1.2 THE SINGLE DWELL SERIAL PN ACQUISITION SYSTEM
1.2.2 Single Dwell Acquisition Time Performance in the
In the absence of a prioriknowledge concerning the relative code phase posi- tions of the received and locally generated codes, the local PN generator is assumed to start the search at any code phase position with equal probability.
Stated in mathematical terms, the probability P1of having the signal present (true hit) in the first cell searched is 1/q, and the probability of it not being there is 1 1/q. For example, if the number of code chips to be searched is denoted by Nuand the search proceeds in half-chip increments, then q2Nu and P11/2 Nu. More generally, if it has been determined that the signal is not present in the first k1 cells, then the a prioriprobability Pkof find- ing it present in the k-th cell is 1/(q1 k), where q1 kis obviously the number of remainingcells to be searched, each possessing an equal prob- ability of having the signal present.
A generating function flow graph for the q-state Markov chain which char- acterizes the acquisition process of the single dwell system is illustrated in Figure 1.8. As is customary in such flow graphs, each branch is labelled with the product of the transition probability associated with going from the node at the originated end of the branch to the node at its terminating end, and an integer (including zero) power of a parameter denoted here by z. The parameter zis used to mark time as one proceeds through the graph and its power represents the number of time units (dwell times) spent in travers-
770 Pseudonoise Code Acquisition in Direct-Sequence Receivers
8If one of the cells corresponds to perfect sync., i.e., the peak of the triangular correlation curve, then there are only three cells which correspond to non-zero code correlation.
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ing that branch. Furthermore, note that the sum of the branch probabilities (letting z1) emanating from each node equals unity.
Using standard signal flow graph reduction techniques [31]—[33], one can show the the generation functionfor the flow graph in Figure 1.8 is given by [1]
(1.1) U1z2 11b2z
1BzHq11z2 c1 q a
q1 l0
Hl1z2 d
The Single Dwell Serial PN Acquisition System 771
Figure 1.8. Generating function flow graph for acquisition time.
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where
(1.2) The mean acquisition time TACQis obtained by differentiating with respect to zand evaluating the result at z1. After some routine algebra, one arrives at the desired result, namely,
(1.3) which for qW1 (the case of practical interest) simplifies to
(1.4) The variance of the acquisition time is determined from the first two deriv- atives of U(z) by
(1.5) or, since U(1) 1, by the equivalent relation
(1.6) Taking the natural logarithm of (1.1) together with its first two derivatives, substituting them into (1.6), and simplifying the resulting algebraic expres- sions, one obtains (for qW1 and KVq) the desired result, namely,
(1.7) We hasten to point out that although Figure 1.6 was drawn for a single dwell system with a non-coherent band-pass detector, the above results apply equally to a single dwell system with a coherent detector, the only difference between the two cases being the interrelation of the parameters td,PFA, and PDfor the detector. This interelationship, which is essential for computing acquisition time performance, will be discussesd later on for the non-coher- ent band-pass detector.
The above analytical results for the mean and variance of the acquisition time can also be obtained by a simple heuristic approach [34] which, although lacking the mathematical elegance of the Markov chain signal flow graph
sACQ2 td211KPFA22q2a 1 12 1
PD2 1 PDb. sACQ2 cd2 ln U1ztd2
dz2 d ln U1ztd2 dz d `
z1
sACQ2 cd2U1ztd2
dz2 dU1ztd2
dz adU1ztd2 dz b2d `
z1
TACQ 12PD2 11KPFA2 2PD 1qtd2. 212PD21q1211KPFA2
2PD td
TACQ d ln U1ztd2 dz `
z1
U1ztd29 H1z2 PFAzK1 11PFA2z.
b1PD
772 Pseudonoise Code Acquisition in Direct-Sequence Receivers
9Note that since the generating function has the property that U(1) 1, we can, if convenient, equivalently differentiate the natural logarithm of U1ztd2 and evaluate it at z1.
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technique, provides additional insight into the acquisition process in terms of the underlying tradeoff between false alarm and detection probabilities.
As before, we start out with the same basic assumptions, namely, that no a prioriknowledge of the currect cell’s location within the total uncertainty region is available and the cell-by-cell search of the entire uncertainty region is repeated until the correct cell is detected. Although not explicitly stated previously, each cell is assumed to be tested only once during each search of the uncertainty region and the order of the cells in the search is inconsequential.
To make matters simple, we assume at first that during each examination of a cell, the detector is characterized by a constant (time-invariant) detec- tion probability PDas before, but now, however, a zero false alarm proba- bility. Then, if k(integer) denotes the particular search of the uncertainty region during which the correct cell is firstdetected, then khas the geometric probability density function (pdf)
(1.8) Note that since, by assumption,PFA0, then the detection probability for each complete search of the entire uncertainty region is equal to the detec- tion probability for the correct cell, namely,PD. Furthermore,k1 repre- sents the number of unsuccessful searches of the uncertainty region, each having passed through qcells. Thus,
(1.9) is an integer random variable which represents the number of cells searched without success of detection prior to the k-th search during which the correct cell will be detected.Since each of these cell examinations occupied a single dwell time td,
(1.10) represents the time expired in passing through this unsuccessful series of searches.
The mean and variance of are readily computed as follows:
(1.11) Since, by definition,p(k) is a probability density function,
(1.12) Furthermore, differentiating both sides of (1.12) with respect to PDand sim- plifying gives
(1.13) a
q
k1
kPD11PD2k1 1 PD . a
q
k1
PD11PD2k11.
qtda
q
k11k12PD11PD2k1. E5Tuœ6 Tuœ a
q
k1
Tuœp1k2 Tuœ
Tuœ ^ Nuœtdqtd1k12 Nuœ Nuœ ^ q1k12
p1k2 PD11PD2k1; k1, 2, 3,p.
The Single Dwell Serial PN Acquisition Systemhttp://jntu.blog.com 773
Thus, combining (1.12) and (1.13) gives the desired result, namely,
(1.14) Similarly,
(1.15) Differentiating (1.13) with respect to PDand simplifying by using (1.13) prior to differentiation gives
(1.16) Finally, expanding the square in (1.15) and substituting (1.12), (1.13), and (1.16) gives the desired result, namely,
(1.17) Having now characterized the period of unsuccessful search, we turn our attention to the k-th search interval during which the acquisition process will terminate at the location of the correct cell. Letting mdenote this correct cell location, the time required to successfully reach this point from the time the k-th search is initiated is clearly
(1.18) Since, for lack of any a prioriinformation regarding correct cell location within the uncertainty region,mwill be uniformly distributed in this region, i.e.,
(1.19) with mean
(1.20) and variance
(1.21) 1q1212q12
6 aq1
2 b2 q21 12 sm2 a
q m1
m2p1m2 aq1 2 b2 m a
q m1
mp1m2 1 q a
q m1
m q1 2 p1m2 1
q ; m1, 2, 3,p, q Tsmtd.
su2q2td2a 1 PD2 1
PDb. a
q
k1
k2PD11PD2k1 2 PD2 1
PD
. q2td2a 1
PD1b2. q2td2a
q
k11k122PD11PD2k1 E51Tuœ Tuœ22^ su2E5Tuœ26Tuœ2
Tu
œ qtda 1
PD1b.
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then the mean and variance of Tsare respectively given by
(1.22) Finally, the total acquisition time is the sum of and Tswith mean
(1.23) and variance11
(1.24) which for qW1 becomes
(1.25) All that remains is to include the effect of a non-zero false alarm proba- bility on the results in (1.23) and (1.25). Since out of the total of
cells searched,kof them are actually correct (one per each of the ksearches of the entire uncertainty region), then there is a possibility of a false alarm only on any of the
(1.26) remaining cells. Equivalently,NFAis the maximumnumber of false alarms that can occur. If ndenotes the actual number of false alarms that occur, each with probability PFAof occurrence, then conditioned on NFA,nhas the bino- mial pdf
(1.27) with conditional mean
(1.28) and conditional variance
(1.29) sn20NFANFAPFA11PFA2.
E5n0NFA6NFAPFA
n0, 1, 2, 3,p, NFA p1n0NFA2 aNFA
n bPFAn 11PFA2NFAn; NFA^ Nuœ mkq1k12mk
Nuœ m sACQ2 0q2td2a 1
12 1 PD2 1
PDb. td2cq2a 1
PD2 1
PDb q21 12 d sACQ2 0su2ss2
tdcqa 1
PD1b aq1
2 b d c12PD2qPD
2PD dtd
TACQ0Tu œ Ts
Tuœ TACQ010
ss2 aq21 12 btd2. Ts aq1
2 btd
The Single Dwell Serial PN Acquisition System 775
10The zero subscript is used to denote the fact that we have assumed PFA0.
11It is reasonable to assume that Tuand Tsare essentially independent.
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Since for each of the nfalse alarms a penalty of Ktdsec is assessed, then the penalty time due to false alarm is
(1.30) Finally, the total acquisition time TACQis the sum of and Tp, i.e.,
(1.31) The mean acquisition time is obtained by averaging TACQof (1.31). Thus, making use of (1.23) and (1.28), we obtain
(1.32) Using (1.26), we have
(1.33) Thus, substituting (1.33) and (1.32) gives the desired result, namely,
(1.34) which agrees identically with (1.3).
To obtain the variance of TACQ, we first rewrite (1.26) as
(1.35) which for qW1 becomes
(1.36) Thus, for large q, we may evaluate the conditional second moment of TACQ as
(1.37) Since from (1.28) and (1.29)
(1.38) and from (1.28)
(1.39) then
(1.40) 5NFA2 11KPFA22NFAK2PFA11PFA26td2
. K23NFAPFA11PFA2NFA2
PFA2 4 6td2
E5TACQ2 0NFA6 5NFA2 2K1NFA2
PFA2 E5nNFA0NFA6NFA2
PFA
E5n20NFA6 NFAPFA11PFA2 1NFAPFA22 E51NFA2 2KnNFAn2K22 0NFA6td2. E5TACQ2 0NFA6 E51NFAnK220NFA6td2
NFAq1k12mNuœ m.
NFA 1q12kqm TACQ c2 12PD2 1q12 11KPFA2
2PD dtd
12PD2qPD 2PD
1 PD
12PD2qPD2 2PD
. NFANuœ mk
TACQ c12PD2qPD
2PD NFAKPFAdtd. TACQTACQ0Tp 3Nuœ mnK4td.
TACQ0 TpnKtd.
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Similarly,
(1.41) Averaging (1.40) and (1.41) over the distribution of NFAgives the uncondi- tional second and first moments of TACQ, namely,
(1.42) Finally, the variance of the acquisition time is obtained as
(1.43) In view of the approximation in (1.36)
(1.44a) and
(1.44b) Thus, using (1.25) and (1.23) in (1.43) gives
(1.45) Finally, if in addition KVq, then (1.45) simplifies to
(1.46) which is in exact agreement with (1.7).
Although somewhat lengthy, the heuristic derivation of (1.34) and (1.46) is important in that false alarms and missed detections are readily identified in terms of their individual contributions to the mean acquisition time and variance.