Chi-Square Distribution Matching in Unambiguous Sine-BOC and Multiplexed-BOC Acquisition

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1 Chi-Square Distribution Matching in Unambiguous Sine-BOC and Multiplexed-BOC Acquisition Md. Farzan Samad and Elena Simona Lohan Department of Communications Engineering, Tampere University of Technology P.O.Box 553, FIN-33101, Finland; BIOGRAPHY Md. Farzan Samad obtained the M.Sc. degree in Communications Engineering from Tampere University of Technology (TUT), Finland, in September Currently, he is a Ph.D. student in the Tampere Unit for Computer-Human Interaction (TAUCHI) unit in the Department of Computer Sciences 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 University of Bucharest, Romania, in 1997, the D.E.A. degree in Econometrics, at Ecole Polytechnique, Paris, France, in 1998, and the Ph.D. degree in Telecommunications from Tampere University of Technology. In 2007 she was nominated as a Docent in the field of Wireless communication techniques for personal navigation. Since November 2003, Simona Lohan has been working as a Senior Researcher at TUT and she has been acting as a group leader for the mobile and satellite-based positioning activities at the Department of Communications Engineering. Her research interests include satellite positioning techniques, CDMA signal processing, and wireless channel modeling and estimation. She has been also involved with the EU FP6 project GREAT and EU FP7 project GRAMMAR. ABSTRACT Multiplexed Binary-Offset-Carrier (MBOC) modulated signals are the main candidates for the future Galileo Open Services (OS) and modernized GPS L1C signals. The Autocorrelation Functions (ACFs) of MBOC signals have additional lobes compared to the ACFs of classical GPS signals (which employ BPSK modulation) and the presence of these lobes introduce new challenges in the signal acquisition process. Several unambiguous acquisition techniques have been previously proposed in order to eliminate or diminish the sidelobes and enhance the acquisition process. The purpose of this paper is to model theoretically, via chisquare distributions, the test statistics of both unambiguous sine BOC-and MBOC modulated signals. The parameter distribution fitting is based on simulations and the resulting theoretical model is compared with the simulation results, in terms of detection probabilities. INTRODUCTION Sine-BOC and MBOC-modulated signals have a narrower main lobe of their ACFs, compared to the BPSK-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 the maximum correlation peak, due to sine and multiplexed BOC modulation. As a result, the ACF becomes ambiguous (i.e., having extra correlation peaks and some low values in the ACF within ±1 chip interval). In order to avoid the ambiguities of the Absolute value of ACF (AACF), several unambiguous or BPSK like acquisition techniques have been proposed in (Martin et al, 2003), (Heiries et al, 2004), (Fishman et al, 2000), (Betz et al, 2004), (Lohan, 2006), (Lohan et al, 2008). These unambiguous acquisition techniques are denoted as: Betz and Fishman (B&F), Martin and Heiries (M&H) and Unsuppressed Adjacent Lobes (UAL) methods, respectively. A theoretical analysis of the ambiguous and B&F unambiguous acquisition in the context of sine BOC modulation has been presented in (Lohan, 2006), based on the statistical modelling of the decision variables in the acquisition process. The purpose of this paper is to extend the analysis presented in (Lohan, 2006) to the other abovementioned unambiguous methods (i.e., M&H and UAL) and to MBOC modulation as well. The deterioration factors for modelling the variance and the non-centrality parameters in the acquisition process are estimated here for both sine BOC and MBOC modulations and for all 4 acquisition methods: one ambiguous (the classical one) and 3 unambiguous (B&F, M&H and UAL).

2 The matching between theory and simulations is validated via detection probability curves versus Carrier-to-Noise Ratio (CNR). The underlying theoretical model of M&H and UAL unambiguous acquisition methods has never been addressed in the literature so far to the best of the authors knowledge. Also, the analysis of chi-square based statistical model of B&F method has been limited only to sine BOC cases so far (Lohan, 2006). The theoretical model of the classical ambiguous acquisition has been however studied 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 comparable with those reported in (Dovis et al, 2008). II. SINE BOC AND MULTIPLEXED BOC MODULATIONS Sine BOC modulation (referred in what follows simply by BOC) is a square sub-carrier modulation (Betz, 1999), where a pseudorandom (PRN) signal at chip rate f c is multiplied by a rectangular sub-carrier of frequency f sc, which splits the signal spectrum into two parts (Barker et al, 2000), (Betz, 1999), (Lohan et al, 2007a). BOC modulation provides a simple and effective way of moving the signal energy away from band center, offering a high degree of spectral separation from conventional Binary Phase Shift Keying (BPSK) signals, whose energy is concentrated near band center. The resulting split-spectrum signal effectively enables frequency sharing, while providing attributes that include simple implementation, good spectral efficiency, high accuracy, and enhanced multipath resolution (Betz, 1999). BOC modulation generalizes the Manchester line coding scheme to more than one zero crossing per spreading symbol or chip (Raghavan et al, 2004), (Saltzberg, 1990). The BOC modulated signal ( ) is the convolution between a BOC waveform ( )and a modulating waveform ( ), as follows (Lohan et al, 2007a): ( ) = ( ) ) ( ) (1) where is the convolution operator, ( ) is the spread data sequence, is the nth complex data symbol (in case of a pilot channel, it is equal to 1), is the symbol period, is the kth chip corresponding to the nth symbol, = 1/f c is the chip period, S F is the spreading factor = / ), and ) is the Dirac pulse. The signals used in GPS and Galileo are wideband signals. Therefore in eq. (1), we assumed to have wideband data, that is, spread via a PRN sequence. MBOC modulation places a small amount of code power at higher frequencies, which improves the code tracking 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 of BOC(1,1) spectrum and BOC(6,1) spectra. It is possible to use a number of different time waveforms to generate MBOC(6,1,1/11) spectrum, which gives some implementation flexibility. Different time waveforms can be used to produce the MBOC(6,1,1/11) PSD. The two main ones are: the Composite BOC (CBOC) and the Time Multiplexed BOC (TMBOC). The CBOC method is based on a weighted sum (or difference) of BOC(1,1) and BOC(6,1)- modulated code symbols (Lohan et al, 2007b). The weighting factors w 1 and w 2 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 use CBOC(+) for even chips and CBOC(-) for odd chips (Avila-Rodriguez et al, 2006). In TMBOC, the whole signal is divided into blocks of N code symbols (Hein et al, 2006). Out of N code symbols, M <N symbols are BOC(1,1)-modulated and the remaining N M code symbols are BOC(6,1) modulated. An example of CBOC( +/- ) and TMBOC modulated waveforms, together with PRN sequence before modulation is shown in Fig 1. III. UNAMBIGUOUS ACQUISITION METHODS Betz & al. (Betz et al, 2004), (Barker et al, 2000) and Fishman & al. (Fishman et al, 2000) introduced the Sideband (SB) techniques, which are denoted here via B&F methods (after the initials of the first authors). In B&F methods, the receiver selects only the first or the first 2 main spectral lobes of the received signal and of the reference code, respectively and it filters out the remaining frequency content. In Single Sideband (SSB) B&F method, only one of the main lobes (either upper or lower) is considered when forming the decision statistic. The Dual SB (DSB) B&F method considers both the upper and lower main spectral lobes. The reference code is the BOC- or MBOC-modulated code sequence. The SSB B&F method suffers from higher non-coherent correlation losses than the DSB B&F method (Fishman et al, 2000).

3 In Unsuppressed Adjacent Lobes (UAL) method proposed and analyzed in (Lohan et al, 2008), (Burian et al, 2006b), (Burian et al, 2006a), the filtering part is completely removed. Therefore, the adjacent lobes of the 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 the complexity of the receiver part is reduced, as no extra filters are required. As for M&H case, the reference code in UAL method is the BPSK-modulated PRN sequence of ±1, held at BOC or MBOC rate. A detailed presentation of the unambiguous methods implementation for MBOC waveforms can be found in (Samad et al, 2009). The normalized AACFs for ambiguous and the non ambiguous BOC and MBOC (TMBOC implementation) algorithms are shown in Fig. 2. From Fig. 2, it can be clearly seen that the ambiguities of the AACF disappear after unambiguous processing. IV. CHI-SQUARE BASED MODELS 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 like techniques, 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 the secondary lobes between the principal lobes (if any). A modification 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) and this is the approach used here (still referred to as M&H for simplicity sake). In M&H, the reference code is not the filtered BOC- or MBOC-modulated code, but the BPSK-modulated code sequence, held at the same rate as BOC or MBOC signal. In signal acquisition, after the reference signal (filtered or not) is correlated with the received signal (filtered or not), coherent integration on N c ms is performed, then the envelope or squared envelope of the coherent correlation is taken and the resulting waveform is further non-coherently integrated over Nnc blocks. The output of non-coherent integration forms the decision statistic Z, which obviously depends on the delay error and Doppler error. In static channels, Z is a central chi-square distribution in an incorrect bin and a noncentral chi-square distribution in a correct bin (Heiries et al, 2004), (Fischer et al, 2004), (Schmid et al, 2004), (Bastide et al, 2002), (Lohan, 2006), (Borio, 2008), (Borio et al, 2008). This is due to the fact that the output of the coherent integration is a complex Gaussian variable, due to the additive white noise real and imaginary parts. The degrees of freedom of the chi-square distributions is 2Nnc and the variance of such distributions is (Lohan, 2006) = (2) ) where is the coherent integration time, is the non-coherent integration length and is the narrowband noise power spectral density (doublesided), which is related to the CNR as follows (Bastide et al, 2002) = 10 [ ] (3) where is the signal energy.

4 filtering effects in the unambiguous SSB and DSB processing. Therefore, the variance and non-centrality parameters 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 DSB method is 4Nnc, because, before the non-coherent integration process, there are 4 real Gaussian variables, coming from the real and imaginary parts of the noise in the upper and the lower bands, respectively (Fischer et 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. Fig. 2. Normalized AACFs for DSB correlation methods. Upper plot: BOC(1,1). Lower plot: MBOC (TMBOC case). We consider first the ambiguous case (i.e., no filtering or shifting of the received signal and reference signal, only plain correlation between the two).in this case, the square-root of the non-centrality parameter of the non-central chi-square distribution is a function )of the 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 autocorrelation value at delay error for the BOC- or MBOC-modulated PRN code. Now, for the unambiguous methods, we observe that the use of a linear filter on a complex Gaussian distribution preserves the same distribution at the output (Lohan, 2006). Therefore, the test statistics of B&F, UAL and M&H methods can also be modeled via chi-square (central and non-central) distributions. For unambiguous algorithms, there are some additional deterioration factors in the variance and non-centrality parameters, denoted here via and, respectively. These parameters account for the correlation losses and Method Variance Square root of noncentrality parameter (if correct bin) aboc/ ambo C SSB B&F, UAL, M&H DSB B&F, UAL, M&H Degrees of freedo m, 2 ( ) ( ), 2 ( ) 2, 4 V. DISTRIBUTION MATCHING RESULTS The values of and depend on the type of the acquisition 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: =1and =1). We noticed that the values for these deterioration factors for unambiguous cases are different for FFT and for time-domain based correlation. Via extensive simulation runs and distribution matching according to minimum Kullback-Leibler divergence () criterion (Kullback et al, 1951), we found the BOC deterioration factors from Tables II and III for time-domain based correlation and FFT based correlation, respectively. The MBOC deterioration factors are shown in Tables IV and V for time-domain based correlation and FFT based correlation, respectively. The simulations were carried out for an oversampling factor, =6. Also, value is shown as a measure of the fitting between theory and simulations. A correct bin is a timefrequency bin where signal is present and an incorrect bin 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 of the total power, if the total

5 power is normalized to 1. And for MBOC, the value is around TABLE II UNAMBIGUOUS BOC METHODS (TIME DOMAIN BASED CORRELATION). B&F UAL M&H TABLE III UNAMBIGUOUS BOC METHODS (FFT CORRELATION). B&F UAL M&H TABLE IV UNAMBIGUOUS MBOC METHODS (TIME DOMAIN BASED CORRELATION). B&F UAL M&H TABLE V UNAMBIGUOUS MBOC METHODS (FFT CORRELATION). B&F UAL M&H The parameter values from tables II, III, IV and V can be explained by the presence of some correlation losses in unambiguous approaches. These correlation losses are associated with the filtering and with the modification of the reference code (which can be seen as a decrease of the non-centrality parameter), together with some decrease in the noise variance (due to the filtering of the signal and noise). Examples of the simulation-based normalized histogram and the theoretical chi-square PDF for correct and incorrect bins are shown in Fig. 3. Similar good matching has been observed for various CNR levels, coherent and non-coherent integration times, and for both FFT and time domain based correlations. Fig. 3. Matching between theoretical and simulationbased distributions of the test statistic Z. Upper plot: BOC(1,1) B&F method, CNR = 25 db-hz, = 20, =2, Time domain correlation. Lower plot: MBOC UAL method, CNR = 30 db-hz, = 10, =3, FFT correlation. Fig. 4 compares the theoretical values with the values from simulations. The upper plot of Fig. 4 shows the comparison for BOC(1,1) modulation, whereas the lower plot shows for MBOC modulation. In the simulations, = 20 ms was used, followed by =2 blocks. The oversampling factor =6 and the time-bin step = 0.5 were considered. From the comparison between theoretical and simulated results, 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 with FFT based correlation of the theoretical model of both BOC(1,1) (upper plot) and MBOC (lower plot) modulations. The simulations were carried out with = 10 ms, =3blocks, =6 and = 0.5 chips. From Fig. 5, it can be observed that the time domain correlations give slightly better values than FFT correlations and the performance difference is not significant. Other values of, and gave similar types of results.

6 Fig. 4. Comparison between theoretical and simulation based results. Upper plot: BOC(1,1); Lower plot: CBOC( +/- ). Fig. 5. Comparison between time domain and FFT based correlations of the theoretical model. Upper plot: BOC(1,1); Lower plot: CBOC( +/- ). VI. CONCLUSIONS In this paper we presented a theoretical model, based on chi square distributions, for the unambiguous acquisition of split spectrum signals, such as sineboc and MBOC signals used in Galileo system. The distribution matching was based on extensive simulation runs and on Kullback-Leibler divergence criterion, used to test the similarity between theoretical and measured distribution. We also verified the obtained parameters via comparing the simulation curves with the theoretical curves of detection probability at various CNRs. Based on the theoretical model it can be seen that the FFT-based correlation is slightly worse than the time-domain correlation (i.e., slightly lower signal energy after unambiguous processing) and that MBOC unambiguous processing is 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. ACKNOWLEDGEMENT This work was carried out in the project Future GNSS Applications and Techniques (FUGAT) funded by the Finnish Funding Agency for Technology and Innovation (Tekes). This work was also supported by the Academy of Finland. REFERENCES Avila-Rodriguez, J. and Wallner, S. and Hein, G. and Rebeyrol, E. and Julien, O. and Macabiau, C. and Ries, L. and DeLatour, A. and Lestarquit, L. and Issler, J. (2006). CBOC -An Implementation of MBOC, in First CNES Workshop on Galileo Signals and Signal Processing, (Toulouse, France), October Barker, C. B. C. and Betz, J. W. and Clark, J. E. and Correia, J. T. and Gillis, J. T. and Lazar, S. and Rehborn, L. K. A. and Straton, J. R. (2000). Overview of the GPS M Code Signal, in Proc. of Intitute of Navigation National Tech. Meeting: Navigating into the New Millennium, pp , January 2000.

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