1.5 RAPID ACQUISITION USING MATCHED FILTER TECHNIQUES
1.5.2 Evaluation of Detection and False Alarm Probabilities for
In order to apply the results for PACQ and of the rapid acquisition non-coherent receiver as per (1.182) and (1.183), one needs to calculate the detection and false alarm probabilities for both the non-coherent correla-
tion detector and the coincidence detector .
Computations of this nature were previously performed in Section 1.2.4 for the non-coherent detector associated with the single dwell time acquisition system. While, in principle, the problems are quite similar, we shall soon see that many of the simplifying assumptions made in the previous analysis are not valid here for the matched filter system. Nevertheless, having gone through the details of such an analysis in Section 1.2.4, our presentation here will be brief with emphasis placed on the differences between the two.
First, unlike (1.58), the signal component of the inputs to the square-law detectors in Figure 1.25b is treated as a random variable reflecting the par- tial correlation effect produced by correlating the input code with the stored reference code over a finitetime interval which is much less than the code period. In particular, if denotes the input PN code delayed bytwith respect to an arbitrary time reference and c(t⫹tˆ) denotes the cor- responding local reference code, then, when the correlation time MTc is much less than the code time period pTc, the correlation
(1.186) is approximately a Gaussian random variable with mean and variance (con- ditioned on the hypothesis Hi;i⫽0, 1) given by [3], [13], [14]
(1.187a) and
(1.187b) In (1.187),
(1.188) denotes the fractional normalized (with respect to a chip) timing offset between the two codes where N is the smallest integer such that lies in the interval (⫺1, 1). Figure 1.29 illustrates the conditional mean and vari- ance of Cfor 0 1t⫺tˆ2>Tc0 ⱕ3. The conditions under which the Gaussian
e⫽ at⫺tˆ Tc
b ;NeTc
var5C0Hi6 ⫽MTc2
Gi1e2 ⫽MTc2e e2; i⫽1 1⫺20e0 ⫹2e2; i⫽0 E5C0Hi6⫽ eMTc11⫺ 0e0 2; i⫽1
0; i⫽0
C⫽^ 冮0MTcc1t⫹t2c1t⫹tˆ2dt
12Ac1t⫹t2
1Pd1,PFA12 1Pd0,PFA02
TACQ
assumption applies to the random variable Chave been discussed in Volume II. In short, for MVp,Cbehaves like a binomial distribution to behave like a Gaussian one.
The second difference concerns the validity of the Gaussian approx- imation to the decision statistic which is the single dwell time system allowed the false alarm and detection probabilities of (1.70) and (1.71) to be replaced by the simpler expressions in (1.79) and (1.80). We recall that the basis for validating this assumption was a central limit theorem argument based on the large values of pre-detection bandwidth-post- detection integration time product Btdtypical of the single dwell time system. Since in a fast-decision rate acquisition system, like the matched filter receiver, post-detection integration is not feasible, the central limit argument made previously does not apply here. Instead one must use the exact probability distributions of the decision statistic under Figure 1.29. The normalized mean and variance of the partial correlation Cas a function of the relative timing offset (normalized to the chip interval) between the incoming and local codes.
hypothesesH0andH1as determined from the solution to the problem of non-coherently detecting a Gaussian random variable (C) in additive
“white” (bandlimited) Gaussian noise. Under certain circumstances, it is possible to simplify matters and obtain results similar to (1.70) and (1.71). Although these simplifications are based on Gaussian approxi- mations to the mixture of two noise processes, one of which is signal dependent [14], it is to be re-emphasized that the assumption of a Gaussian test statistic remains invalid and thus expressions analogous to (1.79) and (1.80) are inappropriate.
1.5.2.1 Exact Results
Defining, analogous to (1.62), the effective signal-to-noise ratio gi under hypothesisHiby
(1.189) whereEc/N0 A2Tc/N0is the chip signal-to-noise ratio and Gi() is the nor- malized variance of the partial correlation Cas defined in (1.186), then the false alarm probability and detection probability of the matched filter (correlation) detector are given by [14]
(1.190) and
(1.191) where
(1.192) is a normalization of the actual detection threshold h(see Figure 1.125b),
(1.193) is the signal-to-noise ratio associated with the partial correlation random variableC, and the coefficients FkandGkare evaluated recursively as
(1.194a) Gk⫹1⫽ 1h*2k⫹1exp1⫺h*2 ⫹ 1k⫹12Gk;G0⫽exp1⫺h*2
gpc⫽ 1E5C0H1622
var5C0H16 ⫽M11⫺ 0e0 22 G11e2 h*⫽^ h
N0MTc
PD0⫽ 1 11⫹2g0
expe⫺a g1
1⫹2g1
bgpcf a
q k⫽0FkGk
PFA0⫽ 11⫹2g0冮h*q>11⫹2g
0
exp3⫺11⫹g02y4I01g0y2dy PD0
PFA0
⫽^
gi⫽ Ec
N0
Gi1e2;i⫽1, 2
Figure 1.30a. versusEc/N0for the matched filter detector with partial corre- lation ( ⫽1,h*⫽10) (reprinted from [14]).
PFA0
Figure 1.30b. versusEc/N0for the matched filter detector with partial corre- lation ( ⫽1,h*⫽12.5) (reprinted from [14]).
PFA0
and
(1.194b)
Figures 1.30a and 1.30b illustrate versus Ec/N0for ⫽1 (g0⫽Ec/N0) and normalized threshold values h*⫽10 and 12.5, respectively. Also shown is an upper bound on given by
(1.195) which over the region of Ec/N0illustrated is very tight. Figures 1.31a and 1.31b are the companion illustrations of versus Ec/N0for ⫽0, .5 and h*⫽7.5, 10, and 12.5. Also, the matched filters are assumed to integrate over 64 chips.
1.5.2.2 Approximate Results
An approximate approach [14] valid for low Ec/N0, where the partial corre- lation “noise” is small with respect to the dominant thermal noise, is to model the quadrature total noise components at the matched filter outputs as inde- pendent Gaussian random variables with variance
(1.196) which reflects the additive contributions of the thermal noise and partial cor- relation “noise” under hypothesis Hi. Under this assumption, the false alarm and detection probabilities of the matched filter detector simplify to
(1.197)
and
(1.198) PD0⫽1⫺ 冮0bexp5⫺1y⫹ ⌫2 6I0122⌫y2dy
PFA0⫽exp•⫺ h* 1⫹ aEc
N0
bG01e2
ả si2⫽ N0
2 c1⫹ aEc
N0
bGi1e2 d;i⫽0, 1, e PD0
PFA0ⱕ 11⫹2g0 expa⫺ 1⫹g0
1⫹2g0
h*bI0a g0
1⫹2g0
h*b PFA0
e PFA0
F0⫽1;E1⫽1
2 a22g1gpc
1⫹2g1
b. Ek⫹1⫽ 1
1k⫹122 c a 2g1
1⫹2g1
bkEk⫹E1Fkd Fk⫹1⫽ k⫹12
1k⫹122 a 2g1
1⫹2g1
bFk⫹2E1Ek⫹1
Figure 1.31a. versusEc/N0for the matched filter detector with partial corre- lation (e ⫽0,M⫽64) (reprinted from [14]).
PD0
Figure 1.31b. versusEc/N0for the matched filter detector with partial corre- lation (e ⫽.5,M⫽64) (reprinted from [14]).
PD0
where
(1.199)
Equations (1.197) and (1.198), which are in the form of (1.70) and (1.71) respectively with NB⫽1, are superimposed (in dotted lines) on the previ- ous exact results in Figures 1.30a, 1.30b and 1.31a, 1.31b. We observe from Figure 1.30 that the approximate analysis of is optimistic by about 1 dB, whereas from Figure 1.31 the exact and approximate results for are in excellent agreement.
It is now a simple matter to compute the false alarm and detection prob- abilities of the coincidence detector. Since a Bout of Amajority logic deci- sion with independent testing is governed by a binomial distribution, then
(1.200)
1.5.2.3 Acquisition Time Performance
At this point one has all the tools needed to compute the mean acquisition time performance of the matched filter system. For the worst case a priori probability distribution, it is possible to make some significant simplifica- tions. The first step in the procedure would be to rewrite the expression for worst case mean acquisition time in (1.183) (valid for a single cell in H1) in terms of the system parameters, recalling that K⫽ANMandtd⫽Tc/N⫽ Tu/NuNwhereNuis the number of chips searched in one pass and Tuis the corresponding time uncertainty.Then, using the approximations n⯝q⫽NuN W1,ANMW1, and the necessary modifications to allow for a multiple cell H1region, it can be shown [13] that
(1.201) which is a very tight approximation to the exact result in the region PACQ⬎ .95 and qPFAV1. In (1.201), PDœ is the detection probability per runof the
TACQ
Tu
⬵1⫹ANMPFA0⫹fr
PDœ
PD1⫽ a
A n⫽BaA
nbPD0
n 11⫺PD02A⫺n. PFA1⫽ a
A n⫽BaA
nbPFA0
n 11⫺PFA02A⫺n
PD0
PFA0
⌫ ⫽
gpcaEc
N0
bG11⑀2 1⫹ aEc
N0
bG1⑀ b⫽ h*
1⫹ aEc
N0
bG11⑀2
multiple cell H1region which from (1.185a) is given by
(1.202) which can easily be generalized for the case of more than two cells in H1. Note that if PD(Q1)⫽PD(Q2), i.e., the fractional offsets for states Q1andQ2 are e ⫽ ⫾1/4 , then , which is identical, as it should be, with (1.82) for the worst case misalignment with half-chip search- ing. Furthermore, in arriving at (1.201), allowance has been made for the pos- sibility of a reset penalty time Tr(fr Tr/Tu is then the penalty time normalized by the uncertainty time) associated with the time required to realign the codes to the initial phase offset at the start of the search after an unsuccessful sweep of the entire uncertainty region. Clearly, if the uncer- tainty region corresponds to the full code period as for short codes or spe- cific acquisition preambles, then realignment is automatic and frwould be zero.
As numerical examples of the application of the foregoing results, Figures 1.32a, 1.32b, and 1.32c illustrate the normalized minimum mean acquisition timeT_
ACQ/Tuand optimized27normalized detection threshold h*versus chip signal-to-noise ratio Ec/N0in dB with N, the number of cells per chip, and M, the number of chips per matched filter integration as parameters. Other parameters assumed were PACQ⬎.95, a code rate rc⫽1/Tc⫽512 kchips/sec, an uncertainty time Tu⫽64 msec corresponding to Nu⫽Tu5c⫽ 32767⫽ 215⫺1 uncertainty chips, a reset time Tr⫽rmsec (fr⫽.078), and best and worst case values of e. For all cases considered, it was found that A⫽4 and B⫽2 (2 out of 4 majority logic decision) was the optimal choice for the coin- cidence detector, although the performance was relatively insensitive to the actualAandBvalues as long as Awas roughly two times B. Furthermore, we observe that even for an optimized system, there exists a rather sharp
“thresholding” effect in the sense that below a certain value of Ec/N0the per- formance degrades rapidly.