HD(1)⫹HM(1)⫽1. Hence, the caseH0(1)⫽1 effectively corresponds to the existence of only one absorb- ing state, namely ACQ.
PACQ⫽ HD112 1⫺HM112 ⫽1 PACQ⫽^ PACQ1q2 ⫽U112⫽ a
q i⫽0
pi
TACQ 1worst
case2
⫽td⫻f PACQ
HD112 eHDœ 112⫹HMœ 112PACQ⫹ 1n⫺12H0œ112 H0112
⫻1HD112⫹HM112PACQ2 f pifH0112 6 1 1
HD112 3HDœ 112⫹HMœ 112⫹ 1n⫺12H0œ112 pifH0112⫽1
T
1uniform2ACQ ⫽td⫻h
PACQeHDœ112
HD112 ⫹ H0œ112
1⫺H0112 ⫺nH0œ112H0n⫺1112 1⫺H0n112
⫹ 1n⫺12HM112H0n⫺2112H0œ112 ⫹HMœ 112H0n⫺1112 1⫺HM112H0n⫺1112 f pifH0112 6 1
1
HD112 cHDœ112 ⫹HMœ 112 ⫹1n⫺12H0œ112 a1⫺HD112 2 b d pifH0112⫽1
TACQ
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 1.1.26Taking 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) withn⫽q(i.e.,H1contains only one cell). Alternately, for the single dwell system with worst case a prioriprobability distribution, we get the additional result
(1.159)
where again we have set n⫽q.
As previously mentioned,nwill almost always exceed qin 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 H1region actually contains 2q/Nu[or (2q/Nu)⫺1] cells rather than just the single (n-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,Q4denote 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-
TACQ
1worst case2⫽ 1
PD
31⫹ 1q⫺12 11⫹KPFA2 4td
26For the multiple dwell case, the gains H(z) should actually be written as H(z) where zis the vector [z1,z2, . . . ,zN]. Also, the prime notation in (1.156) now refers to an N-dimensional par- tial differentiation with respect to the Ncompliments of zwith each component set equal to unity.
Table 1.1
Branch gains for single and multiple dwell systems.
Gain Single Dwell Multiple (N) Dwell
Hp(z) zK
HD(z) PDz
HM(z) (1 ⫺PD)z
HFA(z) PFAz
HNFA(z) (1 ⫺PFA)z
⫹11⫺PFA1>02z1
a
N j⫽2aq
j⫺1
i⫽1PFAi0i⫺1zib11⫺PFAj0j⫺12zj
q
N
i⫽1PFAi0i⫺1zi⫽PFAq
N i⫽1zi
⫹11⫺PD1002z1
a
N j⫽2aq
j⫺1 i⫽1
PDi0i⫺1zib11⫺PDj0j⫺12zj
q
N
i⫽1PDi0i⫺1zi⫽PDq
N i⫽1zi
zNK
lar correlation curve). Then, the H1region of the flow graph expands as in Figure 1.24, where PD(Qi);i⫽1, 2, 3, 4 denotes the detection probability eval- uated via (1.71) or (1.80) at the relative code misalignment corresponding toQi. Thus, instead of the simple entries of HD(z) and HM(z) as in Table 1.1, we now have
(1.160) and
(1.161) As promised, we conclude this section with a discussion of how to apply the flow graph model to the limited acquisition time case. We recall from the introduction that this case is characterized by the requirement to achieve acquisition (with a given probability) within a finite time, say Ts. Thus, the appropriate performance measure to use here is the probability that the acquisition time TACQis less than Ts, which can be calculated by evaluating the cumulative distribution function of TACQatTs. Alternately, if Tscorre- sponds to Jdwells, then we are interested in the probability PACQ(J) of acqui-
⫽ cq
4
j⫽1
11⫺PD1Qj22 dz4. HM1z2⫽ q
4
j⫽1aq
j⫺1
i⫽111⫺PD1Qj22z HD1z2⫽ a
4
j⫽1aq
j⫺1
i⫽111⫺PD1Qi2 2PD1Qj2 bzj
Figure 1.24. Expansion of the H1region for a single dwell serial search system indi- cating the effect of multiple cells with non-zero correlation.
sition in Jor fewer dwells as given by a relation analogous to (1.98), which can be evaluated by finding the generating function corresponding to this cumulative probability distribution. Such a generating function is defined by (1.162) which can be written in the form
(1.163) Comparing (1.163) with (1.94), we observe that
(1.164) SincePACQ(J) is the coefficient of zJin the power series of (1.162), we can evaluate it with a contour integral analogous to that used to evaluate inverse ztransforms, i.e.,
(1.165) whereU(z) is given by (1.154) and ⌫is a counterclockwise closed contour in the region of convergence of V(z) that encircles the origin.