A suitable analytical approach for evaluating the behavior of the SLS is to represent it as a finite Markov chain with absorbing boundaries. Such a Markov chain model for the SLS of Figure 1.59 is illustrated in Figure 1.60.
Note that Figure 1.60 is analogous to the generating function flow graph for the single dwell system (Figure 1.5) except that, for convenience, we have omitted the branch parameter zwhich was previously used to mark time as one proceeded through the graph. Each branch of Figure 1.60, however, is still labelled with the transition probability appropriate for going from one
state to the other. Using straightforward techniques for analyzing Markov chains with absorbing boundaries, it can be shown [9] that the mean and vari- ance of the acquisition time of the combination of the single dwell acquisi- tion system of Figure 1.4 and the SLS of Figure 1.59 are given by
(1.288) and
(1.289) sACQ2 ⫽td2
q2a 1 12 ⫹ 1
PL2 ⫺ 1 PL
b TACQ⫽ 12⫺PL2qtd
2PL
Figure 1.59. A search/lock strategy.
wherePLis the probability of lock, i.e., the probability of entering the lock mode assuming the search has reached the correct cell, and is the mean dwell time for an incorrect cell. In terms of the Markov chain model of Figure 1.60,PLis simply the probability of going from state 1 to state 3 with p1⫽ PD1andq1⫽1⫺PD1wherePD1is the detection probability for the search mode. Furthermore, is the mean time to reach states 0 or 6 (the two absorbing states) from state 1 with p1⫽PFA1,q1⫽1⫺PFA1,p2⫽PFA2, and q2⫽1⫺PFA2wherePFA1andPFA2are, respectively, the false alarm proba- bilities in the search and lock modes.
Before proceeding to an evaluation of PL and for the SLS of Figure 1.59, we draw attention to the similarity of (1.288) and (1.4), the latter being the mean acquisition time for the single dwell acquisition acting alone (i.e., in the absence of the SLS). Clearly without any hit verification, the proba- bility of lock would simply be equal to the detection probability of the detec- tor in the search mode of operation, i.e.,
(1.290) Furthermore, the average dwell time for an incorrect cell without an SLS is the average of the dwell time for a miss (which occurs with probability 1
⫺PFA) and the dwell time for a hit (which occurs with probability PFA) includ- ing the penalty of Kdwell time units to discover the false alarm. Thus,
(1.291) Finally, substituting (1.290) and (1.291) in (1.288) results in (1.4), as it natu- rally should.
⫽td11⫹KPFA2.
td⫽td11⫺PFA2⫹ 1td⫹Ktd2PFA
PL⫽PD.
td
td
td
Figure 1.60. Markov chain model of search/lock strategy.
1.8.1.1 Evaluation of Probability of Lock
The events contributing to an entering of the lock mode starting in search state 1 are as follows:
Since the above events are mutually exclusive, the probability of lock is sim- ply the sum of the probabilities of these events, i.e.,
(1.292)
⫽p12
a
q n⫽0
1p1q12n⫽ p12
1⫺p1q1⫽ p12
1⫺p1⫹p12
PL⫽p12⫹p1q1p12⫹ 1p1q122p12⫹ # # #1p1q12np12⫹ # # #
hit-hit hit-miss-hit-hit hit-miss-hit-miss-hit-hit
⯗
hit-miss-hit-missphit-miss-hit-hit.
n hits, n misses
Figure 1.61. Probability of lock versus search mode detection probability (reprinted from [9]).
à
or in view of the previously mentioned substitution for p1,
(1.293) Figure 1.61 is a plot of PLversusPD1.
1.8.1.2 Evaluation of Mean Dwell Time
The mean dwell time can also easily be computed by first identifying the events contributing to a dismissal of an incorrect cell starting in search state 1, and then assigning the appropriate dwell time and probability of occur- rence to each of these events. Considering first those events which dismiss an incorrect cell without entering the lock mode,we have the following:
Thus, the component of the mean dwell time associated with these events is
(1.294) or, since
(1.295)
(1.296) Considering next the events which dismiss an incorrect cell by first enter- ing the lock mode,we have the following:
hit-hit-lock mode hit-miss-hit-hit-lock mode
⯗
hit-miss-hit-missphit-miss-hit-hit-lock mode.
n hits, n misses
⯗
td112⫽td111⫺PFA12 1⫹PFA111⫺PFA12 31⫺PFA111⫺PFA12 42 . a
q n⫽0
12n⫹12xn⫽ 1⫹x
11⫺x22 ,
⫽td111⫺PFA12a
q n⫽0
12n⫹12 3PFA111⫺PFA12 4n
⫹ # # #12n⫹12td1PFA1n 11⫺PFA12n⫹1⫹ # # #
td112⫽td111⫺PFA12⫹3td1PFA111⫺PFA122 miss
hit-miss-miss hit-miss-hit-miss-miss
⯗
hit-miss-hit-missphit-miss-miss n hits, n misses
PL⫽ PD12
1⫺PD1⫹PD12 .
à à
For the moment, we do not identify, in the above, the possible paths through and out of the lock mode. We merely use the words “lock mode” to denote the collection of these paths and associate with them an average penalty time Tp. Shortly, we shall evaluate Tpin terms of the lock mode false alarm prob- ability and dwell time for the specific SLS of Figure 1.59. Also note that pre- viously, when we considered the single dwell acquisition system alone,Tpwas chosen equal to Ktd.
For the events listed above, the contribution to the mean dwell time is given by
(1.297) or
(1.298) Adding (1.296) and (1.298), the total mean dwell time is given by
(1.299) Using a procedure similar to the above, wherein the dwell time and prob- ability associated with each possible path originating at state 3 and termi- nating at state 6 of Figure 1.60 is identified, it can be shown that
(1.300) Substituting (1.300) in (1.299) gives the desired result for the mean dwell time.
Figure 1.62 is an illustration of the mean dwell time normalized by the search mode dwell time versus the search mode false alarm probability with the lock mode false alarm probability as a parameter. Also the lock mode dwell time is assumed to be five times the search mode dwell time as in the example considered in [9], which has practical application to the Space Shuttle program.
Finally, the mean acquisition time is obtained by substituting (1.293) and (1.299) in (1.288). As a numerical illustration of the result of these substitu- tions, we continue with the example considered in [9] where the following
Tp⫽ c3⫺4PFA2⫹2PFA22
11⫺PFA223 dtd2. td⫽ 11⫹PFA12td1⫹PFA12
Tp
1⫺PFA1⫹PFA12 . td122⫽ 2td1PFA12
31⫺PFA111⫺PFA12 42⫹ TpPFA12
1⫺PFA111⫺PFA12 .
⫹TpPFA12
a
q n⫽0
3PFA111⫺PFA12 4n
⫽td1PFA12
a
q n⫽0
12n⫹22 3PFA111⫺PFA12 4n
⫹ # # #112n⫹22td1⫹Tp2PFA1n⫹211⫺PFA12n⫹ # # #
td122⫽ 12td1⫹Tp2PFA12 ⫹ 14td1⫹Tp2PFA13 11⫺PFA12
additional parameter specifications were given:
td1 ⫽ Search Mode Dwell Time ⫽.91 ms td2 ⫽ Lock Mode Dwell Time ⫽5td1⫽4.55 ms
q ⫽ Number of Search Cells ⫽4094
(PN code of length 2047 chips searched in 1/2 chips increments)
B ⫽ Predetection bandwidth ⫽500 kHz
(P/N0)0 ⫽ Nominal signal-to-noise spectral density ⫽54.7 dB-Hz.
(1.301) Furthermore, a worst case correlation loss L⫽ .5 was assumed during the search mode, while during the lock mode Lwas set equal to .81 to simulate a delay error of .1 chip.
Using the above, we first compute the effective signal-to-noise ratio g⬘in the pre-detection filter bandwidth as defined by (1.62) (with A2 ⫽ P) Figure 1.62. Mean dwell time versus search mode false alarm probability (reprinted from [9]).
multiplied by L, i.e.,
(1.302)
Then, using g⬘of (1.302) for A2/N0Bin (1.81), this equation can be applied to find PD1whenPFA1⫽.01 and Btd1⫽455, and PD2whenPFA2⫽0.5 and Btd2⫽2275. The results of these computations are illustrated in Figure 1.63.
Combining the results of Figure 1.63 for PD1versusP/N0and those of Figure 1.61 for PLversusPD1enables one to determine PLas a function of P/N0. For a lock mode false alarm probability PFA2⫽0.5 and search mode false
g¿⫽ PL N0B⫽ à
P N0
⫻10⫺61search mode2 P
N0
⫻1.62⫻10⫺61lock mode2
Figure 1.63. Detection performance (reprinted from [9]).
alarm probabilities PFA1⫽.001, .01, .03, and 0.1, we obtain from (1.300) and (1.299) the corresponding values of mean dwell time ⫽.912, .934, 1.016, and 1.7 ms which are independent of P/N0. Finally, using the above-deter- mined functional relationship between PLandP/N0together with the val- ues of in (1.288) gives the mean acquisition time of the system, which is illustrated versus P/N0in Figure 1.64. Note that, for smaller values of PFA1, increases faster with decreasing P/N0than is the case for larger PFA1. The effect is primarily due to the increased PD1which results from increased PFA1.
We hasten to add that all of the results of this section have assumed the absence of code Doppler or its derivatives. To include these effects, one
TACQ
td
td
Figure 1.64. Mean acquisition time performance; ⫾300 Hz frequency error (reprinted from [9]).
would simply apply the modifications discussed in Sections 1.2.3 and 1.2.8 to (1.288) and (1.289). For example, the relation for mean acquisition time in the presence of code Doppler analogous to (1.48a) would now be (see (1.288))
(1.303)
where again Nu/q⬘represents the search update in the absence of Doppler, e.g., one-half of a chip, and is the code phase shift due to Doppler dur- ing the mean dwell time. As a numerical illustration of the application of (1.303), Figure 1.64 plots versus P/N0for the previous example with now⌬fc⫽ ⫺300 Hz or ⌬fc⫽ ⫹300 Hz and Nu/q⬘ ⫽1/2.
1.8.2 Another Search/Lock Strategy
Another approach [1] to providing an SLS incorporates a search mode which is modelled after the multiple dwell time acquisition procedure dis- cussed in Section 1.3. In particular, the search mode again consists of two states; however, unlike Figure 1.59, the integration (dwell) times are dif- ferent for the two states. Furthermore, a miss on state 2 does not return the SLS to state 1 but rather immediately dismisses the cell as being incor- rect (see Figure 1.65). The basic philosophy behind the above strategy is to assign a small dwell time td1to the first search state so as to search the code phases quickly and a larger dwell time td2to the second search state to provide a better estimate (higher probability of detection and lower false alarm probability) of whether the correct cell has been found. In this way, some of the false alarm protection is apportioned in the first integration and the remaining (usually greater) protection is placed in the second inte- gration. Finally, the lock mode portion of the SLS uses a third integration timetd3(in practice it could be the same as td2) with the same algorithm as in Figure 1.59, i.e., a reinitiation of the search requires three consecu- tive misses.
Before proceeding with the performance analysis of the SLS in Figure 1.65, we point out its similarity with the multiple (here double) dwell time acquisition system alluded to above. In particular, the double dwell time sys- tem which incorporates the search mode of the SLS in Figure 1.65 employs independent (non-overlapping) integration intervals for its two dwells, whereas a two-dwell version of the multiple dwell time procedure discussed in Section 1.3 uses overlapping integration intervals.
Since the SLS of Figure 1.65 can also be modelled by a Markov chain, the mean and variance of the acquisition time are still given by (1.288) and (1.289), however, with different relationships for PL and . Clearly, sincetd
TACQ
¢fctd
TACQ⫽
a2⫺PL
2PL
bq¿td
1⫹ q¿
Nu
¢fctd
⫽ TACQ0no code Doppler
1⫹ q¿
Nu
¢fctd
the only event which causes an entering of the lock mode starting in search state 1 is hit-hit, the probability of lock is now given by
(1.304) where PD1andPD2are, respectively, the detection probabilities for search states 1 and 2. To compute the mean dwell time we again separate the events into those that cause a cell dismissal without entering the lock mode and those that do likewise by passing through the lock mode. In the case of the former, we observe from Figure 1.65 that there are only two appropriate
PL⫽PD1PD2
Figure 1.65. Another search/lock strategy.
events, namely, miss and hit-miss. Thus, the corresponding component of the mean dwell time is
(1.305) wherePFA1andPFA2are, respectively, the false alarm probabilities for search states 1 and 2. For the latter case, we have already observed that only two successive hits cause an entering of the lock mode, which is accompanied by a penalty of Tpsec before rejecting the cell. Thus, this corresponding com- ponent of the mean dwell time is
(1.306) Finally, adding (1.305) and (1.306) gives the mean dwell time as
(1.307) Note that by letting Tp⫽Ktd2and substituting (1.304) and (1.307) in (1.288) and (1.289), we obtain the results for the mean and variance of the acquisi- tion time given in [1], which have also been modified as per our previous discussions to include the effects of code Doppler. Using the actual lock mode of the SLS in Figure 1.65,Tpwould in reality be given by (1.300) with, in general,PFA2replaced by PFA3andtd2bytd3.
We conclude our discussion of this strategy by pointing out that measured values of acquisition time taken on an actual implementation of Figure 1.65 in the code acquisition portion of the Space Shuttle Orbiter system were accurate to within 1 dB of the theoretical results as predicted by the above equations.