1.6 PN SYNC SEARCH PROCEDURES AND SWEEP STRATEGIES FOR A NON-UNIFORMLY DISTRIBUTED SIGNAL LOCATION
1.6.1 An Example—Single Dwell Serial Acquisition with an Optimized Expanding Window Search
As a means of illustration, we shall reduce the scope of the general search problem discussed above by considering first the specific, but practical, case of a single dwell PN acquisition system with a symmetric
“expanding window” search centered around the mean of a symmetric, unimodal,a priorisignal location pdf px(x) [20]—[24]. An example of such a search strategy for Nsw 3 sweeps prior to acquisition is illustrated
in Figure 1.33. Let denote the lengths (in
number of cells) of uncertainty regions to be searched during the Nsw sweeps. (We shall set L00 for convenience.) Then if, as before,k(inte-
L1L2L3 # # # L
Nsw
838 Pseudonoise Code Acquisition in Direct-Sequence Receivers
Figure 1.33. A three-window optimized sweep strategy.
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ger) denotes the particular search of the uncertainty region during which the correct cell is firstdetected, then the generalization of (1.8) for the pdf of kis [20]
(1.203) where P(Ln) is the probability that the signal location xis within the set Ln, i.e.,
(1.204) with P(L0) 0 and PDis still the probability of detection given that the sig- nal is there. Note that for a uniform a priori signal location pdf and
then and only
the n1 term has a non-zero contribution to the sum in (1.203), whereupon p(k) immediately reduces to the result in (1.8).
Although (1.203) can be formally derived using Bayes’ probability rule, it can also be obtained by a simple heuristic argument as follows. Let Lkbe divided into knon-overlapping regions, namely,L1,L2L1,L3L2, . . . , LkLk1. Each of these regions represents the additional number of cells searched on a given sweep relative to the previous sweep. By the initiation of the k-th sweep, the region LnLn1has been searched k2 ntimes;n 1, 2, . . . ,k 1. Thus the jointprobability of firstdetecting signal in Ln Ln1and that indeed it was in that region is PD(1 PD)k2nP(LnLn21) or, in view of (1.204),PD(1 PD)kn[P(Ln) P(Ln1)]. Since the region Lk Lk21has not as yet been searched, the joint probability of detecting the sig- nal in this region and that it is indeed there is simply PD[P(Lk) P(Lk21)].
Since the above set of joint probabilities corresponds to mutually exclusive events, (1.203) follows immediately.
To define an optimum symmetric search strategy, we must first define a criterion of optimization along with any additional constraints imposed by the physical system. As suggested earlier, we shall choose as an optimum search strategy that which minimizes the total acquisition time for Nsw sweeps where the constraint is to accomplish this minimization subject to a given desired probability of acquiring by the end of these Nsw sweeps.
Letting denote this probability, i.e., the probability of acquiring in Nsw or fewer sweeps, then since krepresents the sweep (search) at which the sys- tem is firstacquired, clearly we would have
(1.205) where p(k) is the pdf of kas given in (1.203). Substituting (1.203) into (1.205)
PNsw a
Nsw
k1p1k2 PNsw
P1L12 P1L22 pP1LNsw21 L1L2pLNswq
P1Ln2 LLn>2
n>2
px1x2dx20Ln>2px1x2dx
p1k2 PDa
k
n111PD2kn3P1Ln2P1Ln12 4; k1, 2, 3,p
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and simplifying yields
(1.206) Analogous to (1.10), the acquisition time for the first Nswsweeps is28
(1.207) Thus, in mathematical terms, the optimization problem can be stated as:
For a given PDand px(x), choose the search lengths so as to minimize LTwith an acquisition probability equal to .
This type of problem is most easily solved by the method of LaGrange multipliers. In particular, the set corresponds to the sta- tionary points of the function
(1.208) where l is the LaGrange multiplier (as yet unknown). The stationary points of Fcorrespond 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)
(1.211) For a given px(x), one can for each iimplicitly solve for Lias a function of l, say fi(l). Then lcan, in principle, be eliminated by satisfying the constraint in (1.206), i.e.,
(1.212) Typically, for a given , and form of pdf px(x), the solution of (1.212) for l(and thus Li:i1, 2, . . . ,Nsw) will exist only for certain val- ues of Nsw. To show this, lower and upper bounds on Nswcan be obtained as follows. Since P(Lk) 1 for all k1, 2, . . . ,Nsw, then from (1.212) we
PD, PNsw PNswPDa
Nsw k1
P1fk2 1l2 2 11PD2Nswk. PD11PD2Nswipx1Li>22l; i1, 2,p, Nsw.
PD11PD2NswidP1Li2
dLi l0 0F
0Li0; i1, 2,p, Nsw. FPNswlLT
L1, L2,p, LNsw
PNsw
L1, L2,p, LNsw TNswtda
Nsw
i1
Li^ tdLT. PNswPDa
Nsw
k1
p1Lk211PD2Nswk.
840 Pseudonoise Code Acquisition in Direct-Sequence Receivers
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 unified approach discussed in Section 1.4.
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have
(1.213) Satisfaction of (1.213) with the equality sign gives a lower bound on Nsw, namely,
(1.214) Similarly, since , then
Hence,
(1.215) or
(1.216) Before presenting a specific example of the application of the previous results and the benefits gained therein by using an optimum search strategy, we must first discuss a measure of improvement which can be used in com- paring the optimized scheme to the uniform (full sweeps across the entire uncertainty region) sweep scheme. We recall that for a uniform a prioricell location pdf, the probability of first acquiring during the k-th full sweep is given by p(k) of (1.8). Thus, for a given desired probability of acquisition , the number NFof full sweeps (each having qcells) would be obtained from the solution to
(1.217) or
(1.218) The solution to (1.218) is easily shown to be
(1.219) where again :x; denotes the largest integer less than or equal to x. Since within a given sweep interval, say the k-th, the cumulative probability of acquisition would be a linear interpolation between the upper and lower bounds of (1.218), the total search length LTU(in cells) for the uniform sweep
NF jln11PNsw2 ln11PD2 k
111PD2NFPNsw1 11PD2NF1. a
NF1 i0
PD11PD2iPN
sw a
NF
i0
PD11PD2i PNsw
Nsw
lnc1 PNsw P1L12 d ln11PD2 . PNswP1L12 31 11PD2Nsw4 P1LNsw2.
P1L12P1L22 # # #
L1L2 # # # L
Nsw
Nswln11PNsw2 ln11PD2 . PNswPDa
Nsw
k111PD2Nswk111PD2Nsw.
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842 Pseudonoise Code Acquisition in Direct-Sequence Receivers
Table 1.2 Improvement in acquisition time performance of optimized non-uniform sweep relative to uniform sweep. % Reduction Uniform SweepOptimized SweepAcquisition Time NFLTU/s*Li/sLT/sI (Nsw3) PD.252.548.831.6640 214.672.96 3.33 (Nsw4) 1.19 1.938.471.7342 2.46 2.89 (Nsw4) PD.52.63 3.53 320.44.2415.251.3425 4.85 (Nsw5) .56 2.4215.221.3425 3.38 4.12 4.74 *For the uniform sweep case,we assume the search region corresponds to 3s,i.e.,q6s.
PNsw.9 PNsw.5
a11 Ib100
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strategy is given by
(1.220) or upon simplification
(1.221) The improvement factor of the optimized sweep strategy over the uniform sweep strategy is then
(1.222) where LTis determined from the solution of (1.211) using the same PDand
.
As an example, consider a truncated Gaussian a prioripdf for px(x). Then, the solution to (1.211) for the optimized sweep lengths can be expressed in the form [20]
(1.223) where s is the standard deviation of px(x). Using (1.212) to eliminate l requires solution of the transcendental equation
(1.224) which although impossible analytically can be accomplished numerically on a digital computer for each allowable value of Nsw. Several numerical exam- ples were worked out in [20] with the results shown in Table 1.2.