Evaluation of Detection and False Alarm Probabilit- 123docz.net

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 PACQand 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 partial 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 by twith respect to an arbitrary time reference and c(t tˆ) denotes the corresponding 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;i0, 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 variance of Cfor 0 1ttˆ2>Tc0 3. The conditions under which the Gaussian

e attˆ

Tc b ; NeTc

var5C0Hi6MTc2Gi1e2 MTc2e e2; i1 120e0 2e2; i0 E5C0Hi6 eMTc11 0e0 2; i1

0; i0

C^ 0MTcc1tt2c1ttˆ2dt

12Ac1tt2

1Pd1, PFA12 1Pd0, PFA02

TACQ

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assumption applies to the random variable Chave been discussed in Volume II. In short, for M Vp,Cbehaves like a binomial distribution to behave like a Gaussian one.

The second difference concerns the validity of the Gaussian approximation 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

828 Pseudonoise Code Acquisition in Direct-Sequence Receivers

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.

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hypotheses H0and H1as 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 approximations 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 hypothesis Hiby

(1.189) where Ec/N0 A2Tc/N0is the chip signal-to-noise ratio and Gi() is the normalized 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 variable C, and the coefficients Fkand Gkare evaluated recursively as

(1.194a) Gk11h*2k1exp1h*2 1k12Gk; G0exp1h*2

gpc 1E5C0H1622

var 5C0H16 M 11 0e0 22 G11e2 h*^ h

N0MTc PD0 1

112g0 expea g1

12g1bgpcf a

FkGk

PFA0 112g0h*>q112g

exp311g02y4I01g0y2dy PD0 PFA0

gi Ec

N0 Gi1e2; i1, 2

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830 Pseudonoise Code Acquisition in Direct-Sequence Receivers

Figure 1.30a. versus Ec/N0for the matched filter detector with partial correlation ( 1,h*10) (reprinted from [14]).

PFA0

Figure 1.30b. versus Ec/N0for the matched filter detector with partial correlation ( 1,h*12.5) (reprinted from [14]).

PFA0

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and

(1.194b)

Figures 1.30a and 1.30b illustrate versus Ec/N0for 1 (g0Ec/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 correlation “noise” is small with respect to the dominant thermal noise, is to model the quadrature total noise components at the matched filter outputs as independent Gaussian random variables with variance

(1.196) which reflects the additive contributions of the thermal noise and partial correlation “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) PD01 0bexp51y 2 6I0122y2dy

PFA0exp• h*

1 aEc

N0bG01e2ả si2 N0

2 c1 aEc

N0bGi1e2 d; i0, 1, e PD0

PFA0 112g0 expa 1g0 12g0

h*bI0a g0 12g0

h*b PFA0

e PFA0

F01; E1 1

2 a22g1gpc 12g1b. Ek1 1

1k122 c a 2g1

12g1bkEkE1Fkd Fk1 k12

1k122 a 2g1

12g1bFk2E1Ek1

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832 Pseudonoise Code Acquisition in Direct-Sequence Receivers

Figure 1.31a. versus Ec/N0for the matched filter detector with partial correlation (e 0,M64) (reprinted from [14]).

PD0

Figure 1.31b. versus Ec/N0for the matched filter detector with partial correlation (e .5,M64) (reprinted from [14]).

PD0

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where

(1.199)

Equations (1.197) and (1.198), which are in the form of (1.70) and (1.71) respectively with NB1, are superimposed (in dotted lines) on the previous 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 probabilities of the coincidence detector. Since a Bout of Amajority logic decision 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 simplifications. 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 KANMand tdTc/N Tu/NuNwhere Nuis the number of chips searched in one pass and Tuis the corresponding time uncertainty.Then, using the approximations nqNuN 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 1ANMPFA0fr

PDœ PD1 a

A nBaA

nbPDn011PD02An. PFA1 a

A nBaA

nbPFAn 011PFA02An

PD0

PFA0

gpcaEc

N0bG11 2 1 aEc

N0bG1

b h*

1 aEc N0bG11 2

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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 Q1and Q2

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 uncertainty 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 time T_

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 rc1/Tc512 kchips/sec, an uncertainty time Tu 64 msec corresponding to Nu Tu5c32767 2151 uncertainty chips, a reset time Trrmsec (fr.078), and best and worst case values of e. For all cases considered, it was found that A4 and B2 (2 out of 4 majority logic decision) was the optimal choice for the coincidence detector, although the performance was relatively insensitive to the actual Aand Bvalues 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 performance degrades rapidly.

Evaluation of Detection and False Alarm Probabilities for

A BASIS FOR A JAMMING GAME

ENERGY GAIN CALCULATIONS FOR TYPICAL SYSTEMS