1.6 PN SYNC SEARCH PROCEDURES AND SWEEP STRATEGIES FOR A NON-UNIFORMLY DISTRIBUTED SIGNAL LOCATION
1.6.2 Application of the Circular State Diagram Approach
In Section 1.6.1 we considered the optimization and performance of a par- ticular expanding window serial search strategy as applied to a single dwell acquisition system. This strategy was but one of a class of search strategies that attain improved acquisition performance when a priori probabilistic information about the true code sync position is available.
In this section, we generalize these results by allowing for an arbitrary ser- ial search strategy and an arbitrary detector configuration. The approach taken is based upon the circular state diagram method introduced in Section
PNsw⫽PDa
Nsw k⫽1
11⫺PD2Nsw⫺ke1⫺2Qa
B2 lncPD11⫺PD2Nsw⫺k
l12ps d b f
⫽^ fi1l2;i⫽1, 2,p,Nsw
Li⫽212s
BlncPD11⫺PD2Nsw⫺i l12ps d PNsw
I⫽LTU
LT
LTU⫽qeNF⫹ 11⫺PD2NF⫺ 11⫺PNsw2 11⫺PD2NF⫺ 11⫺PD2NF⫹1f. LTU⫽qeNF⫹ PNsw⫺ 31⫺ 11⫺PD2NF4
1⫺ 11⫺PD2NF⫹1⫺ 31⫺11⫺PD2NF4f
1.4 as a tool for modelling and analyzing the complete acquisition behavior of straight (uniform) serial search schemes. The advantage of this approach in the application being considered here is that it circumvents complicated combinatorial arguments used in [21]—[23] to characterize the performance of such systems by employing a transform domain description of the sto- chastic acquisition process. Such a description allows a simple and more sys- tematic evaluation of the generating function of the process using well-known flow graph reduction techniques.
While the method to be described applies to arbitrary serial search strate- gies, the focus here will be on the two classes of non-uniform strategies depicted in Figure 1.2, namely, the z-search and the expanding window search.These two classes can be further subdivided into brokenorcontinu- oussearches (depending on whether the receiver employs rewinding in order to skip certain cells), and edgeorcentersearches (depending on where the search and each subsequent sweep are initiated).
Recall from Section 1.4.1 that for a straight serial search, the process can be modelled by a circular state diagram (see Figure 1.23) with n⫹2 states, wheren⫺ 1 of these correspond to out-of-sync cells (hypothesis H0), one to the collecting state (hypothesis H1), one to the absorbing correct state (ACQ), and one to the possibly absorbing false alarm state. Along the branches between these various states are found generalized gains H(z) that represent the generating functions of the individual discrete-time detection processes associated with the corresponding paths. Applying standard flow graph reduction techniques to the circular state diagrams then allows eval- uation of the moment generating function U(z) of the underlying acquisi- tion process.
To apply this approach to the non-uniform search case, one merely trans- lates the motion of the specific search strategy under consideration into a circular motion along an equivalent circular state diagram analogous to Figure 1.23. To demonstrate how this is accomplished, let us consider first the continuous/center zserial search illustrated in Figure 1.34. Here the search is initiated at the center of the code phase uncertainty region and proceeds following the arrows in the manner shown; i.e., it reverses direc- tion every time the boundaries are reached. Assuming that the location of the true sync state (H1) is at the shaded cell, the search process will meet it once during each sweep at the dotted positions. We indicate the starting cell by n⫺kwhere, for the H1cell to be in the indicated side,kmust sat- isfy 1 ⱕ kⱕ(n⫺ 1)/2 (for convenience, we assume nto be odd). Similar diagrams can be drawn for k⫽0 or (n⫹1)/2ⱕkⱕn⫺1; however, that will not be necessary due to the symmetry of the problem. Furthermore, we note that since the search is always initiated at the center,pjshould be interpreted as the probability that the central (entrance) cell is not the n-th (H1) but the j-th; in other words,pjstands for the probability that H1is actu- allyn⫺jpositions to the right (if jⱖ(n⫹1)/2) or jpositions to the left (if jⱕ(n⫺1)/2).
Figure 1.34.Cell numbering for the continuous/center zserial search with 1 ⬉k⬍(n⫺1)/2.
Translating the zmotion 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(z) from state n⫺kto state ACQcan 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⬍(n⫺1)/2)
(1.225) Finally, averaging Un⫺k,ACQ(z) over the a prioriprobability distribution of the
Un⫺k,ACQ1z2⫽ H0k1z2HD1z2 31⫹HM1z2H0n⫺2⫺2k1z2 4 1⫺HM2 1z2H021n⫺221z2
Figure 1.35. Equivalent circular state diagram for the continuous/center zserial search, 1 ⬉k⬍(n⫺1)/2.
Figure 1.36.Flow graph and corresponding path gains for the continuous/center zserial search entering at node n⫺k,0⬍k⬍(n⫺1)/2.
code phase uncertainty, and taking certain symmetries into account, gives the desired result for the acquisition process generating function U(z), namely,
(1.226) Equation (1.226) can be combined with any a prioridistribution to provide specific results. For example, for the symmetric triangular distribution
(1.227)
which in [22] is used as an approximation to a truncated Gaussian distribu- tion,U(z) of (1.226) becomes
(1.228) From the generating function U(z) of (1.228), we can obtain the mean acquisition time 苶TACQby a relation identical to that in (1.3). For example, for a single dwell system, we use the branch gains of Table 1.1 together with the relation of (1.153) in (1.228), whereupon performing the required
⫹1⫹HM1z2H0n⫺21z2 f.
⫺H0n⫺31z2HM1z2 a1⫺ 2
n⫺1 a 1⫺H01n⫺12>21z2
H01n⫺32>21z211⫺H01z22bb d
⫻ eH01z211⫹H0n⫺11z22
1⫺H01z2 c1⫺ 2
n⫺1 a1⫺H01n⫺12>21z2
1⫺H01z2 b U1z2⫽ a 2
n⫺1b a HD1z2
1⫺HM2 1z2H021n⫺221z2b pj⫽f
a 2
n⫺1b c1⫺ a 2
n⫺1bjd j⫽1,p,n⫺1 2
pn⫺j j⫽n⫹1
2 ,p,n⫺1
a 2
n⫺1b j⫽n
⫹ H01n⫺12>21z2 3p1n⫹12>2⫹p1n⫺12>2H0n⫺11z2
1⫺HM1z2H02n⫺31z2 ⫹ pn
1⫺HM1z2H0n⫺21z2s.
⫽HD1z2 • a
1n⫺32>2
j⫽1
H0j1z2 3pn⫺j⫹pjH0n⫺11z2 4 31⫹HM1z2H0n⫺2⫺2j1z2 4 1⫺HM2 1z2H021n⫺221z2
U1z2⫽ a
n⫺1 k⫽0
pkUn⫺k,ACQ1z2
differentiation and evaluating the result at z⫽1 gives
(1.229) which for large nreduces to
(1.230) Consider now another variation of the class of z-search strategies, namely, the broken/center zsearch. This is similar to the continuous/center z, with the exception that the same cells are not searched twice in a row. Instead, when one of the two boundaries is reached, the local code is quickly rewound to the center and the search continues in the opposite direction. Clearly, for a prioridistributions which are peaked around the center, an improvement in acquisition performance should be expected with respect to the continu- ous/centerzstrategy. The magnitude of this improvement will be demon- strated shortly by comparing the mean acquisition time performance of the broken/ and continuous/center zsearch strategies for the triangular a priori distribution. First, however, we present the generating function U(z) for the broken/centerzwhich is obtained from its circular state diagram by steps identical to those used in arriving at (1.226). As in our previous discussions, lettingTrdenote the reset penalty time required to rewind the code,U(z) is found to be
(1.231) For the triangular a prioridistribution, (1.231) evaluates to
(1.232)
⫻ c1⫺ 2
n⫺1 a1⫺H01n⫺12>21z2
1⫺H01z2 b d ⫹ 1
1⫺zTr>tdHM1z2H01n⫺12>21z2f.
U1z2 ⫽ a 2
n⫺1bHD1z2 e 1⫹zTr>tdH01n⫹12>21z2
1⫺z2Tr>tdHM1z2H0n1z2 a H01z2
1⫺H01z2b
⫹ pn
1⫺zTr>tdHM1z2H01n⫺12>21z2ả.
U1z2⫽HD1z2 • a
1n⫺12>2
j⫽1
3pn⫺j⫹pjzTr>tdH01n⫹12>21z2 4H0j1z2
1⫺z2Tr>tdHM1z2H0n1z2
TACQ⫽tde 1 PD
⫹ 211⫹KPFA2n13⫺PD⫹PD22 3PD12⫺PD2 f.
⫺PD16n2⫺15n⫹92⫹PD212n2⫺4n2 4 f TACQ⫽tde 1
PD
⫹ 1⫹KPFA
3PD12⫺PD2 1n⫺12 36n2⫺18n⫹12
Once again applying the necessary differentiation to arrive at (1.232) the mean acquisition time of a single dwell system, we obtain from (1.232) the result
(1.233) which for large nreduces to
(1.234) To illustrate the improvement in mean acquisition time performance by using a broken/ rather than a continuous/center zsearch, we can take the ratio of the latter terms in (1.230) and (1.234) since, for large enough n, the first terms in these equations can be ignored. Thus, to a good approximation (1.235) Figure 1.37 is a plot of this mean acquisition time improvement factor ver- sus PD. We observe that the maximum improvement occurs for PD ⫽ 1 (absolute probability of detecting the correct cell once it is reached) in which case (1.235) reduces to
(1.236) i.e., a 37.5 percent saving in acquisition time.
In the more general case where the a prioriprobability distribution of the code phase uncertainty is arbitrary (but symmetric), for PD⫽1 and nlarge, it is simple to show that29
(1.237) TACQ0cont.
TACQ0broken
⫽ 2a
n>2 j⫽1
jpj⫹ n 2 2a
n>2
j⫽1jpj⫹ n 4 TACQ0cont.
TACQ0broken
⫽ 8 5⫽1.6 TACQ0cont.
TACQ0broken
⫽ 213⫺3PD⫹PD22 312⫺PD2 11⫺127PD2 . TACQ⫽tde 1
PD
c1⫹ a4⫺3PD
2 b Tr
tdd⫹ 11⫹KPFA2n PD
a1⫺ 7
12PDb f.
⫹ 11⫹KPFA2 1n⫺12PD
c 5
121n2⫺2n⫺32PD⫹ 1n2⫺2n⫺1211⫺PD2 ds TACQ⫽td• 1
PD
⫹ Tr
td
c an⫺3
2 bPD⫹21n⫺2211⫺PD2 d 1n⫺12PD
29When nis large, the assumption of nodd is inconsequential.
Figure 1.37.Mean acquisition time improvement factor versus detection probability for single dwell acquisition system with tri- angulara prioricode phase uncertainty distribution.
which is lower and upper bounded by
(1.238) corresponding to the a prioridistributions
(1.239a) and
(1.239b) Thus, regardless of the a priori probability distribution of the code phase uncertainty, the broken/center z search potentially offers an improvement of at least 20percent and at most 100percent over the continuous/center z search.
Of course, for PD⬍1, these improvements will decrease accordingly.
Finally, we consider the class of expanding window search strategies, two representative cases of which (A and B) are shown in Figure 1.38. The two cases differ in the way the search is continued once the entire uncer- tainty region has been covered without success. In particular, case A repeats the search starting from the R1window,30while case B continues by repeat- ing the window. We note that the strategies analyzed in [21]—[23]
constitute slight variations of case B. The equivalent circular state diagrams for the two cases are shown in Figure 1.39. We observe that, after complet- ing the first sweep, case B is indistinguishable from the continuous/
centerzsearch. Furthermore, the diagrams are composed of hexagons with an inscribed Ri; those represent portions of the state diagram which corre- spond to the partial sweeps Ri;i⫽1, . . . ,Nsw. As seen from Figure 1.38, for each starting cell n⫺kthere exists a minimum index jksuch that the first jk⫺1 partial windows do not contain the H1state, while the remaining Nsw
⫺jk⫹1 do. This is manifested in Figure 1.39 by the fact that only the hexa- gons after (and including) the jk-th can lead to the ACQstate. The portions of the state diagram included in the hexagon Rjcan be derived from Figure 1.35 with a proper modification. Typical forms are shown in Figures 1.40a and 1.40b for the two possibilities, i.e., 1 ⱕjⱕjk⫺1 (Figure 1.40a) and jk ⱕ jⱕNsw(Figure 1.40b). It has been assumed, without loss of generality, that 1 ⱕkⱕ(n⫺1)/2. Cascading the successive hexagons of Figures 1.40a and 1.40b as per Figure 1.39 will result in the equivalent circular state dia- grams for those expanding window search strategies. It is then a matter of systematically following the steps established previously (i.e., diagram con-
RNsw
RNsw
pj⫽0;j⫽n>2,n>2⫹1.
pn>2⫽pn>2⫹1⫽12
pj⫽0;j⫽1,n p1⫽pn⫽12
6
5ⱕ TACQ0cont.
TACQ0broken
⫽2
30Note that the radius Riof the i-th partial window is one-half of the length Liof the sweep for that window.
Figure 1.38. Definitions for the expanding window search technique; cases A and B (reprinted from [38]).
solidation, gain calculation via Mason’s formula, and averaging) in order to arrive at the final expressions of interest. For the two cases A and B, they are given by
(Case A) (1.240)
and (Case B)
(1.241)
⫹ a
Nsw
i⫽1
HDH04S11,i⫺12
a
Nsw
j⫽iF21i,j2 a
Ri⫺1⫹1⬉k⬉Ri
F31k,j2 U1z2⫽ aHD11⫹H0n2H04S11,Nsw2
1⫺HM2
H02n ba
Nsw i⫽1
H021Nsw⫺i⫹12
Ri⫺1⫹a1⬉k⬉Ri
F31k,Nsw2 U1z2 ⫽ a
Nsw
i⫽1
F11i2a
Nsw
j⫽iF21i,j2 a
Ri⫺1⫹1⬉k⬉Ri
F31k,j2
Figure 1.39. Equivalent state diagrams for the two cases A and B of the expand- ing window serial search of Figure 1.38 (reprinted from [38]).
where
(1.242a) (1.242b) (1.242c) and
(1.242d) In deriving the above, the simplifying assumptions were made that the
S1m,n2 ⫽ • a
n
l⫽mRl, if m⬉n 0, if m 7 n. F31k,j2⫽pk1H0k⫹HMH02Rj⫺k2 F21i,j2 ⫽HM21j⫺12H04S1i,j⫺1211⫹H02Rj2
F11i2 ⫽ HDH04S11,i⫺12
1⫺HM21Nsw⫺i⫹12H04S11,Nsw2
Figure 1.40a. Portion of state diagram corresponding to Rj, expanding window search, 1⬉j⬉jk⫺1 (reprinted from [38]).
a prioridistribution is symmetric (pk⫽pn⫺k),nis large, and RiⰇNsw;i⫽1, 2, . . . ,Nsw, all of which are met in practical systems. Also, the dependence of the gains on zhas been dropped.
As a particular case of interest, let us consider the “equiexpanding” win- dow search, whereby the radii Riincrease by the same amount of code chips, i.e.,Ri⫽(n/2Nsw)i;i⫽1, . . . ,Nsw. Arbitrarily assuming case A, the moment generating function of (1.240) becomes
(1.243)
⫻ a
Ri⫺1⫹1ⱕkⱕRi
pk1H0k⫹HMH01l⫹i2n>Nsw⫺k2
⫻ a
Nsw⫺i l⫽0
HM2lH0l12i⫹l⫺12n>Nsw11⫹H01l⫹i2n>Nsw2 U1z2⫽HDa
Nsw
i⫽1
H0i1i⫺12n>Nsw
1⫺HM21Nsw⫺i⫹12H0n1Nsw⫹12
Figure 1.40b. Portion of state diagram corresponding to Rj, expanding window search,jⱖjk(reprinted from [38]).
and, for the single dwell detector, the corresponding mean acquisition time is given by
(1.244) where
(1.245a)
(1.245b)
(1.245c) with
(1.246a)
(1.246b) E21k,Nsw2⫽^ a
RNsw⫺k⫹1ⱕlⱕRNsw⫺k⫹1
lpl; k⫽1, 2,p,Nsw. E11k,Nsw2⫽^ a
RNsw⫺k⫹1ⱕlⱕRNsw⫺k⫹1
pl; k⫽1, 2,p,Nsw
⫹11⫺PD2 c 11⫺PD22
1⫺ 11⫺PD22⫺k 11⫺PD22k 1⫺ 11⫺PD22kd f
⫻ e 1Nsw⫹1⫺k211⫺PD2⫹ N n PD
E21k,Nsw2 E11k,Nsw2 g1k,Nsw;PD2⫽ 1⫺11⫺PD22k
2⫺PD
⫺k1k⫺1211⫺PD22k
⫻ c1⫺ 11⫺PD22k
1⫺ 11⫺PD22 ⫺k11⫺PD22k⫺2d
⫹ e1
2341Nsw⫹1⫺k2⫺14 ⫹ 1⫹11⫺PD22
1⫺11⫺PD22f 11⫺PD22 b1k,Nsw;PD2⫽ 1
21Nsw⫹1⫺k211⫺ 11⫺PD22k2
⫹ 1Nsw⫺k21Nsw⫹1⫺k211⫺11⫺PD22k2 a1k,Nsw;PD2⫽Nsw1Nsw⫹1211⫺PD22k
⫻ 3a1k,Nsw;PD2⫹b1k,Nsw;PD2⫹g1k,Nsw;PD2 4r TACQ⫽tde 1
PD
⫹211⫹KPFA2n Nsw a
Nsw
k⫽1
E11k,Nsw2 1⫺ 11⫺PD22k
For large n, it is convenient to replace the discrete a prioriprobability dis- tributionpkby a continuous distribution corresponding to its envelope px(x) and evaluate the sums in (1.246) as integrals, namely,
(1.247a)
(1.247b) For the symmetric triangular distribution of (1.227), we have
(1.248) E21k,Nsw2 ⫽ n
4 c3Nsw12k⫺12⫺213k2⫺3k⫹12
3Nsw3 d
E11k,Nsw2⫽ 1 Nsw
a2k⫺1
2Nsw
b
E21k,Nsw2 ⫽ 冮1n>2211⫺k>N1n>2211⫺1k⫺sw122>Nsw2xpx1x2dx.
E11k,Nsw2⫽ 冮1n>2211⫺1k⫺12>Nsw2
1n>2211⫺k>Nsw2
px1x2dx
Figure 1.41. Normalized mean acquisition time versus number of partial windows for expanding window search strategy; single dwell system with triangular a priori distribution for code phase uncertainty.
whereas for the truncated Gaussian distribution considered in Section 1.6.1
(1.249)
⫽s
expc⫺a n
212sb2a1⫺ k Nsw
b2d ⫺expc⫺a n
212sb2a1⫺k⫺1 Nsw
b2d 1⫺2Qa n
2sb
t. E21k,Nsw2
E11k,Nsw2 ⫽ Qa n
2s a1⫺ k Nsw
b b ⫺Qa n
2s a1⫺ k⫺1 Nsw
b b 1⫺2Qa n
2sb
Figure 1.42a. Normalized mean acquisition time versus number of partial windows for expanding window search strategy, single dwell system with Gaussian a prioridis- tribution for code phase uncertainty,n/2⫽3s.
As before, we can ignore the 1/PD term in (1.244) when nis sufficiently large. Doing so, Figures 1.41 and 1.42a and 1.42b are plots of normalized acquisition time versus the number of sweeps Nswin the uncertainty region with detection probability PD as a parameter. We observe from these figures that except for PD⫽1, there always exists an opti- mum number of partial windows in the sense of minimizing mean acquisition time. For PD⫽1, one window, i.e., a continuous/center zsearch, is optimum.
Furthermore, the more peaked the distribution, e.g., Gaussian rather than tri- angular, or Gaussian with n/2⫽5srather than Gaussian with n/2⫽3s, the more there is to be gained by using an expanding window rather than a z- type search. Also, the sensitivity of using more than the optimum number of partial windows decreases as the distribution becomes more peaked.