MBOC Chi Square v06

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Chi-Square Distribution Matching in

Unambiguous Sine-BOC and

Multiplexed-BOC AcquisitionMd. Farzan Samad and Elena Simona Lohan

Department of Communications Engineering, Tampere University of Technology

P.O.Box 553, FIN-33101, Finland;[email protected]; [email protected]

BIOGRAPHY

Md. Farzan Samad obtained the M.Sc. degree inCommunications Engineering from TampereUniversity of Technology (TUT), Finland, inSeptember 2009. Currently, he is a Ph.D. student in the

Tampere Unit for Computer-Human Interaction(TAUCHI) unit in the Department of ComputerSciences at the University of Tampere (UTA), Finland.His research interests include satellite positioning

techniques and mobile haptics.

Elena Simona Lohan obtained the M.Sc. degree in

Electrical Engineering from the Politehnica Universityof Bucharest, Romania, in 1997, the D.E.A. degree inEconometrics, at Ecole Polytechnique, Paris, France, in

1998, and the Ph.D. degree in Telecommunicationsfrom Tampere University of Technology. In 2007 shewas nominated as a Docent in the field of Wirelesscommunication techniques for personal navigation.

Since November 2003, Simona Lohan has beenworking as a Senior Researcher at TUT and she hasbeen acting as a group leader for the mobile and

satellite-based positioning activities at the Departmentof Communications Engineering. Her research interestsinclude satellite positioning techniques, CDMA signalprocessing, and wireless channel modeling andestimation. She has been also involved with the EUFP6 project GREAT and EU FP7 project GRAMMAR.

ABSTRACT

Multiplexed Binary-Offset-Carrier (MBOC) modulatedsignals are the main candidates for the future GalileoOpen Services (OS) and modernized GPS L1C signals.The Autocorrelation Functions (ACFs) of MBOCsignals have additional lobes compared to the ACFs ofclassical GPS signals (which employ BPSKmodulation) and the presence of these lobes introduce

new challenges in the signal acquisition process.Several unambiguous acquisition techniques have beenpreviously proposed in order to eliminate or diminish

the sidelobes and enhance the acquisition process. Thepurpose of this paper is to model theoretically, via chi-square distributions, the test statistics of both

unambiguous sine BOC-and MBOC modulated signals.

The parameter distribution fitting is based onsimulations and the resulting theoretical model iscompared with the simulation results, in terms of

detection probabilities.

INTRODUCTION

Sine-BOC and MBOC-modulated signals have anarrower main lobe of their ACFs, compared to theBPSK-modulated signal. This feature is known to

improve the accuracy in the delay tracking process(Hein et al, 2006), (Avila-Rodriguez et al, 2006).However, additional peaks and some gaps or deep

fades appear within 1 chip interval around themaximum correlation peak, due to sine andmultiplexed BOC modulation. As a result, the ACFbecomes ambiguous (i.e., having extra correlation

peaks and some low values in the ACF within 1 chipinterval). In order to avoid the ambiguities of theAbsolute value of ACF (AACF), several unambiguous

or BPSK like acquisition techniques have beenproposed in (Martin et al, 2003), (Heiries et al, 2004),(Fishman et al, 2000), (Betz et al, 2004), (Lohan,

2006), (Lohan et al, 2008). These unambiguousacquisition techniques are denoted as: Betz andFishman (B&F), Martin and Heiries (M&H) andUnsuppressed Adjacent Lobes (UAL) methods,

respectively. A theoretical analysis of the ambiguousand B&F unambiguous acquisition in the context of

sine BOC modulation has been presented in (Lohan,2006), based on the statistical modelling of thedecision variables in the acquisition process. Thepurpose of this paper is to extend the analysis

presented in (Lohan, 2006) to the other above-mentioned unambiguous methods (i.e., M&H andUAL) and to MBOC modulation as well. Thedeterioration factors for modelling the variance and the

non-centrality parameters in the acquisition process areestimated here for both sine BOC and MBOCmodulations and for all 4 acquisition methods: one

ambiguous (the classical one) and 3 unambiguous(B&F, M&H and UAL).
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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The matching between theory and simulations isvalidated via detection probability curves versusCarrier-to-Noise Ratio (CNR). The underlying

theoretical model of M&H and UAL unambiguousacquisition methods has never been addressed in theliterature so far to the best of the authors knowledge.

Also, the analysis of chi-square based statistical modelof B&F method has been limited only to sine BOCcases so far (Lohan, 2006). The theoretical model of

the classical ambiguous acquisition has been howeverstudied before for both sine BOC and MBOC cases in(Schmid et al, 2004), (Bastide et al, 2002), (Dovis et al,2008), (Borio, 2008), (Borio et al, 2008) and our

results regarding the ambiguous case are comparablewith those reported in (Dovis et al, 2008).

II. SINE BOC AND MULTIPLEXED BOC

MODULATIONS

Sine BOC modulation (referred in what follows simplyby BOC) is a square sub-carrier modulation (Betz,1999), where a pseudorandom (PRN) signal at chip

rate fc is multiplied by a rectangular sub-carrier offrequencyfsc, which splits the signal spectrum into twoparts (Barker et al, 2000), (Betz, 1999), (Lohan et al,

2007a). BOC modulation provides a simple andeffective way of moving the signal energy away fromband center, offering a high degree of spectralseparation from conventional Binary Phase Shift

Keying (BPSK) signals, whose energy is concentratednear band center. The resulting split-spectrum signaleffectively enables frequency sharing, while providing

attributes that include simple implementation, goodspectral efficiency, high accuracy, and enhancedmultipath resolution (Betz, 1999).

BOC modulation generalizes the Manchester linecoding scheme to more than one zero crossing per

spreading symbol or chip (Raghavan et al, 2004),

(Saltzberg, 1990). The BOC modulated signal ( ) isthe convolution between a BOC waveform ( ) anda modulating waveform ( ) , as follows (Lohan et al,2007a):

( ) =

( ) )

( ) ( 1 )

where is the convolution operator, ( ) is thespread data sequence, is the nth complex datasymbol (in case of a pilot channel, it is equal to 1),

is the symbol period, is the kth chipcorresponding to the nth symbol, = 1/fc is the chip

period, SF is the spreading factor

= / ) , and ) is the Dirac pulse. Thesignals used in GPS and Galileo are wideband signals.

Therefore in eq. (1), we assumed to have widebanddata, that is, spread via a PRN sequence.

MBOC modulation places a small amount of codepower at higher frequencies, which improves the codetracking performance (Hein et al, 2006), (GJU, 2006),(Avila-Rodriguez et al, 2006). The Power Spectral

Density (PSD) of MBOC(6,1,1/11) is a combination ofBOC(1,1) spectrum and BOC(6,1) spectra. It ispossible to use a number of different time waveforms

to generate MBOC(6,1,1/11) spectrum, which givessome implementation flexibility.

Different time waveforms can be used to produce theMBOC(6,1,1/11) PSD. The two main ones are: theComposite BOC (CBOC) and the Time Multiplexed

BOC (TMBOC).

The CBOC method is based on a weighted sum (ordifference) of BOC(1,1) and BOC(6,1)- modulated

code symbols (Lohan et al, 2007b). The weighting

factors w1 and w2 are chosen such that = 1 .There are 3 proposed implementations of CBOC:CBOC(+), CBOC(-) and CBOC(+/-). The last one is a

combination of the 2 previous ones, i.e., we useCBOC(+) for even chips and CBOC(-) for odd chips(Avila-Rodriguez et al, 2006). In TMBOC, the wholesignal is divided into blocks ofNcode symbols (Hein

et al, 2006). Out ofNcode symbols,M


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Fig. 1. A snapshot of PRN sequence (upper plot) and

of CBOC(+/-) (middle plot) and TMBOC (lower plot)-modulated signals.

Martin & al. (Martin et al, 2003) and Heiries & al.

(Heiries et al, 2004) proposed the BPSK liketechniques, which are denoted here as M&H methods.In M&H methods, the filter bandwidth includes the

two principal lobes of the spectrum and all thesecondary lobes between the principal lobes (if any). Amodification to the original M&H algorithm of

(Heiries et al, 2004) was proposed in (Lohan et al,2008), (Burian et al, 2006b), (Burian et al, 2006a) andthis is the approach used here (still referred to as M&Hfor simplicity sake). In M&H, the reference code is not

the filtered BOC- or MBOC-modulated code, but theBPSK-modulated code sequence, held at the same rateas BOC or MBOC signal.

In Unsuppressed Adjacent Lobes (UAL) methodproposed and analyzed in (Lohan et al, 2008), (Burian

et al, 2006b), (Burian et al, 2006a), the filtering part iscompletely removed. Therefore, the adjacent lobes ofthe main lobes are fully unsuppressed in UAL and they

may affect the performance of the acquisition block(Lohan et al, 2008). The advantage is that thecomplexity of the receiver part is reduced, as no extrafilters are required. As for M&H case, the reference

code in UAL method is the BPSK-modulated PRN

sequence of1, held at BOC or MBOC rate.

A detailed presentation of the unambiguous methodsimplementation for MBOC waveforms can be found in(Samad et al, 2009).

The normalized AACFs for ambiguous and the nonambiguous BOC and MBOC (TMBOC

implementation) algorithms are shown in Fig. 2. FromFig. 2, it can be clearly seen that the ambiguities of theAACF disappear after unambiguous processing.

IV. CHI-SQUARE BASED MODELS

In signal acquisition, after the reference signal (filteredor not) is correlated with the received signal (filtered or

not), coherent integration on Nc ms is performed, thenthe envelope or squared envelope of the coherentcorrelation is taken and the resulting waveform is

further non-coherently integrated over Nnc blocks. Theoutput of non-coherent integration forms the decision

statisticZ, which obviously depends on the delay error and Doppler error . In static channels, Z is acentral chi-square distribution in an incorrect bin and anoncentral chi-square distribution in a correct bin

(Heiries et al, 2004), (Fischer et al, 2004), (Schmid etal, 2004), (Bastide et al, 2002), (Lohan, 2006), (Borio,2008), (Borio et al, 2008). This is due to the fact thatthe output of the coherent integration is a complex

Gaussian variable, due to the additive white noise realand imaginary parts. The degrees of freedom of thechi-square distributions is 2Nnc and the variance of

such distributions is (Lohan, 2006)

=

) ( 2 )where is the coherent integration time, is thenon-coherent integration length and is thenarrowband noise power spectral density (double-

sided), which is related to the CNR as follows (Bastideet al, 2002)

= 1 0 [ ]

( 3 )

where is the signal energy.


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Fig. 2. Normalized AACFs for DSB correlationmethods. Upper plot: BOC(1,1). Lower plot: MBOC

(TMBOC case).

We consider first the ambiguous case (i.e., no filteringor shifting of the received signal and reference signal,only plain correlation between the two).In this case, thesquare-root of the non-centrality parameter of thenon-central chi-square distribution is a function ) ofthe delay and Doppler errors as follows (Heiries et al,2004), (Bastide et al, 2002), (Lohan, 2006), (Borio,2008), (Borio et al, 2008):

=

= ( ) ( ) )

(

)

( 4 )

where ) and ( ) is the auto-correlation value at delay error for the BOC- orMBOC-modulated PRN code.

Now, for the unambiguous methods, we observe thatthe use of a linear filter on a complex Gaussian

distribution preserves the same distribution at theoutput (Lohan, 2006). Therefore, the test statistics ofB&F, UAL and M&H methods can also be modeled

via chi-square (central and non-central) distributions.For unambiguous algorithms, there are some additionaldeterioration factors in the variance and non-centrality

parameters, denoted here via and, respectively.These parameters account for the correlation losses and

filtering effects in the unambiguous SSB and DSBprocessing. Therefore, the variance and non-centralityparameters of each of the acquisition methods studied

here can be modeled according to table I. From table I,it can be seen that the degrees of freedom for DSBmethod is 4Nnc, because, before the non-coherent

integration process, there are 4 real Gaussian variables,coming from the real and imaginary parts of the noisein the upper and the lower bands, respectively (Fischeret al, 2004).

TABLE I

PARAMETERS OF THE CENTRAL AND NON-CENTRAL

CHI-SQUARE DISTRIBUTIONS OF THE TEST STATISTIC Z

IN AMBIGUOUS (aBOC/aMBOC) AND UNAMBIGUOUS

ACQUISITION.

Method Variance Square root of non-centrality parameter

(if correct bin)

Degreesoffreedo

maBOC/aMBO

C

( )

, 2

SSBB&F,UAL,M&H

( ) , 2

DSB

B&F,UAL,

M&H

( ) 2 , 4

V. DISTRIBUTION MATCHING RESULTS

The values of and depend on the type of theacquisition algorithm and on the modulation types(BOC or MBOC). For ambiguous acquisition, there is

no deterioration (i.e., the ambiguous case is taken as

reference: = 1 and = 1 ).We noticed that the values for these deteriorationfactors for unambiguous cases are different for FFT

and for time-domain based correlation. Via extensivesimulation runs and distribution matching according tominimum Kullback-Leibler divergence (KL) criterion(Kullback et al, 1951), we found the BOC deterioration

factors from Tables II and III for time-domain basedcorrelation and FFT based correlation, respectively.The MBOC deterioration factors are shown in Tables

IV and V for time-domain based correlation and FFTbased correlation, respectively. The simulations were

carried out for an oversampling factor, = 6 . Also,KL value is shown as a measure of the fitting between

theory and simulations. A correct bin is a time-frequency bin where signal is present and an incorrectbin is a time-frequency bin where only noise is present.

From tables II, III, IV, V, it can be observed that for

B&F method,

and

values are related to the

power per main lobe (). The of BOC(1,1)signal is around 0.427 of the total power, if the total


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power is normalized to 1. And for MBOC, the value is around 0.3896.

TABLE II

UNAMBIGUOUS BOC METHODS (TIME DOMAIN BASED

CORRELATION).

Method KL(incorrectbins)

KL(correctbins)

B&F 0.427 0.444 0.0028 0.0180

UAL 1 0.655 0.0020 0.0215

M&H 1 0.655 0.0147 0.0196

TABLE III

UNAMBIGUOUS BOC METHODS (FFT CORRELATION).

Method KL(incorrect

bins)

KL(correct

bins)

B&F 0.427 0.427 0.0019 0.0498UAL 1 0.620 0.0021 0.022

M&H 1 0.616 0.0139 0.0378

TABLE IV

UNAMBIGUOUS MBOC METHODS (TIME DOMAIN BASED

CORRELATION).


KL(correctbins)

B&F 0.3896 0.3561 0.0248 0.0129

UAL 1 0.527 0.0021 0.0157

M&H 1 0.531 0.0171 0.0097

TABLE V

UNAMBIGUOUS MBOC METHODS (FFT CORRELATION).


KL(correctbins)

B&F 0.3896 0.3896 0.0025 0.0175

UAL 1 0.590 0.0019 0.0377

M&H 1 0.585 0.0156 0.0379

The parameter values from tables II, III, IV and V can

be explained by the presence of some correlation lossesin unambiguous approaches. These correlation lossesare associated with the filtering and with themodification of the reference code (which can be seenas a decrease of the non-centrality parameter), together

with some decrease in the noise variance (due to thefiltering of the signal and noise).

Examples of the simulation-based normalizedhistogram and the theoretical chi-square PDF forcorrect and incorrect bins are shown in Fig. 3. Similar

good matching has been observed for various CNRlevels, coherent and non-coherent integration times,

and for both FFT and time domain based correlations.

Fig. 3. Matching between theoretical and simulation-based distributions of the test statistic Z. Upper plot:

BOC(1,1) B&F method, CNR = 25 dB-Hz, = 2 0 ,

= 2 , Time domain correlation. Lower plot: MBOC

UAL method, CNR = 30 dB-Hz, = 1 0 , = 3 ,FFT correlation.

Fig. 4 compares the theoretical values with the values from simulations. The upper plot of Fig. 4shows the comparison for BOC(1,1) modulation,whereas the lower plot shows for MBOC modulation.

In the simulations, = 2 0 ms was used, followed by = 2 blocks. The oversampling factor = 6 andthe time-bin step = 0 . 5 were considered. Fromthe comparison between theoretical and simulatedresults, it can be said that the theoretical values match

quite well the simulated values. Similar good

matchings were observed for various, and values and for both DSB and SSB methods.

Fig. 5 compares time-domain based correlation withFFT based correlation of the theoretical model of bothBOC(1,1) (upper plot) and MBOC (lower plot)modulations. The simulations were carried out with

= 1 0 ms, = 3 blocks, = 6 and = 0 . 5chips. From Fig. 5, it can be observed that the time

domain correlations give slightly better values thanFFT correlations and the performance difference is not

significant. Other values of, and gavesimilar types of results.


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Fig. 4. Comparison between theoretical and simulationbased results. Upper plot: BOC(1,1); Lower plot:

CBOC(+/-).

VI. CONCLUSIONS

In this paper we presented a theoretical model, basedon chi square distributions, for the unambiguous

acquisition of split spectrum signals, such as sineBOCand MBOC signals used in Galileo system. Thedistribution matching was based on extensivesimulation runs and on Kullback-Leibler divergencecriterion, used to test the similarity between theoreticaland measured distribution. We also verified the

obtained parameters via comparing the simulationcurves with the theoretical curves of detectionprobability at various CNRs. Based on the theoreticalmodel it can be seen that the FFT-based correlation is

slightly worse than the time-domain correlation (i.e.,slightly lower signal energy after unambiguousprocessing) and that MBOC unambiguous processingis slightly worse than the sine BOC(1,1) unambiguous

processing. Based on the detection probability curves,we can also state that the unambiguous processing,

especially when used in dual sideband configuration,offers a greater advantage over the ambiguous

processing for both sine BOC and MBOC cases.

Fig. 5. Comparison between time domain and FFTbased correlations of the theoretical model. Upper plot:BOC(1,1); Lower plot: CBOC(+/-).

ACKNOWLEDGEMENT

This work was carried out in the project Future GNSSApplications and Techniques (FUGAT) funded by theFinnish Funding Agency for Technology and

Innovation (Tekes). This work was also supported bythe Academy of Finland.

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