Data and Pilot Combining for Composite GNSS Signal Acquisition

8/3/2019 Data and Pilot Combining for Composite GNSS Signal Acquisition

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Hindawi Publishing CorporationInternational Journal of Navigation and ObservationVolume 2008, Article ID 738183, 12 pagesdoi:10.1155/2008/738183

Research Article

Data and Pilot Combining for CompositeGNSS Signal Acquisition

Daniele Borio and Letizia Lo Presti

Dipartimento di Elettronica, Politecnico di Torino, Corso Duca degli Abruzzi, 10129 Torino, Italy

Correspondence should be addressed to Daniele Borio, [email protected]

Received 1 August 2007; Revised 20 December 2007; Accepted 18 March 2008

Recommended by Olivier Julien

With the advent of new global navigation satellite systems (GNSS), such as the European Galileo, the Chinese Compass and themodernized GPS, the presence of new modulations allows the use of special techniques specifically tailored to acquire and trackthe new signals. Of particular interest are the new composite GNSS signals that will consist of two different components, the dataand pilot channels. Two strategies for the joint acquisition of the data and pilot components are compared. The first technique,noncoherent combining, is from the literature and it is used as a comparison term, whereas the analysis of the second one, coherentcombining with sign recovery, represents the innovative contribution of this paper. Although the analysis is developed with respectto the Galileo E1 Open Service (OS) modulation, the obtained results are general and can be applied to other GNSS signals.

Copyright 2008 D. Borio and L. Lo Presti. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

With the advent of the new global navigation satellitesystem (GNSS), such as the European Galileo and theChinese Compass, new signals and new modulations havebeen introduced in order to guarantee the coexistence andinteroperability with existent systems, like the AmericanGPS, and to fully exploit the technologies currently available.

An example of those new signals is the coherent adaptivesubcarrier modulation (CASM) that will be used for thetransmission of the Galileo signal on the E1 frequency.CASM combines three different signals in a constant ampli-

tude modulation that allows the use of efficient class Camplifiers [1, 2].

The three signals combined in the CASM are denoted asthe A, B, and C channels. The first one is used for public-regulated service (PRS), whereas the latter two will providethe open service (OS). The B and C channels have twodifferent roles: the first one, denoted data channel, will carrythe navigation message whereas the second one, denotedpilot channel, will be used for determining the pseudorangesbetween the satellites and the receiver. The B and C signalswill be transmitted at the same time, at the same frequency,and they will be separated only by different codes [3]. TheA signal is not completely defined by the galileo interface

control document [1] but will probably be separated infrequency as for the A signal emitted by Giove-A [4], the firstexperimental satellite of the Galileo constellation.

The presence of new modulations allows one to adoptspecial techniques specifically tailored to acquire and tracknew signals.

One solution, when acquiring composite GNSS signalssuch as the Galileo E1 OS modulation, consists in ignoringthe data channel and processing only the pilot signal. In thisway only half of the useful power is employed and the GNSSreceiver could not be able to acquire and track signals thatwould be easily processed if all the useful power were used.

Pilot and data channel combining allows recovering all theavailable power improving the acquisition performance andproviding more reliable signal detection. For these reasonsthe design of signal combing techniques is critical for theefficient acquisition of the new GNSS signals.

In this paper, Galileo E1 OS signals are consideredand in particular two acquisition strategies for the efficientcombining of data and pilot channels are analyzed. In thefirst strategy, called noncoherent combining, the receivedsignal is correlated separately with the pilot and data localreplicas. The correlation outputs are then squared andsummed. This strategy essentially exploits the principleof noncoherent integration [57] employed to extend the
mailto:[email protected]:[email protected]


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2 International Journal of Navigation and Observation

integration period over the bit duration. Its use for data andpilot combining is reported by [8], however its performancein terms of false alarm and detection probabilities has beenonly marginally investigated.

The second strategy consists in multiplying the receivedsignal by two new signals, obtained by summing and

subtracting the data and pilot local replicas, respectively.Then, the maximum of the two correlations is adopted asdecision variable. This strategy can be seen as an extensionof the (bc) technique proposed by [9, 10] in which onlythe correlation with the difference between data and pilotcodes was considered. This second strategy implements theMaximum Likelihood estimator for the code delay, theDoppler frequency, and the relative phase between data andpilot channels, and, to the best of our knowledge, has neverbeen applied to the E1 OS signals. This strategy allowsthe coherent combining of data and pilot channels and itwill be denoted as coherent channel combining with signrecovery. The proposed method is the adaptation of theacquisition strategy proposed in [11] to the Galileo E1 OSsignal. In particular [11] considered the case of data and pilotcomponents transmitted with a90 degree phase difference.Moreover, [11] only proposed the method without providingany analytical characterization of its performance.

Both strategies have been analyzed in terms of false alarmand detection probabilities and closed-form expressions forthe probabilities of coherent channel combining with signrecovery have been derived. To the best of our knowledgethese expressions have never been derived before and repre-sent one of the innovative contributions of this paper. More-over, these formulas are general and can be easily adapted tothe case of other modulations, for example, for the GPS L5case that has been analyzed only by simulations [11].

In the analysis the coherent integration time is limitedto a single code period. In this way the acquisition blockhas not to deal with the problem of bit transitions thatcan occur every 4 milliseconds. Moreover, an integrationtime of 4 milliseconds should be sufficient to acquire GNSSsignals in high to moderate C/N0 conditions. The problemof bit transitions can be overcome by using noncoherentintegrations. The integration time can be also increasedby considering the pilot channel alone and exploiting thestructure imposed by its secondary code [11]. These issuesare however out of the scope of this paper.

Monte Carlo techniques have been used for supportingthe theoretical analysis: simulations and analytical expres-

sions agree well proving the effectiveness of the developedtheory.

From the analysis it emerges that the coherent combiningalways outperforms noncoherent combining and thus itshould be adopted for the joint acquisition of data and pilotchannels.

This work is organized as follows: Section 2 introducesthe CASM and provides a model for Galileo E1 OS signals.In Section 3 the noncoherent and coherent combiningalgorithms are described and the expressions for false alarmand detection probabilities are derived. In Section 4 thederived formulas are validated by Monte Carlo simulations.Finally, the conclusions are presented in Section 5.

2. SIGNAL AND SYSTEM MODEL

The signal at the input of a Galileo receiver, in one-pathadditive Gaussian noise environment, can be written as

rRF(t) =L

i=12PR,iyi(t) + RF(t) (1)

that is the sum of L useful signals, emitted by L differentsatellites and with power PR,i, and ofa noise term RF(t). Eachsignal yi(t), in the Galileo E1 band, is given by [1, 2]:

yi(t)

=

2

3

eB,i

t 0,i eC,it 0,icos2fRF + fd,it+ i

13

2eA,i

t0,i

+eA,i

t0,i

eB,i

t0,i

eC,i

t0,i]

sin

2

fRF + fd,i

t+ i

,

(2)

where

(i) eA,i(t), eB,i(t) and eC,i(t) are the three useful signalsemitted on the E1 frequencies and corresponding tothe A, B, and C channels, respectively, and eA,i(t) isa restricted access signal for PRS whereas eB,i(t) andeC,i(t) are the data and pilot signals for OS;

(ii) 0,i, fd,i, and i are the delay, the Doppler frequency,and the phase introduced by the transmission chan-nel;

(iii) fRF = 1575.42 MHz is the E1 central frequency.

eA,i(t), eB,i(t), and eC,i(t) are real binary sequences thatassume value in the set {1, 1}. These three signals arecombined according to the CASM in order to obtain aconstant envelope signal yi(t). This result is achieved byintroducing the term proportional to the three useful signalsin (2).

The input signal (1) is recovered by the receiver antenna,downconverted, and filtered by the receiver front end. Alow-IF receiver is assumed. In mass-market receivers thebandwidth of the front-end filter is generally limited to a fewMHz and thus the components depending on the PRS signalin (2) can be considered eliminated by the filtering process.This is due to the fact that a BOC(15,2.5) modulation [ 2, 4]

is employed for the PRS signal. This modulation splits thePRS signal power far from the E1 central frequency wherethe power spectral densities (PSD) of the OS signals areconcentrated. In this way, the received signal, before theanalog-to-digital (AD) conversion, is given by

rIF(t) =L

i=1

2PR,iyi(t) + IF(t)

=L

i=1

2PR,i

2

3

eB,i

t 0,i eC,it 0,i

cos(2fIF + fd,it+ i + IF(t),

(3)


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D. Borio and L. Lo Presti 3

where fIF is the receiver intermediate frequency and IF(t)is the downconverted and filtered noise component. In (3)the effect of the front-end filtering on the OS components isconsidered negligible.

Finally, (3) is sampled and AD is converted, obtaining,by neglecting the quantization impact, the following signal

model:

rIF(nTs) =L

i=1

2PR,iyi

nTs

+ IF

nTs

=L

i=1

2PR,i

2

3

eB,i

nTs 0,i eC,inTs 0,i

cos2fIF + fd,inTs + i + IFnTs.(4)

In the following, the notation x[n] = x(nTs) will indicatea discrete-time sequence x[n], obtained by sampling a

continuous-time signal x(t) with a sampling frequency fs =1/Ts. For this reason (4) can be rewritten as

rIF[n] =L

i=1

Ci

eB,i

n 0,i

Ts

eC,i

n 0,i

Ts

cos2FiD,0n + i + IF[n],(5)

where Ci = (4/9)PR,i and FiD,0 = (fIF + fd,i)Ts. Ci representsthe total power of the i th received signal since the term

eB,i

n 0,i

Ts eC,i

n 0,i

Ts cos

2FiD,0n + i

(6)

has unitary power. The spectral characteristics of IF[n]depend on the type of filtering along with the sampling anddecimation strategy adopted in the front end. A convenientchoice is to sample the IF signal with a sampling frequencyfs = 2BIF, where BIF is the one-sided front-end bandwidth.In this case, it is easily shown that the noise variance becomes

2IF = E

2IF(t) = E2IFnTs = N0 fs2 = N0BIF, (7)

where N0/2 is the Power Spectral Density of the IF noise. Theautocorrelation function

RIF[m] = E

IF[n]IF[n + m] = 2IF[m] (8)

implies that the discrete-time random process IF[n] is aclassical independent and identically distributed (iid) wide-sense stationary (WSS) random process, or a white sequence.[m] is the Kronecker delta. From now on, this signal modelis adopted and the system performance is expressed in termsof Carrier-to-Noise Ratio Ci/N0.

Both data and pilot signals in (5) can be represented asthe product of three terms

eB/C ,i[n] = cB/C ,i[n]sB[n]dB/C ,i[n], (9)

where cB/C ,i[n] is a spreading code of length Nc = 4092, sB[n]is the subcarrier signal that in the Galileo OS signal is theBOC(1,1), and dB/C ,i[n] is the navigation message for thedata signal and the secondary code in the pilot case. dB/C ,i[n]is constant over one code period. Hereinafter the productbetween the code and the subcarrier will be denoted as

eB/C ,i[n] = cB/C ,i[n]sB[n]. (10)

As a result of code orthogonality, the different Galileo codesare analyzed separately by the receiver, and thus hereinafterthe case of a single satellite is considered and the index i isdropped. Thus the resulting signal is

rIF[n]

=

C

eB

n 0

Ts

eC

n 0

Ts

cos

2FD,0n+

+IF[n].

(11)

3. OS SIGNALS ACQUISITION

In traditional GPS receivers the delay and the Dopplerfrequencies of received signals are estimated by using thecorrelation with local signal replicas opportunely delayedand modulated. In the acquisition stage, the code delay andDoppler frequency are estimated as the ones that make thecorrelation with the local replica pass a fixed threshold. Inthe Galileo E1 case however this strategy cannot be directlyemployed since two different signals are present. Moreover,due to the navigation message on the data channel and to thesecondary code on the pilot channel, these two componentscan be either summed or subtracted.

In this Section, we analyze two possible strategies forsignal acquisition for Galileo OS signals. The performanceof the two algorithms is evaluated in terms of false alarmand detection probabilities that are the probabilities that thedecision variable passes a fixed threshold under two differenthypotheses:

(H0) the signal is present and correctly aligned with thelocal replica;

(H1) the signal is absent or not correctly aligned with localreplica.

The plot of the detection probability versus the false alarmprobability is called receiver operating characteristic (ROC)

and it completely defines the system performance [12]. TheROCs are usually employed for comparative analysis, aseffective metric for characterizing an acquisition system [5,6]. For these reasons they are adopted in this work and usedas basis for the analysis of the two acquisition algorithmsconsidered in the paper.

3.1. Pilot and data noncoherent combining

The conceptual scheme for the acquisition of Galileo E1signals, with the noncoherent combination of data and pilotchannels, is depicted in Figure 1. The received signal ismultiplied by two orthogonal sinusoids at the frequencyFD,


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1

N

N1n=0

() ()2

Codegenerator

Datachannel

cos(2FDn)

rIF[n]

Frequencygenerator

90

sin(2FDn)

1

N

N1n=0

() ()2

1

N

N1n=0

() ()2

Codegenerator

Pilotchannel

1

N

N1

n=0() ()2

Snc(FD , )

Decisionvariable

Figure 1: Conceptual scheme of the Galileo OS signals acquisition with noncoherent combining.

leading to the in-phase (I) and quadrature (Q) components.These signals are then split in two different brancheswhere the correlations with the data and pilot codes areevaluated. The cross-correlations from the two branches arenoncoherently combined leading to the decision variable

Snc(FD , ) =RBFD, 2 + RCFD, 2, (12)

where

RB

FD, = 1

N

N1n=0

rIF[n]eB

n

Ts

exp

j2FDn

,

RC

FD , = 1

N

N1n=0

rIF[n]eC

n

Ts

exp

j2FDn

(13)

are the cross-correlations of the received signal with the dataand pilot replicas delayed by and modulated by FD . In(13) N denotes the number of samples used to integrate thereceived signal over one-code period. More specifically, N isgiven byTcfs, where Tc = 4 milliseconds is the durationof one code period, fs is the sampling frequency, and

denotes the floor operator. Since the cross-correlationsRB(FD, ) and RC(FD , ) have been obtained from the inputsignal rIF[n], they also depends on 0, FD,0, and , theparameters that characterizerIF[n]. The dependence on theseparameters has not been explicitly reported in (13) for theease of notation.

The multiplication by the complex exponential in (13) isimplemented in Figure (1) by the multiplication with the twoorthogonal sinusoids.

In Appendix A it is shown that RB(FD, ) and RC(FD, )are two complex Gaussian random variables that, due to theorthogonality properties of Galileo codes [3], are approx-

imatively independent. Thus |RB(FD, )|2 and |RC(FD, )|2

are two 2 random variables with 2 degrees of freedom.From this consideration and (12), Snc(FD, ) is a 2 randomvariable with 4 degrees of freedom.

In order the determine the ROC, the two followingprobabilities have to be evaluated:

Pncfa () = P

Snc

FD,

> |H1

,

Pncd ()

=PSncFD, > |H0.

(14)

When the signal is not present or not correctly aligned it ispossible to assume [5, 6] that both RB(FD, ) and RC(FD , )are zero mean. Thus Snc(FD, ) is a central 2 random vari-able, whose distribution is completely characterized by thevariance ofRB(FD, ) and ofRC(FD, ). From (13) we have

Var

RB

FD,

= VarRCFD, = Var

1

N

N1i=0

rIF[n]eB

n

Ts

exp

j2FDn

= 1N2

N1i=0

VarrIF[n] = 1N2

N1i=0

VarIF[n]= N0 fs2N

= N0BIFN

.

(15)

Equation (15) directly derives from the noise model reportedin Section 2. In this case it is assumed that the input noiseis a white sequence. In a real GNSS receivers the front endcan introduce some correlation among the different noisesamples. This correlation causes a correlation loss [13] thatwould affect both noise and signal components. This effectcould be included in the analysis but it would introduceno further insight and thus, for the sake of clarity, it is notconsidered in this context.


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Since both RB(FD , ) and RC(FD, ) are complex randomvariables with iid real and imaginary parts, the variance (15)is equally divided between the two components and thus wecan define

2n =N0 fs4N

= N0BIF2N

, (16)

2n is the variance of the real and of the imaginary parts ofRB(FD, ) and RC(FD, ). Given these premises and exploitingthe fact that Snc(FD, ) is a central 2 random variable it ispossible to derive the false alarm probability [14]

Pncfa () = exp

22n

1 +

22n

. (17)

When the signal is present and correctly aligned Snc(FD, )is a noncentral 2 random variable with noncentralityparameter

=ERBFD, 2 +ERCFD, 2=2ERBFD, 2(18)that assumes the following expression [13, 15]:

= C2

sin2(NF)

(NF)2K2() C

2, (19)

where

(i) F= FD,0FD is the difference between the Dopplerfrequencies of the received signal and of the localreplica;

(ii) =

(0

)/Ts is the difference between thedelays of the received signal and of the local replica,normalized with respect to the sampling interval;

(iii) K() is the correlation between the incoming code,filtered by the frontend, and the code generated at thereceiver.

When the Doppler frequency and the delay of the localreplica match the ones of the received signal, the loss

(sin2(NF)/(NF)2)K2() can be assumed negligibleand C/2.

From these considerations it is possible to evaluate thedetection probability [14, 16]

Pncd () = Q2

2n,

2n

Q22CNN0 fs

,

4N

N0 fs

,

(20)

where Q2(, ) is the Generalized Marcum Q function oforder 2 [16, 17], defined as

QK(a, b) =1

aK1

+b

xK exp x

2 + a2

2

IK1(ax)dx

(21)

where IK1() is the modified Bessel function of first kindand order K 1 [18].

3.2. Pilot and data coherent combining withsign recovery

By considering (11) and by collecting the navigation messageof the data channel, it is possible to rewrite the received signalrIF[n] as

rIF[n]

=

CdB

n 0

Ts

eB

0Ts

dC

n(0/Ts)

dB

n(0/Ts) eCn 0

Ts

cos

2FD,0n+

+ IF[n].

(22)

Expressed in this form, (22) indicates that the navigationmessage dB[n] is spread by the equivalent pseudorandomsequence

eeq[n] = eB[n] dC[n]dB[n]

eC[n]. (23)

Since the navigation message dB[n] and the secondary codedC[n] are constant over one code period and since theycan assume only two values, 1 and 1, only two equivalentspreading sequences are possible

eeq[n] =

eB[n] eC[n],eB[n] + eC[n].

(24)

In the coherent combining scheme, the received signal is

correlated with both equivalent codes: the equivalent codethat maximizes the cross-correlation is likely the correct one.Based on this principle the decision variable is given by

Sml(FD , ) = maxR+FD , 2, RFD, 2, (25)

where

R+(FD, )

= 1N

N1n=0

rIF[n]

eB

n

Ts

+ eC

n

Ts

exp

j2FDn

,

R(FD, )

= 1N

N1n=0

rIF[n]

eB

n

Ts

eC

n

Ts

exp

j2FDn

.

(26)

This kind of algorithm is based on the Maximum Likelihoodestimator for the code delay, Doppler frequency, and relativesign between data and pilot channels, since, as shown inAppendix B, the maximization of the correlation function,with respect to the delay , the frequency FD, and theratio dC[n]/dB[n], corresponds to the maximization of thelikelihood function evaluated for the received signal rIF[n].


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1

N

N1n=0

() ()2

Codegenerator

Data + pilot

channel

cos(2FDn)

rIF[n]Frequencygenerator

90

sin(2FDn)

1

N

N1n=0

() ()2

1

N

N1n=0

() ()2

Codegenerator

Data pilotchannel

1

N

N1n=0

() ()2

Sml(FD , )

Max

Decisionvariable

Figure 2: Conceptual scheme of the Galileo OS signals acquisition with coherent combining.

The acquisition scheme with coherent combining isreported in Figure 2: the received signal is correlated with thetwo equivalent codes and the decision variable is obtained bychoosing the maximum of the two cross-correlations. It hasto be noted that the variables R+(FD, ) and R(FD, ) can beobtained by opportunely combining the cross-correlationswith the data and the pilot codes

R+(FD, ) = 1NN

1

n=0rIF[n]eBn Ts expj2FDn

+1

N

N1n=0

rIF[n]eC

n

Ts

exp

j2FDn

= RB

FD ,

+ RC

FD,

,

(27)

R(FD, ) = 1N

N1n=0

rIF[n]eB

n

Ts

exp

j2FDn

1

N

N1n=0

rIF[n]eCn Ts exp j2FDn= RB

FD ,

RCFD, .(28)

In this way the coherent combining algorithm can beimplemented with a computational load similar to the onerequired by the noncoherent combining strategy.

From (28) and (27) it clearly emerges that R+(FD, )and R(FD, ) are linear combinations of RB(FD, ) andRC(FD, ). Thus, since RB(FD , ) and RC(FD, ) are complexGaussian random variables, R+(FD, ) and R(FD, ) are alsocomplex and Gaussian.

The ROC for the coherent combining strategy can beeasily obtained by determining the false alarm and detection

probability of the variables |R+(FD, )|2 and |R(FD, )|2.In fact it can be easily shown that the false alarm and thedetection probabilities of the decision variable Sml(FD, ) aregiven by

P

Sml(FD, ) > = Pmax R+(FD, )2,R(FD, )2 > = 1 Pmax R+(FD, )2, R(FD, )2 < = 1 PR+(FD, )2 < ,R(FD , )2 < = 1 PR+(FD, )2 < PR(FD , )2 < .

(29)

The last line in (29) has been obtained by exploiting the

independence between |R+(FD, )|2 and |R(FD , )|2 thatderives from the independence of R+(FD, ) and R(FD, ).

In fact we have

E

R+

FD,

R

FD ,

= ERBFD, + RCFD, RBFD, RCFD, = ERBFD , 2 RCFD, 2= 0.

(30)

Equation (30) is zero since |RB(FD, )|2 and |RC(FD, )|2 areequally distributed and the difference of their mean cancelsout. From (30) R+(FD, ) and R(FD , ) are uncorrelated


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and, since they are both Gaussian random variables, inde-

pendent. For these reasons |R+(FD, )|2 and |R(FD, )|2 arealso independent.

Since R+(FD, ) and R(FD, ) are Gaussian randomvariables, |R+(FD , )|2 and |R(FD , )|2 are 2 distributedwith two degrees of freedom. When the signal is absent, or

the local replicas are not alignedwith the receivedsignal, then|R+(FD, )|2 and |R(FD, )|2 are both central 2 randomvariables [14] and the false alarm probability of Sml(FD, )is

Pmlfa () = 1

1 exp

42n

2. (31)

Equation (31) has been obtained by substituting the cumu-lative density function (cdf) of central 2 random variables[14] into (29). It can be noted that the exponential in (31)depends on 42n instead of 2

2n as for (17). This is due to the

fact that the equivalent code (24) has twice the power of the

single pilot and data codes.When the Galileo signal is present and correctly aligned

with the local replica, |R+(FD, )|2 and |R(FD, )|2 are nomore central 2 random variables, and the noncentralityparameters + and have to be determined. + and can be obtained by determining the mean of R+(FD , ) andR(FD, ) under the hypothesis that the local equivalentcode matches or not the navigation bit. In particular wehave

E

R+

FD,

=ERBFD, + RCFD,

=

2E

RB

FD, =C sin(NF)

(NF)K()

C if dC[n]dB[n]

=1,

0 ifdC[n]

dB[n]=1

(32)

and similarly

ER

FD, = ERBFD, RCFD,

=

2E

RB

FD, =C sin(NF)

(NF)K()

C if dC[n]dB[n]

=1,

0 ifdC[n]

dB[n]=1.

(33)

From these considerations it emerges that the decisionvariable Sml(FD, ), under the hypothesis of presence of signal

Table 1: Simulation parameters.

Parameter Value

Sampling frequency fs = 4.092MHzIntermediate frequency fIF =

fs4= 1.023MHz

Receiver bandwidth BIF =fs2 = 2.046MHz

Code rate 1.023106 chip/sCode length 4092 chip

Integration time NTs = 4 ms

and correct alignment, is given by the maximum betweena central 2 and a noncentral 2 random variables withtwo degrees of freedom. The noncentrality parameter of thenoncentral 2 random variable is given by

=

Csin2(NF)

(NF)2K2()

C. (34)

Given these premises it is finally possible to express thedetection probability

Pmld () = 1

1exp

42n

1Q1

22n,

22n

1

1exp

42n

1Q1

2CN

N0 fs,

2NN0 fs

,

(35)

where Q1(

,

) is the Marcum Q function of order 1.

4. SIMULATION ANALYSIS

In order to validate the theoretical analysis developed inthe previous sections, the Galileo E1 OS has been simulatedaccording to the parameters reported in [1] and acquiredaccording to the two methods considered in this work. Thesimulation parameters are reported in Table 1. More in detaila GNSS signal has been simulated according to model (11).The spreading codes from [1] has been used to generate theGalileo E1 OS signals, and the acquisition systems depictedin Figures 1 and 2 have been implemented in order toevaluate the decision statistics (12) and (25). The false

alarm and detection probabilities have been estimated byverifying if the decision variables, under both (H1) and (H0)hypotheses, passed or not the decision threshold obtained byinverting (20) and (31). 2105 trials have been used for theestimation process. Frequency and delay errors have not beensimulated and, for the evaluation of the detection probability,the incoming signal has been considered perfectly alignedwith local replica. These errors were not considered in thetheoretical model and thus they have not been considered inthe simulation scheme as well. The impact of frequency anddelay residual errors has been extensively studied in [15] andcan be easily included in the models developed in previoussections. This topic is however out of the scope of this paper.


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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

False alarm probability

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Detectionprobability

Noncoherent combining-theoretical

Noncoherent combining-simulated

Coherent combining-theoretical

Coherent combining-simulated

Figure 3: Simulated and theoretical ROCs for noncoherent andcoherent acquisition systems. C/N0 = 25 dB-Hz.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Noncoherent combining-theoreticalNoncoherent combining-simulated




The simulations results are reported in Figures 3, 4, and5, where the cases of C/N0 = 25,30, and 35 dB-Hz havebeen considered. These C/N0 values have been chosen sincethey represent marginally strong and weak signal conditions.The probabilities evaluated by Monte Carlo simulationsmatch the theoretical models developed in previous sectionsand the theoretical and simulated ROCs overlap in all

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1


0.85

0.9

0.95

1


Noncoherent combining-theoretical

Noncoherent combining-simulated




the considered cases. Simulations effectively support thetheoretical models proving the effectiveness of the analysisdeveloped in Section 3.

It can be noted, from Figures 3, 4, and 5, that the coherentacquisition algorithm always outperforms the noncoherentcombining strategy. Moreover, it can be observed that thetwo methods tend to have similar performance for low C/N0.This performance degradation of coherent combing withsign recovery is probably due to the fact that, for low C/N0,the estimation of the relative phase between data and pilotbecomes unreliable preventing an effective combination ofthe two channels transmitted for the Galileo E1 OS.

5. CONCLUSIONS

In this paper, two different acquisition strategies for theacquisition of the Galileo E1 OS signals have been consideredand deeply analyzed. The first strategy, the noncoherentcombining, is from the literature whereas the analysis of the

second one, coherent combining with sign recovery, is newand represents the innovative contribution of this paper. Ananalytical model for the false alarm and detection probabil-ities of both algorithms has been derived and Monte Carlosimulations have been used for supporting the theoreticalanalysis. From the theoretical model and simulations it isshown that the coherent combining algorithm outperformsthe noncoherent combing proving the effectiveness of theproposed method. Thus, provided that both algorithmsrequire similar computational loads, the coherent combiningacquisition algorithm results in preferable to noncoherentcombining and should be adopted for the joint acquisitionof the E1 OS data and pilot channels.


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APPENDICES

A. INDEPENDENCE OF THE RANDOMVARIABLES AT THE OUTPUT OF THE DATA ANDPILOT CORRELATORS

In this appendix we show that the random variables,obtained by correlating the input Galileo signal with the localreplicas of the data and pilot channels, are approximativelyindependent. The proof is based on the correlation proper-ties of Galileo memory codes [3].

From (11) the signal at the input of the Galileo receiveris the sum of a useful term and of white Gaussian noise.This input sequence is correlated with local replicas of thedata and pilot primary codes, opportunely delayed andmodulated, and for each satellite, a pair of random variablesis obtained:

RB, FD =

1

N

N1

n=0 rIF[n]eBn

Ts expj2FDn,RC

, FD = 1

N

N1n=0

rIF[n]eC

n

Ts

exp

j2FDn

.

(A.1)

Since rIF[n] is the sum of a deterministic component andwhite Gaussian noise and since both RB(, FD) and RC(, FD)are linear transformations ofrIF[n], then their independencecan be proven by considering

XB, FD =1

N

N1

n=0 IF[n]eBn

Ts expj2FDn,XC

, FD

= 1N

N1n=0

IF[n]eC

n

Ts

exp

j2FDn

(A.2)

that are obtained from the noise component ofrIF. Since theinput noise is assumed to be zero mean and since the usefulsignal component determines the mean of the correlationfunction, it is possible to write XB(, FD) = RB(, FD) E{RB(, FD)} and XC(, FD) = RC(, FD) E{RC(, FD)}.Thus the independence of XB(, FD) and XC(, FD) impliesthe independence ofRB(, FD) and RC(, FD). By defining

M=

IF[0]

IF[1]

. . .

IF[N 1]

;

Ex =

1 0 . . . 0

0 expj2FD

. . . 0

. . . . . . . . . . . .

0 0 . . . expj2FD(N 1)

;

B =

eB

Ts

eB

1

Ts

. . .

eBN 1 Ts

;

C =

eC

Ts

eC

1

Ts

. . .

eC

N 1

Ts

;

(A.3)

it is then possible to rewrite (A.2) as

XB

, FD = 1

NMTExB,

XC

, FD = 1

NMTExC,

(A.4)

and the covariance between XB and XC becomes

EXB

, FD

XC

, FD

= EXHC , FDXB, FD= E

1

N2CHEHx M

MTExB

= 1N2

CHEHx EMMTExB

= 2IF

N2CHEHx ExB =

2IFN2

CHB

0.(A.5)

The covariance (A.5) is almost zero for the quasi-orthogonality of the Galileo memory codes. In (A.5) the factthat EHx Ex = IN and E{MMT} = IFIN has been used. IN isthe NN identity matrix.

From (A.5) XB(, FD) and XC(, FD) can be considereduncorrelated and thus, since they are Gaussian randomvariables, independent.

B. MAXIMUM LIKELIHOOD ESTIMATOR

In this appendix, the Maximum Likelihood estimator for thedelay 0, the Doppler frequencyFD,0, and the relative phasebetween the Galileo E1 OS data and pilot channel is derived.Its connection with the coherent acquisition block with signrecovery is also shown.

In this context the signal presence is assumed and thesignal parameters are treated as unknown constants. By


10/12


considering (22) and by assuming IF[n] a white Gaussiansequence, it is possible to derive the joint probability density

function of the set {rIF[n]}N1n=0 :

f(r)=

1

22IFN/2 exp

N1i=0

ri i

, FD, S

2

22IF ,(B.1)

where

(i)

i

, FD, S

= CdBi

Ts eBi

Ts S eCi

Ts cos2FDi+,(B.2)

(ii) S = dC[i / Ts]/dB[i /Ts] is the relative phasebetween the data and the pilot channels and canassume two values: 1 and 1,

(iii) ri is the i th component of the vector r.

The Likelihood function for the set (, FD, S) is given by

L

, FD, S

= 122IF

N/2 expN1i=0 rIF[i] i, FD, S2

22IF

(B.3)

and its maximization is obtained by maximizing the argu-ment of the exponential in (B.3). Thus the MaximumLikelihood estimator for (, FD, S) is given by

, FD , S =arg min

,FD ,S

N1i=0

rIF[i] i

, FD, S

2

=arg min,FD ,S

N1i=0

r2IF[i]2N1i=0

i

, FD, S

rIF[i]+N1i=0

i

, FD, S2

.

(B.4)

The term

N1i=0 r

2IF[i] does not depend on (

,

FD,

S ) and thus

can be dropped. Moreover,

N1i=0

i

, FD , S2

=

N1

i=0 CeBi

Ts S eCi

Ts 2

cos22FDi + =C

N1i=0

22S eB

i

Ts

eC

i

Ts

1

2+

1

2cos

4FDi+2

= 2CN1i=0

1

2+

1

2cos

4FDi + 2

2SCN1i=0

eB

i

Ts

eC

i

Ts

1

2+

1

2cos

4FDi+

CN,

(B.5)

where the orthogonality between eB[i/Ts] and eC[i/Ts]and the fact that the summation in (B.5) acts as a lowpassfilter have been exploited. Equation (B.5) shows that also the

termN1

i=0 i(, FD, S)2 is approximatively independent from

the parameters (, FD, S) and thus (B.4) becomes

, FD, S = arg max,FD ,S

N1i=0

i

, FD, S

rIF[i]

= arg max,FD ,S

N1i=0

rIF[i]dB

i

Ts

eBi

Ts S eCi

Ts cos2FDi + = arg max

,FD ,SdB

N1i=0

rIF[i]

eB

i

Ts

S eC

i

Ts

cos

2FDi +

.

(B.6)

Since the navigation bit dB is supposed constant over one-code period it can be moved out the summation in (B.6).Equation (B.6) represents the expression for the maximumlikelihood estimator for (, FD, S), however it can be appliedonly under the hypothesis of knowing the bit dB and thephase . In order to remove the dependence from those twoparameters the cost function considered in (B.6) is usuallymodified according to the following methodology. Since allthe terms in the last summation of (B.6) are real signals, it ispossible to rewrite (B.6) as follows:, FD, S

= arg max,FD ,S

Re

dB

N1i=0

rIF[i]

eB

i

Ts

S eC

i

Ts

exp j2FDi + j

,

(B.7)


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that is, the real part of a complex scalar product. Equation(B.7) can be further expanded as

(, FD, S )=

arg max,FD ,S

ReN1

i=0 rIF[i]eBi

Ts S eCi

Ts expj2FDi

exp

j +j

1dB

2

,

(B.8)

where the terms in square brackets do not depend on thephase and on the sign dB . Since dB takes values in {1, 1}it has been expressed as

dB = exp

j(1 dB) 2

. (B.9)

The summation in (B.8) is a complex number that can beexpressed in terms of phase and amplitude as

N1i=0

rIF[i]

eB

i

Ts

S eC

i

Ts

exp

j2FDi

=AS

, FD, S

expjS

, FD, S

.

(B.10)

In this way (B.8) can be rewritten as

,

FD,

S

= arg max,FD ,S

AS, FD, SRe

exp

jS

, FD, S

exp

j + j(1 dB)

2

.

(B.11)

If the phase and the sign dB are unknown, they can beestimated by maximizing (B.11) also with respect to thesetwo parameters. Equation (B.11) is maximized with respectto and dB when the condition

exp

jS(, FD, S)

exp

j+ j(1

dB)

2 =1 (B.12)

is verified. By substituting (B.12) into (B.11), the jointestimator for the delay0, the Doppler frequency FD,0, andthe relative phase S becomes

, FD, S =arg max

,FD ,SAS

, FD, S

=

N1i=0

rIF[i]

eB

i

Ts

S eC

i

Ts

exp

j2FDi

(B.13)

that is a form of quadrature or incoherent matched filter[19]. The estimator (B.13) is equivalent to the coherentacquisition block with sign recovery, FD, S

=arg max,FD ,SN1

i=0r

IF[i]eBi Ts S eCi Ts expj2FDi2

.

(B.14)

ACKNOWLEDGMENT

The authors would like to thank Laura Camoriano for hersupport during the development of this work.

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Documents

Data and Pilot Combining for Composite GNSS Signal Acquisition