OVERSAMPLING LIMITS FOR BINARY OFFSET CARRIER MODULATION FOR THE ACQUISITION OF GALILEO SIGNALS Adina Burian∗ Institute of Communications Engineering Tampere University of Technology P.O. Box 553, FIN-33101 Tampere, Finland
[email protected]
Elena Simona Lohan Institute of Comm. Engineering Tampere University of Technology P.O. Box 553, FIN-33101 Tampere, Finland
[email protected]
A BSTRACT Code and frequency acquisition are important stages of today’s satellite systems, such as the European system Galileo. Newly proposed modulation types, such as Binary Offset Carrier (BOC) modulation, have triggered new challenges in the acquisition process. The features and properties of BOC-modulated code sequences are still not wellunderstood in the context of fast acquisition algorithms. Moreover, no research studies have been dedicated so far to the effects of oversampling in the presence of BOCmodulated pseudorandom signals. However, the oversampling factor is typically dictated by hardware limitations, and cannot be assumed to be always equal to unity. In this paper, we study the effects of oversampling on the BOCmodulated pseudorandom codes during the CDMA code acquisition process. We also focus on the design of the code-Doppler bin size and we give the conditions to be fulfilled in order to achieve good detection probabilities and low mean acquisition times. Simulation results are shown for realistic signals of the Galileo satellite system. I.
I NTRODUCTION
BOC modulation has been proposed in [2], in order to get a more efficient sharing of the L-band spectrum by multiple signals of civilian and military users, using together the modernized Global Positioning System (GPS) with the already-existing GPS signals. The spectral efficiency is achieved by moving the signal energy away from the band center, thus offering a higher degree of spectral separation between BOC-modulated signals and the other signals which use traditional shift-keying modulation. BOC modulation performance was studied for the GPS military M-signal in [3, 4]. Later, it has also been proposed for the use with the new Galileo signals and modernized GPS signals in [6, 11]. The BOC-modulation is an extension of the so-called Tricode Hexaphase Modulation, which is based on Manchester coding of each chip symbol. Both are square-wave modulation schemes which employ Manchester coding for a pseudorandom (PRN) code and use the typical Non-Return-to-Zero (NRZ) format [7]. Another proposed modulation method is Alternative-BOC, which has different PRN codes in the lower and the upper main lobes, and allows up to four signal to be transmitted on the same spectrum [11]. BOC and alternative-BOC modulations are conceptually closed, therefore the focus of this ∗ This
work was carried out in the project ”Advanced Techn. for Mobile Positioning” funded by the National Technology Agency of Finland (Tekes).
Markku Renfors Institute of Comm. Engineering Tampere University of Technology P.O. Box 553, FIN-33101 Tampere, Finland
[email protected]
paper will be on the BOC-modulated signals. The usage of BOC modulated signals for Galileo system was proposed in [2, 3]. Since BOC modulation can be seen as a generalization of Manchester line coding [2], the results presented here are not restricted to Galileo and modernized GPS applications, but they may regard as well some other applications which use Manchester or BOC codes. The line codes are typically used to provide particular spectral characteristics of the data signals. One important parameter that affects the performance of the baseband receiver is the oversampling factor, or the number of sub-samples per BOC-sample. The effect of the oversampling factor in the presence of BOC modulation is the main concern of our paper. The sampling resolution affects the timing accuracy. For example, with an oversampling factor of 2 and a chip rate of 10.23 MHz, the timing error is of the order of 48 nsec, which translates into a distance error of the order of 15 meters. If BOC modulation is also used, this error decreases further with a factor equal to the BOC modulation order. In a conventional hardware implementation, the correlation spacing can also be seen as the resolution at which the correlation function is sampled [13]. Most current GPS receivers use a sampling rate of two samples per chip for either a C/A code or P(Y) code [5]. Moreover, the initial sampling point varies randomly according to the channel delay. Therefore, assuming that the sampling point does not always correspond to the maximum eye opening (i.e. the peak of the correlation function), some correlation losses are expected. This paper is organized as follows: in Section II, the BOC modulation is briefly presented. Section III introduces the received signal model and the acquisition problem. The impact of oversampling factor on BOC-modulated PRN codes is presented in Section IV. Section V concludes the paper. II.
B INARY O FFSET C ARRIER (BOC) MODULATION
BOC modulation is a square subcarrier modulation, where a signal d(t) is multiplied by a rectangular subcarrier of frequency fsc , which splits the spectrum of the signal into two parts. Formally, the BOC-modulated signal dBOC (t) can be written as [2], [4]: dBOC (t) = d(t) ∗ sign(sin(2πfsc t)).
(1)
The usual notation for BOC modulation is BOC(fsc , fc ), where fsc is the sub-carrier frequency, and fc is the chip rate. For Galileo signals, the notation BOC(n1 , n2 ) is also used, where n1 and n2 are two indices satisfying the
BOC(10,5) (i.e., BOC(4,2)) modulation − time−domain waveform
a baseband signal with BOC modulation is [3]:
Spreading sequence BOC modulated code
πf 2fsc
µ
¶ 2 πf fc
sin µ ¶
.
πf 2fsc
πf cos
(3)
0
−0.5
−1 5
10
15
20
25
30
35
40
Chips
Figure 1: Example of BOC-modulated PRN code in time domain. relationships n1 = fsc [MHz]/1.023 [MHz] and n2 = fc [MHz]/ 1.023 [MHz], respectively (we remark that the sub-carrier frequency and the chip rate are always integer multiples of 1.023 MHz frequency, hence, n1 and n2 have integer values) [2, 6]. It should be mentioned that BOC modulation generalizes the Manchester scheme (i.e., BOC(fsc , fsc )) to more than one zero crossing on spreading symbol. An example showing the BOC-modulated waveform is shown in Fig. 1 for BOC(10, 5). Both the spreading code (solid line) and the modulated spreading code (dashed line) are plotted. We define the BOC-modulation order NBOC as: 2fsc NBOC , . (2) fc We notice that from the baseband analysis point-of-view, the BOC-modulated waveforms depend only on NBOC . Here, the NBOC factor also represents the number of BOCsamples per chip. The BOC signals with NBOC of four or greater provide a better displacement of signal energy from band center [2]. Representative values for BOC modulation orders are NBOC = 4 (e.g., BOC(8, 4), BOC(10, 5)), NBOC = 6 (e.g., BOC(9, 3)), together with Manchester modulation NBOC = 2 (e.g., BOC(5, 5) or BOC(1, 1)) [2]. A particular case of BOC is BPSK with NBOC = 1.
The PSD can be also computed as the absolute value of the Fourier transform of the modulated-signal autocorrelation function. The theoretical and simulated PSDs are shown in Fig. 2 for BOC(10, 5) modulation. The theoretical curves are obtained with Eq. 3. The simulation curves are obtained for a PRN sequence of length 50000. Clearly, BOC modulation has weaker frequency components around the carrier frequency, insuring a low interference in C/A and P(Y) signal bands of GPS. We also remark from Fig. 2 that the theory and simulation results match reasonably well. The real part of the autocorrelation function (ACF) for a PRN spreading code of length SF = 10230 and BOC modulation is shown in Fig. 3 for BOC(10,5) modulation. If we compare the real part of the ACF of a BOC-modulated code with the real part of the ACF of a BPSK-modulated code, we notice that the main lobe is narrower, but we do have some fades in the ACF, which could make the synchronization process more difficult. The impact of oversampling will be considered in Section IV. 4
2.5
x 10
Autocorrelation function, pn code of length 10230+ BOC(2,2) BOC ACF BPSK ACF
2
1.5
1
ACF
Code sequence
sin Gs (f ) = fc
0.5
¶
µ
1
0.5
0
−0.5
−1
−1.5 −10
−8
−6
−4
−2
0
2
4
6
8
10
Sub−chips
Figure 3: Real part of ACF for BOC-modulated random sequences and for BPSK-modulated random sequences.
BOC(10,5) (i.e., BOC(4,2)) modulation − PSD −60
PSD (simulated) PSD (theoretical) −70
PSD
−80
Data symbols bn
−90
Spreading
−100
−110
−120
PN code −10
−5
0
5
10
BOC modulation
Ns
Channel
Doppler Code acquisition
Towards despreading, data recovery and position estimation blocks
15
Frequency (MHz)
Figure 2: Example of PSD for BOC-modulated PRN code. The normalized power spectral density (PSD) Gs (f ) of
Figure 4: Main operations performed at transmitter and receiver.
Correct time/frequency window
R ECEIVED SIGNAL MODEL AND ACQUISITION PROBLEM
The simplified baseband block diagram of the transmitter and receiver of a BOC-modulated CDMA signal is shown in Fig. 4. After spreading and BOC-modulation, the data sequence is further upsampled, with a sampling factor of Ns , representing the number of sub-samples per BOC sample. Therefore, one chip will consist of NBOC Ns subsamples. The reason for using further oversampling might be twofold: first, in order to achieve the desired delay accuracy when performing the acquisition and tracking in digital domain (we remark that the alternative solution would be to use interpolation); second, because the sampling rate might be dictated by some hardware constraints. Therefore, we might expect that some applications do require an oversampling factor higher than 1. Moreover, this oversampling factor might be non-integer. Mathematically, the received signal r(t) via a channel which introduced the delay τ and the Doppler shift fD can be written as: n=+∞ X p +j2πfD t r(t) = Eb e bn sn (t − τ ) + η(t)
(4)
Autocorrelation after non−coherent integration
III.
1 0.8 0.6 0.4 0.2
0 500 10 8
0
6 4
Doppler error (Hz)
−500
2 0
Code phase [chips]
Figure 6: Illustration of a time-frequency window.
processing can be moved to the tracking stage. The codeDoppler acquisition is done via correlation with a reference signal sref (t, τb, fc D , n1 ) including the PRN code and the BOC modulation: sref (t, τb, fc D , n1 )
c
= e−j2πfD t
SF X k=1
n=−∞
ck,n1
NX BOC
(−1)m
m=1
× p(t − nSF NBOC Ns Ts − kNBOC Ns Ts − mNs Ts − τb).(6)
where Eb is the bit energy (incorporating also the channel gain), bn are the data symbols, η(·) is the additive white noise added by the channel, and sn (t) is the BOCmodulated and spread code sequence, corresponding to nth data symbol:
The incoming signal is correlated with a reference code with a certain estimated delay τb and Doppler shift fbD . In the sequel, it is coherently averaged over NC msec, and SF NX BOC afterwards non-coherent averaging over NN C blocks may X sn (t) = ck,n (−1)m p(t − nSF NBOC Ns Ts be further used (this is especially needed in fading environm=1 k=1 ments, when the maximum coherent integration length is − kNBOC Ns Ts − mNs Ts ), (5) dictated by the coherence time of the channel). The decision statistic is then formed and compared with a detection where ck,n is the k-th chip value corresponding to the n-th threshold γ. data symbol, SF is the spreading factor or the code epoch The decision variable is formed in a serial, hybrid or length (e.g., for C/A signal of GPS, SF = 1023 and for parallel manner [12], by splitting the code-Doppler search Galileo signals, SF = 10230), p(·) is a train of rectangular space into several code-Doppler windows, as illustrated in pulses, Ns is the oversampling factor (i.e., the number of Fig. 6 and explained in more detail in [9]. Each window has samples per BOC sub-sample), and Ts is the sampling rate. one or several test cells (or bins) N , according to the search We used here a continuous-time model for clarity purpose. strategy. The length of a bin in time direction is denoted by However, the model can be easily extended to digital-time (∆t)bin (in chips) and the length of a bin in frequency direcdomain and all the simulations were performed in digital- tion is denoted by (∆f )bin . The time-frequency bin defines time domain. the final time-frequency error after the acquisition process and is characterized by one correlator output. The acquisition speed and performance depends on the step (∆t)bin of rx sign Form Yes Coherent Non-coherent 2 scanning of all the possible code phases. Typically, a step of decision > . integration integration Declare Z z1 statistic a half chip is used in order to get a reasonably fast acquisiacquisition tion. [5]. The Doppler-bin length depends on the coherence ref. code at Doppler integration period and should satisfy ∆fbin ≤ 1000/NC (in Readjust No frequency fD est. delay and Hz). est. fD Figure 5: Acquisition block diagram.
We focus on the acquisition block from Fig. 4, shown in detail in Fig. 5. The acquisition refers to the coarse estimation of the Doppler shift fD and the channel delay τ . If the channel delay is acquired with an error less than one chip, we declare that the acquisition is succesful, and the
IV.
I MPACT OF OVERSAMPLING FACTOR ON BOC- MODULATED PRN CODES
We have built a simulation model based on Galileo satellite system specifications [6, 1]. The spreading factor was SF = 10230 and several BOC modulation orders have been compared. We have assumed that no Doppler-frequency information exists (i.e. there is no GPS assistance) and the frequency range ∆fmax is 5 KHz. We have modeled
CNR=18 dB−Hz, Nc=100, Nnc=10, Pfa=0.001
ACF for various combinations of NBOC and Ns 1
−0.2
10
(∆ (∆ (∆ (∆ (∆ (∆
t)bin=0.25 t)bin=0.25 t)bin=0.33 t)bin=0.33 t) =0.5 bin t)bin=0.5
0.8
−0.3
Pd
NBOC=1, N =2, BOC N =2, BOC NBOC=2,
0.9
Autocorrelation function
Ns=1, N =2, s Ns=1, Ns=2, N =1, s Ns=2,
−0.1
10
10
−0.4
10
−0.5
10
0.7
integer Ns N =1 s N =1.8 s Ns=2.5
0.6 0.5 0.4 0.3 0.2
−0.6
10
0.1 0
5
10 NBOC
15
20
0 −2
−1.5
−1
CNR=18 dB−Hz, Nc=100, Nnc=10, Pfa=0.001
−0.5 0 0.5 Delay lag in chips
1
1.5
2
400 350
Figure 8: Absolute value of ACF for various oversampling factors and NBOC = 3.
300
MAT [s]
250 200 150 Ns=1, N =2, s Ns=1, Ns=2, N =1, s Ns=2,
100 50 0 0
5
10 NBOC
(∆ (∆ (∆ (∆ (∆ (∆
t)bin=0.25 t)bin=0.25 t)bin=0.33 t)bin=0.33 t)bin=0.5 t)bin=0.5 15
20
Figure 7: Impact of BOC modulation order on acquisition performance in the absence and in the presence of oversampling (upper plot: detection probability, lower plot: MAT).
the non-integer oversampling factors via upsampling and downsampling. The initial sampling point was taken as a random variable, varying according to the channel delay. The detection probability Pd is the probability to have the estimated delay τb within one chip from the true delay τ . The simulations were carried out in a static channel, such that only the parameters Ns , NBOC and (∆t)bin were influencing the system performance. The target false alarm was fixed to some Pf a value, and the detection threshold γ was computed adaptively in order to meet this target. Formulas which relate Pf a , Pd and γ can be found in [12]. The mean acquisition time (MAT) was computed for a single-dwell structure using: M AT =
(2 − Pd )(1 + KPf a ) τD q, 2Pd
(7)
where q is the number of bins in the time-frequency uncertainty space, and K is the penalty time, which corresponds to the time lost if a false alarm occur. Above, τD is the dwell time needed to form the test statistic and it is equal to NC NN C Ns Ts , where NC is the coherent integration length (in msec) and NN C is the non-coherent integration length (in blocks of symbols) [10]. We remark that the procedure can be extended to multiple-dwell structures, whose MAT is given in [8] (and not reproduced here due to the lack of space). The penalty factor used in the simulations was set to K=10. Fig. 7 shows the Pd and MAT for various BOC modulation orders, in the absence and in the presence of
oversampling, for three time-bin steps values. The performance slightly deteriorates when (∆t)bin increases, as expected, because the grid of scanning the possible code phases becomes less dense (and therefore, the probability to find the correct peak decreases). The time-bin length cannot be higher than one chip, in order to be able to detect a correlation peak. We observe that the oversampling factor Ns does not have much impact on the detection performance, as long as it is integer. We also notice that, in the presence of BOC modulation (i.e, NBOC > 1), there are periodical deep fades at certain modulation orders. Therefore, the next condition should be satisfied when choosing a pair of NBOC and (∆t)bin : (∆t)bin [chips] 6=
k , (∀) k integer, k ≥ 1. NBOC
(8)
The condition (8) can be explained by the fact that the absolute value of the ACF of oversampled and BOCmodulated signal exhibits deep fades, as can be seen in Fig. 8 and these fades depend on the pair (NBOC , (∆t)bin ). On the other hand, when no BOC modulation is used, for any Ns order, the absolute value of the ACF looks exactly as the absolute value of an unmodulated random sequence. This explain the best performance obtained for NBOC =1 (i.e. BPSK case) from in Fig. 7 and Fig. 9. Therefore for any NBOC strictly higher than 1, there always will be some fades in the absolute value of the correlation function, which are prone to deteriorate the performance. Fig. 9 shows the impact of Ns on the detection probability. We note that the MAT curves can be obtained in a straightforward manner from this figure. We also observe, as expected, that the higher the (∆t)bin , the smaller the detection probability is. Moreover, for integer Ns values the detection probability is higher than for neighbor non-integer oversampling factors, which suggests that noninteger oversampling factors should be avoided. The performance in terms of CNR impact is illustrated in Fig. 10 for various NBOC values. The performance in terms of detection probability and MAT given by NBOC = 2 (Manchester coding) is slightly superior than those of NBOC = 4 and NBOC = 6, as can be seen from Fig. 10. We mention that the poor performance for NBOC = 5 is due to fact that condition from eq. (8) is not fulfilled.
V.
C ONCLUSIONS
The purpose of this paper was to analyze the behavior of BOC-modulated signals in the presence of oversampling, for target applications such as Galileo satellite system and modernized GPS. We showed that the performance, in terms of both MAT and detection probability, is typically deteriorated in the presence of oversampling, but sufficient performance can be achieved if the time-bin step is designed properly, according the BOC modulation order. Moreover, non-integer oversampling factors are prone to give worse performance than their rounded values. The results were argued via simulation results with Galileo satellite system signals modeled according to the current proposals. CNR=18 dB−Hz, NBOC=4, Nc=100, Nnc=10, Pfa=0.001
(∆ (∆ (∆ (∆
−0.3
10
−0.4
10
t)bin=0.2 t) =0.4 bin t)bin=0.5 t)bin=0.6
Pd
−0.5
10
[2] J.W. Betz. ”The offset carrier modulation for GPS modernization”. In Proc. of ION Tech. meeting, pages 639–648, 1999. [3] J.W. Betz. ”Design and performance of code tracking for the GPS M code signal”. In MITRE Technical Papers, Sep. 2000. [4] J.W. Betz and D.B. Goldstein. ”Candidate designs for an additional civil signal in GPS spectral bands”. In MITRE Technical Papers, February 2002. [5] P.A. Dafesh and J.K. Holmes. ”Practical and theoretical trade-offs of active parallel correlator and passive matched filter acquisition implementation”. In Proc. of IAIN World Congress in association with ION Annual Meeting, pages 352–366, 2000. [6] G.W. Hein, J. Godet, J.L. Issler, J.C. Martin, T. Pratt, and R. Lucas. ”Status of Galileo frequency and signal design”. In Proc. of ION GPS, 2002. [7] J.K. Holmes, S.H. Raghvan, and S. Lazar. ”Acquisition and tracking performance of NRZ and square wave modulated symbols for use in GPS”. In Proc. of ION GPS Meeting, pages 611–625, 1998.
−0.6
10
[8] A. Lohan E.S., Lakhzouri and M. Renfors. ”Selection of the multiple-dwell hybrid search strategy for the acquisition of Galileo signals in fading channels”. In Proc. of IEEE PIMRC, 2004.
−0.7
10
−0.8
10
1
2
3
4
5
6
7
8
9
10
Ns
Figure 9: Impact of the oversampling factor for different time-bin steps.
[9] E. Pajala, Lohan E.S., and M. Renfors. ”On the choice of the parameters for fast hybrid-search acquisition architectures of GPS and Galileo signals”. In Proc. of FWCW/NRS, 2004. [10] G.J. Povey. ”Spread spectrum PN code acquisition using hybrid correlator architectures”. Wireless Personal Comm., 8:151–164, 1998.
Ns=2, (∆ t)bin=0.4, Nc=100, Nnc=10, Pfa=0.001 300 N =2 BOC NBOC=4 N =5 BOC NBOC=6
250
[11] L. Ries, L. Lestarquit, E. Armengou-Miret, F. Legrand, W. Vigneau, C. Bourga, and P. Erhard. ”A software simulation tool for GNSS2 BOC signal analysis”. In Proc. of ION GPS Meeting, pages 2225–2239, 2002.
MAT [s]
200
150
[12] M.K. Simon, R.A. Omura, R.A. Scholtz, and B.K. Levitt. Spread Spectrum Communications Handbook. Revised Edition, McGraw-Hill, 1994.
100
50
0 18
19
20 21 CNR [dB−Hz]
22
23
Figure 10: Impact of CNR on MAT for different BOC modulation orders.
R EFERENCES [1] F. Bastide, O. Julien, Macabiau C., and B. Roturier. ”Analysis of L5/E5 acquisition, tracking and data demodulation thresholds”. In Proc. of ION GPS Meeting, pages 2196–2207, 2002.
[13] C. Yang. ”FFT acquisition of periodic, aperiodic, puncture and overlaid code sequences in GPS”. In Proc. of ION GPS Meeting, pages 137–146, 2001.
