1.2 THE SINGLE DWELL SERIAL PN ACQUISITION SYSTEM
1.2.3 Single Dwell Acquisition Time Performance in the
When code Doppler is present, the acquisition time performance of the sys- tem of Figure 1.6 is affected in two ways. First, the code Doppler causes the relative code phase between received and locally generated PN codes to be time varying during the dwell time of the integrate-and-dump. This has the
sACQ2 td211KPFA22q2a 1 12 1
PD2 1 PDb td2c12PD2qPD
2PD dK2PFA11PFA2. sACQ2 td211KPFA22q2a 1
12 1 PD2 1
PDb TACQ0NFAtd.
sACQ2 0sN2FAtd2
3sN2FA11KPFA22NFAK2PFA11PFA2 4td2. sACQ2 TACQ2 1TACQ22
E5TACQ6NFA11KPFA2td.
E5TACQ2 65NFA2 11KPFA22NFAK2PFA11PFA26td2 NFA11KPFA2td.
1NFAKNFAPFA2td E5TACQ0NFA6E51NFAnK2 0NFA6td
The Single Dwell Serial PN Acquisition Systemhttp://jntu.blog.com 777
effect of increasing or decreasing the probability of detection PD, depend- ing on whether the code Doppler is causing the relative code phase to increase or decrease.The second and potentially more dominant effect is that code Doppler affects the average search rate. In fact, if the code phase shift caused by the Doppler over a single dwell time is equal to the step size (phase update) of the search, then the average search rate is reduced to zero.
To take both of these effects into account when computing mean acqui- sition time is indeed a difficult, if not impossible, analytical task. However, it is possible to account for just the effect of Doppler on average search rate in a way which represents an obvious extension of the previous results.
Letting fcdenote the code Doppler in chips/sec, then the mean search (code phase) update min chips is given by [1]:
(1.47) where12Nu/qrepresents the search update in the absence of Doppler or, equivalently, the step size of the search in fractions of a chip,fctdis the code phase shift due to Doppler during the dwell time, and fcKtd is the code phase shift during verification caused by a false alarm. Thus, replacing qby Nu/min (1.4) and (1.7) gives expressions for the mean and variance of the acquisition time in the presence of code Doppler, namely [1],
(1.48a)
(1.48a)
Since fccan be either positive or negative, depending upon its sign, the code Doppler can either speed up or slow down the search. With regard to the
sACQ2 0no code Doppler
c1 q¿
Nu ¢fctd11KPFA2 d2 sACQ2
td211KPFA22Nu2a 1 12 1
PD2 1 PDb cNu
q¿ ¢fctd11KPFA2 d2 TACQ0no code Doppler
1 q¿ Nu
¢fctd11KPFA2 TACQ 12PD211KPFA2Nutd
2PDcNu
q¿ ¢fctd11KPFA2 d m Nu
q¿ ¢fctd ¢fcKtDPFA,
778 Pseudonoise Code Acquisition in Direct-Sequence Receivers
12The prime on qis used here to denote the number of cells searched in the absence of Doppler.
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magnitude of the code Doppler, we shall assume that 0fc0td(1 KPFA) V Nu/q, so that the denominator of (1.48a) and (1.48b) never approaches zero and the search always proceeds in the direction dictated by the code phase update provided by the local PN generator. Finally, note that, when fc0, then q qand (1.48a) and (1.48b) reduce to (1.4) and (1.7), respectively as they should.
The way in which code Doppler is accounted for in extending the results of (1.4) and (1.7) to those given in (1.48a) and (1.48b) can also be applied to further extend them to include the effect of code Doppler rate. In par- ticular, one computes the mean search update analogous to (1.47) and again replaces qby Nu/min the expression for mean acquisition time and acquisi- tion time variance.
When code Doppler alone was present, we observed that the mean (sta- tistical) search update was time invariant, i.e., the expression in (1.47) char- acterizes every cell being searched. When, in addition, code Doppler rate is present, the mean search update is now time dependent in the sense that it is now a function of the cell being searched.
In general, the mean search update min any given search cell is equal to the nominal search update Nu/q(typically, 1/2 for half-chip search incre- ments) of the local PN code generator plus the mean change in phaseof the received code over the search time of that cell.13Thus, letting mndenote the mean search update in the n-th cell being searched, and the code Doppler rate in chips/sec2, then from the Markov model previously estab- lished for the single dwell time system, we have that
(1.49) Note from (1.49) that the mean search update is a linear function of the search cell. Because of this dependence on n, we cannot directly replace q by Nu/min (1.4) to arrive at a formula for mean acquisition in the presence of code Doppler and Doppler rate. Rather, we should first find the average mean search updatemobtained by averaging mn1of (1.49) over all qsearch cells, i.e.,
(1.50) and then make the above suggested replacement in (1.4). Thus, from (1.49)
m^ 1 q a
q1 n0
mn1
n¢f
#
c3td11KPFA2 42; n0.
mn1 Nu
q¿ ¢fctd11KPFA2 1 2¢f
#
ctd231PFA1K22K2 4
¢f
#
c
The Single Dwell Serial PN Acquisition System 779
13We continue to assume, as before, that the code phase derivatives are positive when they are in such a direction as to aid the search (reduce the acquisition time).
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and (1.50), we have that
(1.51) Since the total mean search update must correspond to the total number of chips searched, we have that14
(1.52) Thus, substituting Nu/mfor qin (1.51) results in a quadratic equation in m, namely,
(1.53) Letting
(1.54) then
(1.55) Finally, substituting N/mfor qin (1.4), with mdefined by (1.55), gives the resulting expression for mean acquisition time in the presence of code Doppler and code Doppler rate, namely,
(1.56) TACQ 12PD2 11KPFA2Ntd
2PDCBa1B1 4AC B2 b 2A
S . m B
2Ac1B1 4AC B2 d. C12¢f
#
ctd2Nu11KPFA22, B Nu
q¿ ¢fctd11KPFA2 1 2¢f
#
ctd2K2PFA11PFA2 A1
1 2¢f
#
ctd2N11KPFA220.
m2mcNu
q¿ ¢fctd11KPFA2 1 2¢f
#
ctd2K2PFA11PFA2 d qm a
q1 n0
mn1Nu. 1
2¢f
#
ctd25q11KPFA22K2PFA11PFA26. m Nu
q¿ ¢fctd11KPFA2
780 Pseudonoise Code Acquisition in Direct-Sequence Receivers
14Here we have made the assumption that 0fc0and 0fcare small enough such that mn10 for all n0, 1, . . . ,q1 and thus the search proceeds only in one direction, namely, that dictated by the local PN generator code phase update.
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Also, the acquisition time variance in the presence of code Doppler and code Doppler rate becomes
(1.57)
Note again that, when , we have that BN/q N/q,C 0, whereby (1.56) and (1.57) reduce to (1.4) and (1.7), respectively.
1.2.4 Evaluation of Detection Probability PDand False Alarm Probability PFAin Terms of PN Acquisition System Parameters
The formulas for mean acquisition time and acquisition time variance devel- oped in the previous section are all functions of the detection probability PD, false alarm probability PFA, and dwell time td. Thus, it would appear at first glance that, for specified values of detection probability and false alarm probability, one could arbitrarily select the dwell time to achieve any desired mean acquisition time. Upon closer examination, one realizes that indeed this is not possible with the fallacy lying in the fact that, for a given PFAand pre-detection signal-to-noise ratio,PDis implicitly a function of td. To place this statement in evidence, we begin by evaluating PDand PFAin terms of the PN acquisition system parameters for the simple case of no code Doppler or Doppler derivatives.
When signal is present (i.e., the cell being searched corresponds to a sam- ple value on the PN correlationc curve), then the input to the square-law envelope detector can be expressed in the form15
(1.58) where
(1.59) R1t2 21Anc1t2 22ns21t2; u1t2tan1 ns1t2
Anc1t2 . 12R1t2cos1v0tcu1t22,
12ns1t2sin1v0tc2
x1t2s1t2n1t2 12A cos1v0tc2 12nc1t2cos1v0tc2
¢fc ¢f
#
c0 sACQ2
td211KPFA22Nu2a 1 12 1
PD2 1 PDb
£Ba1B1 4AC
B2 b 2A
§2 .
The Single Dwell Serial PN Acquisition System 781
15To keep the presentation simple, we shall, at this point, ignore partial correlation effects pro- duced by the filtering of the product of incoming and local PN waveforms over less than a full code period. Later in the chapter, we shall present both exact and approximate approaches for accounting for these effects and their significance.
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In (1.58),Ais the rms signal amplitude,16v0the radian carrier frequency, and nc(t),ns(t) are band-limited, independent, low-pass, zero-mean Gaussian noise processes with variance s2N0B/2, where N0is the single-sided noise spectral density and Bis the noise bandwidth of the pre-detection band- pass filter.
The output of the square-law envelope detector in Figure 1.6, in response to the input x(t) of (1.58), is (ignoring second harmonics of the carrier):
(1.60) and has a non-central chi-squared pdf which is given by
(1.61) where
(1.62) is the pre-detection signal-to-noise ratio. In the absence of signal, i.e.,A 0, (1.61) reduces to the central chi-squared pdf:
(1.63) which characterizes the square-law output in all search cells that contain noise only.
If y(t) is sampled at intervals T 1/B, then these samples are approxi- mately independent, and furthermore, the integrate-and-dump output can be approximated by a summation over these sampled values, namely,17
(1.64) where
(1.65) NB^ td
T Btd. Z^ 1
td0tdy1t2dt N1B a
NB1 k0
y1kT2 p1y2 •
1
2s2 expa y
2s2b; y0,
0; otherwise
g^ A2 N0B A2
2s2 p1y2 •
1
2s2 expca y
2s2gb dI0a2 B
gy
2s2b; y0
0; otherwise
y1t2 ^ x21t2R21t2 1Anc1t2 22ns21t2
782 Pseudonoise Code Acquisition in Direct-Sequence Receivers
16For the moment, we shall not enter into a discussion concerning the various system losses and gains which enter into the calculation of the effective signal amplitude to be used in predict- ing true system signal-to-noise ratio behavior. Such a discussion will be given later on in the development.
17For simplicity of notation, we assume that i0 in Figure 1.6 and set Z0Z. Furthermore, it is convenient to assume that NBis integer, although the results which follow are, for large NB, valid for NBnon-integer.
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Using the approximation in (1.64) and the first order pdf’s of (1.61) and (1.63), the pdf of Z, namely,p(Z), for signal present is given by
(1.66) and for signal absent is given by
(1.67)
Normalizing Zby 2s2/NBN0/tdor, equivalently, letting Z* ZNB/2s2, we can rewrite (1.66) and (1.67), respectively, in the simpler form,
(1.68)
and
(1.69) The probability of false alarm,PFA, is the probability that Zexceeds the threshold hwhen signal is absent or, equivalently, in terms of the normal- ized random variable Z*and the normalized threshold h* hNB/2s2,
(1.70) The detection probability PDis the probability that Zexceeds the threshold
eh* a
NB1 k0
1h*2k k! .
PFA h*qp1Z*2dZ*1 0h*11Z*NB21NB121!2 exp1Z*2dZ*
^ p1Z*2 •
1Z*21NB12
1NB12! exp1Z*2; Z*0
0; otherwise.
p1Z*2 à a Z*
NBgb1NB12>2 exp1Z*NBg2
INB1322NBgZ*4; Z*0
0; otherwise
^ p1Z2 à aNB
2s2b aZNB
2s2 bNB1
1NB12! expaZNB
2s2 b; Z0
0; otherwise.
p1Z2e NB
2s2 a Z
2gs2b1NB12>2 expcNBa Z
2s2gb d INB1c2
BNB2g Z
2s2d; Z0
0; otherwise
The Single Dwell Serial PN Acquisition Systemhttp://jntu.blog.com 783
hwhen signal is present. Thus, using (1.68) rather than (1.69), we get
(1.71) which, if desired, can be expressed in terms of a generalized Marcum’s Q- function.
For large NB(the case of most practical interest), things become quite a bit simpler. Defining , then from (1.61) and (1.63), the pdf’s of in the presence and absence of signal are, respectively,
(1.72) (1.73) Also, from (1.64) and the definition of Z*in terms of Z, we have
(1.74) Since, by previous assumption, the s are independent random variables, then for large NB, Z* is approximately Gaussian distributed with mean and variance . The means and variance of the pdf’s in (1.72) and (1.73) are well known [35] to be
(1.75) (1.76) Thus,
(1.77) (1.78) Using the Gaussian assumption, the false alarm probability is
(1.79) where Q(x) is the Gaussian probability integral. Thus, if PFAis specified,b
^ Q1b2 Qah*NB
2NB b PFA h*q22pN1 B
expc1Z*NB22 2NB ddZ*
Z*NB; sZ*2 NB; signal absent.
Z*NB11g2; sZ*2 NB112g2; signal present y*1; sy*2 1; signal absent.
y*1g; sy*2 12g; signal present sZ*2 NBsy*2
Z*NBy*
yk* Z* a
NB1 k0
yk*. p1yk*2 eey*k; yk*0
0; otherwise.
p1yk*2 ee1yk*g2I0122gyk*2; yk*0
0; otherwise
yk*
yk*y1kT2>2s2
PD1 0h*aNZ*Bgb1NB12>2exp1Z*NBg2INB1322NBgZ*4dZ*
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can be determined. The corresponding detection probability under the same assumption is
(1.80) Combining (1.79) and (1.80) and reidentifying NBand gin terms of the sys- tem parameters gives the final relation
(1.81)
Thus, given PD,PFA,A2/N0and B, the dwell time tdis determined.
Before we use (1.81) and the dwell time determined from it in the formulas derived in the previous section for mean acquisition time and acquisition time variance, several modifications based upon practical con- siderations must be made.