Correlation properties of Multiplexed Binary Offset Carrier (MBOC) modulation Elena Simona Lohan and Markku Renfors Institute of Communications Engineering, Tampere University of Technology P.O.Box 553, FIN-33101, Finland;
[email protected],
[email protected]
A BSTRACT Recently, a Multiplexed Binary Offset Carrier (MBOC) modulation has been recommended by the GPS-Galileo Working Group on Interoperability and Compatibility in order to increase the tracking abilities of Galileo Open Service signals and of GPS L1 civil signal. MBOC modulation also ensures a better spectral separation with C/A codes. The purpose of this paper is to analyze the MBOC properties in time and frequency domain and to present a unitary theoretical model of MBOC waveforms. First, several spectral properties of MBOC signals are presented, namely: spectral separation coefficients with GPS signals, root-mean-square bandwidth and maximum value of the spectrum. Second, two implementations of MBOC signals are discussed and a generic baseband model for them is introduced. Third, a unified theoretical formula for the autocorrelation function of MBOC-modulated pseudorandom codes is derived. Finally, the advantage of a theoretical framework for the MBOC correlation properties is emphasized via an example of tracking multipath performance in multipath channels. The theoretical findings are also validated via simulations. I. I NTRODUCTION The current interface control document for Galileo signal in space [1] specifies the use of sine Binary Offset Carrier Modulation (SinBOC) for Open Service signals. A SinBOC(1,1) modulation was selected, meaning that a square wave sub-carrier of 1.023 MHz frequency is used to create separate spectra on each side of the transmitted carrier and that the chip rate of the pseudorandom (PRN) codes is also set to 1.023 MHz. The same SinBOC(1,1) modulation was selected as the baseline for the future GPS L1C signals [2]. Significant work has been dedicated during the past couple of years to the optimization of SinBOC modulation. Various solutions have been proposed, such as Binary Coded Symbols (BCS) and Composite BCS (CBCS) [3], [4], modified BOC waveforms [5], or Multiplexed BOC (MBOC) modulations [6], [7]. The main optimization criterion has been the multipath tracking performance and the underlying principle of the above proposals has been the placement of higher frequency components in the modulated signal, in such a way that the multipath performance is enhanced, due to a narrower main lobe of the envelope of the correlation functions. Also, the compatibility with BOC(1,1) receivers was an important factor for the choice of the optimized modulation. The main candidates This work has been supported by the Academy of Finland and by the project ”Advanced Techniques for Personal Navigation (ATENA)” funded by the Finnish Funding Agency for Technology and Innovation (Tekes).
for the future Galileo Open Service (OS) and GPS L1C signals are currently the MBOC-modulated signals [8]. MBOC signal is defined in frequency domain (via the overall power spectral densities of data and pilot signals) and, thus, it allows for several time-domain implementations. Two of the current proposals are the Time-Multiplexed BOC (TMBOC) and Composite BOC (CBOC) implementations [6], [7]. Momentarily, there is rather scarce information regarding the correlation properties of MBOC-modulated satellite signals and their performance in multipath scenarios (especially in channels with more than 2 paths). The purpose of this paper is to present a generic theoretical model of MBOC signals, to derive their correlation properties based on this generic model and to show some examples of the multipath tracking performance of MBOC signals both via semi-analytical and simulation-based models. II. MBOC MODULATION OVERVIEW The main idea behind the MBOC modulation is to place some (small amount) of code power at higher frequencies in order to improve the code tracking performance [6], [7], [8]. According to GJU recommendation [8], the MBOC Power Spectral Density (PSD) was fixed to: 10 1 GSinBOC(1,1) (f ) + GSinBOC(6,1) (f ), 11 11 (1) where GSinBOC(m,n) (f ) is the normalized1 PSD of a sine BOC(m,n)-modulated PRN code, given by [9], [10]: GM BOC (f ) =
1 GSinBOC(m,n) (f ) = Tc
!2 sin πf NTBc sin πf Tc πf cos πf NTBc
(2)
Above, the indices m and n are related to sub-carrier frequency fsc and chip frequency fc via m = fsc /fref and n = fc /fref , respectively, fref = 1.023 MHz is the reference C/A code frequency [9], and NB = 2 ffscc = 2 m n is the BOC modulation index (see [10] for more details). For SinBOC(1,1), NB = 2, while for SinBOC(6,1), NB = 12. Incidentally, SinBOC(6,1) modulation has the same modulation index as CosBOC(15,2.5)-modulated signal (proposed for Publicly Regulated Service PRS signal). The MBOC PSD of eq. (1) is the total PSD of pilot and data signals together, thus many practical ways of implementing it are possible, as it will be discussed in Section II.B. The PSD of MBOC signal (given by eq. (1)) is shown in Fig. 1. It is noticed that extra spectral peaks compared to SinBOC(1,1) case appear around ±6 MHz (in the equivalent baseband model, or 1 Normalization is done with respect to signal power over infinite bandwidth, or, equivalently, with respect to chip interval [9]
around fcarrier ± 6 MHz for the passband model, with carrier frequency fcarrier ), due to the SinBOC(6,1) component.
the RMS BW is, the smaller the variance of the delay tracking process is [11]): sZ BT /2
βRM S =
Normalized PSD of SinBOC(1,1) and MBOC signals −60
−BT /2
−65
PSD [dBW−Hz]
−70 −75 −80 −85 −90 −95 −100 −20
−15
−10
−5 0 5 Frequency [MHz]
10
15
20
Fig. 1. Power Spectral Density for MBOC and SinBOC(1,1)-modulated signals; equivalent baseband model.
A. Spectral properties of MBOC signals One important spectral measure for any satellite signal is its Spectral Separation Coefficient (SSC) κSSC with other satellite signals. The lower the SSC is, the better spectral separation between signals we have. SSC between two signals within a complex (double-sided) finite bandwidth BT is defined as [9]: κSSC =
Z
BT /2
−BT /2
R
Pe1 (f )Pe2 (f )df,
BT /2 P (f )df −BT /2 υ
(3)
where Peυ (f ) = Pυ (f )/ , υ = 1, 2 is the PSD of the υ-th signal, normalized to the unit power over the bandwidth of interest. Pυ (f ), υ = 1, 2 is the PSD of the υ-th signal (for example, for MBOC signal, P1 (f ) = PM BOC (f ) is given by eq. (1)). The interfering signal (defined by the PSD P2 (f )) can be, for example, any of the followings: C/A code, military M code, or P(Y) code, of GPS system, or PRS codes (CosBOC(15,2.5)-modulated) of Galileo system. Generic formulas for the PSDs of such signals can be found in [10]. Also, a low self-interference coefficient (κSSC,self = R BT /2 2 Pe (f )df ) is typically desired when choosing a certain −BT /2 1 modulation among the others. Another spectral value typically used to characterize the ability of a satellite signal with PSD Pυ (f ) to work at low power levels is the Maximum Value of the Spectrum (MVS) mM V S , defined as [9], [10]: mM V S = max Pυ (f ) f ∈BT
(4)
The smaller the MVS, the better the corresponding modulation is, because it enables the modulated signals to be transmitted at a higher power with less disturbance of the noise floor [9]. Finally, the Root Mean Square (RMS) bandwidth βRM S is a measure of the delay tracking abilities of the signal (the higher
(5)
Table I compares the values of SSC, MVS, and RMS bandwidth coefficients for SinBOC(1,1) and MBOC-modulated signals (MBOC PSD is defined in eq. (1) and the PSDs of the other considered signals are taken from [10]). Numerical integration over a double-sided bandwidth BT was used to compute these results (BT is specified in the table and two limit cases were considered: a small bandwidth of 12 MHz, for strong bandwidth limitation requirements, and a large bandwidth of 40 MHz. Here, κSSC,C/A is the SSC of MBOC signal with C/A code of GPS, κSSC,M is the SSC with M-code of GPS, κSSC,P (Y ) is the SSC with P(Y)-code of GPS, and κSSC,P RS is the SSC with PRS-code of Galileo. Based on Table I, it is clear that MBOC signals provide slightly better separation with C/A and P(Y) codes, slightly worse separation with Mcode, and similar separation with PRS signals compared with SinBOC(1,1) signals. Also, MBOC signals have slightly lower self-interference, better MVS and better RMS bandwidth (thus, the potential of better tracking ability). Power containment per B for different modulations T
100 90
Percentage of power containment [%]
SinBOC(1,1) MBOC
f 2 Peυ (f )df
80 MBOC SinBOC(1,1) SinBOC(6,1)
70 60 50 40 30 20 10 0 0
10
20 30 40 50 60 Double−sided bandwidth B [MHz]
70
80
T
Fig. 2. Power containment for MBOC and BOC-modulated signals.
Fig. 2 illustrates the power containment factors (i.e., the percentage of the signal power contained within a certain bandwidth) for MBOC, SinBOC(1,1) and SinBOC(6,1) signals. For example, the double-sided bandwidth BT required to contain 90% of the signal power is 11.86 MHz for MBOC signal given in eq. (1), 6.1 MHz for SinBOC(1,1) signal, and 36.68 MHz for SinBOC(6,1). B. MBOC implementation structures There are two main ways to achieve a PSD as given in eq. (1) [6], [7]: 1. The Composite BOC (CBOC) method: via a weighted sum of SinBOC(1,1) and SinBOC(6,1)-modulated code symbols (where SinBOC(1,1) part is passed through a hold block in order to match the rate of SinBOC(6,1) part). Following the
TABLE I C OMPARISON BETWEEN S IN BOC(1,1) AND MBOC SIGNALS FROM THE POINT OF VIEW OF THEIR SPECTRAL COEFFICIENTS .
Modulation type & BT [MHz] SinBOC(1,1) at 12 MHz MBOC at 12 MHz SinBOC(1,1) at 40 MHz MBOC at 40 MHz
κSSC,C/A [dBW/Hz] −67.58 −67.73 −67.79 −68.16
κSSC,M [dBW/Hz] −73.09 −73.07 −82.26 −81.97
κSSC,P (Y ) [dBW/Hz] −69.54 −69.64 −70.26 −70.53
BOC model and derivations of [10], this composite sum can be written as:
i=0
X
−1
−1
X
k=0
NB2 −1
+ w2
NB 2 NB 1
Tc Tc i −k (−1) c t − i NB 1 NB 2
(−1)i c t − i
i=0
Tc NB 2
,
(6)
where NB1 = 2 is the BOC modulation order for SinBOC(1,1) signal, NB2 = 12 is the BOC modulation order for SinBOC(6,1) signal, and w1 and w2 are amplitude weighting factor chosen in such a way to match the PSD of eq. (1) and w12 + w22 = 1. For example, if 50% or 100% percentage of the total signal power is placedq on pilot channel, qthen it is straight1 forward to show that w1 = 10 11 and w2 = 11 . The second sum in the first right-hand term of eq. (6) is due to rate preservation between the two signals (since SinBOC(6,1) has NB2 /NB1 = 6 times higher rate than SinBOC(1,1) signal, a hold block is added after SinBOC(1,1) modulation). Above, c(t) is the pseudorandom code, including data bits (the model applies for both pilot and data channels):
c(t) =
p
Eb
∞ X
n=−∞
bn
SF X
PRN code
2
4
6
8
10
CBOC signal
0 −1
2
4
6
8
10
1 TMBOC signal
0 −1
2
4
6 Chip interval
8
10
q ) and TMBOC (50% pilot power and Fig. 3. Example of CBOC (w1 = 10 11 SinBOC(6,1) spreading symbols in locations 2 and 6 inside blocks of 11 spreading symbols or chips) time waveforms for binary spreading symbols.
division may be applied for both pilot and data channels, individually. The choice of N and M parameters depends on the power percentage of pilot channels with respect to the data channels [6]. Analytically, using similar derivations as in [10], TMBOC waveforms may be written as:
cm,n pTB2 (t − nTc SF − mTc ). (7)
2 S is not necessarily equal with the code length. For example, the code F length for Galileo signals is 4SF = 4092 chips.
RMS [MHz] 1.41 1.73 2.53 3.36
1
N
m=1
where bn is the n-th code symbol (it may be equal to 1, ∀ n if pilot channel is considered), Eb is the code symbol energy, SF is the spreading factor or number of chips per code symbol (e.g., SF = 1023 chips in GPS and Galileo2 ), cm.n is the m-th chip corresponding to the n-th symbol, Tc is the chip rate, and pTB2 (·) is a rectangular pulse of support Tc /NB2 and unit amplitude. In eq. (6), the first term comes from the SinBOC(1,1)-modulated code (held at rate 12/Tc in order to match the rate of the second term), and the second term comes from a SinBOC(6,1)-modulated code. q An example of CBOC-modulated signal with w1 = 10 11 and using binary spreading symbols is shown in the middle plot of Fig. 3. 2. The Time Multiplexed BOC (TMBOC) method: the whole signal is divided into blocks of N code symbols and M < N of of N code symbols are SinBOC(1,1)-modulated, while N − M code symbols are SinBOC(6,1)-modulated. This
MVS [dBW/Hz] −62.89 −63.30 −62.89 −63.30
0
MBOC waveforms
= w1
X
κSSC,self [dBW/Hz] −64.42 −64.72 −64.73 −65.45
1
sCBOC (t) = w1 sSinBOC(1,1),held (t) + w2 sSinBOC(6,1) (t) NB1 −1
κSSC,P RS [dBW/Hz] −79.03 −73.07 −91.6 −91.34
sT M BOC (t) =
n∈S
NB 2 −1 NB 1 X
(−1)i pTB
k=0
+
−1
B1 SF X X X p cm,n bn Eb
2
„
m=1
t−i
i=0
Tc Tc −k NB1 NB2
«
NB2 −1 « „ SF X X X p Tc t−i (−1)i pTB cm,n bn Eb 2 NB2 m=1 i=0
(8)
n∈S /
where S is the set of chips (or spreading symbols) which are SinBOC(1,1) modulated (i.e., if I is the total set of symbols, then card(S)/card(I)=M/N, where card(·) stands for the cardinality of a set). The choice of M and N parameters is dependent on the pilot/data power split [6]. There are many solutions which satisfy eq. (1). For example, if we assume that all the data spreading symbols are SinBOC(1,1) modulated and that xpow power fraction is placed on the pilot channel (and 1 − xpow power fraction on data channel), then it is straightforward to show that M and N values satisfy the condition (the 11 factor here comes from
the GJU recommendation as seen in eq. (1)): 1 M =1− N 11xpow
(9)
An example of TMBOC-modulated signal with 50/50 % power split between pilot and data channels (i.e., xpow = 0.5, M = 9 of N = 11 spreading symbols are SinBOC(1,1) modulated, and N − M = 2 out of 11 spreading symbols are SinBOC(6,1) modulated) and using binary spreading symbols is shown in the lower plot of Fig. 3. The placement of SinBOC(6,1)-modulated symbols is a design parameter, but preliminary simulation results showed that the actual positions do not affect significantly the multipath performance. An equivalent unified model of CBOC and TMBOC modulations can be derived as follows, based on eqs. (6) and (8) and using the facts that M, N 2 paths. The solution adopted here is a semi-analytical one: we generated 3000 random realizations of multipath profiles (amplitudes and phases) and we looked at the particular case with constant spacing xct between successive paths τl = lxct Tc . Then, the MEEs versus the delay spacing xct were computed over the 3000 random points via eq. (20). The results are shown in Fig. 6 for narrowband EML and for HRC. The dashed lines show the lower MEEs, and the continuous lines show the upper MEEs. The x-axis label is the successive path separation xct . It was noticed that, for more than 2 paths, the worst combinations of a and θ (which give the upper and lower MEE) depend on xct and no analytical rule could be found for them. The mapping between delay estimates ∆b τ in chips and the corresponding delay error d∆bτ in meters (also used for RMSE values in Section V) is: d∆bτ = c∆b τ /fc , where c is the speed of light and fc = 1.023 MHz is the chip rate. The results in 4 path channels for the narrow correlator are simNarrow correlator, ∆
=0.025 chips, L
EL
=4
paths
100 80
upper MEE, TMBOC, x
=0.5
60
lower MEE, TMBOC, x
=0.5
40
upper MEE, SinBOC(1,1) lower MEE, SinBOC(1,1)
MEE [m]
pow
pow
20 0 −20 −40 −60 −80 −100 0
0.2
0.4 0.6 0.8 1 Successive path separation [chips]
High resolution correlator, ∆
1.2
=0.025 chips, L
EL
1.4
=4
paths
4 2
MEE [m]
0 −2 −4
upper MEE, TMBOC, x
=0.5
lower MEE, TMBOC, x
=0.5
pow
pow
−6
upper MEE, SinBOC(1,1) lower MEE, SinBOC(1,1)
−8 −10 0
0.2
0.4 0.6 0.8 1 Successive path separation [chips]
1.2
1.4
Fig. 6. MEEs for BOC and TMBOC signals (xpow = 0.5) in 4-path static channels.Upper plot: narrow correlator; lower plot: high resolution correlator. Early-late spacing ∆EL = 0.025 chips.
ilar with the results reported in [3] for 2 path channels: MBOC waveforms outperform the SinBOC(1,1) waveforms (clearly, TMBOC and CBOC signals are equivalent of suitable weighting factors are used, as shown in Section II.B, thus the results from Fig. 6 are also valid for CBOC signals). When HRC discriminators are used, MBOC does not have an advantage over SinBOC(1,1)-modulated signals. It is also to be mentioned the asymmetry in the MEE upper and lower curves, which might be explained by the fact that adding several paths in anti-phase is more difficult to deal with than adding several paths in phase. It is also worth mentioning here that, usually, MEE curves give
only a limited view of multi-path performance. More realistic results are via RMSE criterion as shown in the next section. V. MBOC TRACKING PERFORMANCE IN FADING MULTIPATH CHANNELS
The semi-analytical results from Section IV have also been validated via simulations in fading multipath channels. Pseudorandom codes with SinBOC(1,1) and MBOC modulation have been generated at the receiver. An oversampling factor Ns = 4 samples per smallest BOC interval (Tc /12) was used. The channel paths were assumed to obey Nakagami-m fading distribution (with a Nakagami-m factor of 0.5), according to indoor channel models of [18]. An exponential decaying power delay profile with random number of paths (uniform distributed between 1 and 4) was assumed. The LOS delay was assumed to vary random (uniform distribution within ±0.01 chips from the previous LOS delay realization). Coherent integration of 10 ms has been used. The successive paths separation was also assumed to be uniformly distributed in the interval [0.02; 0.35] chips (i.e., very closely spaced path situation). The results are for rectangular pulse shaping and they are shown in Fig. 7 for narrow EML (NEML) and in Fig. 8 for HRC. The first comparison criterion is the Root Mean Square Error (RMSE) on the convergence region (i.e., on those delay estimates which are within less than 0.35 chips from the true LOS delay, because the width of the main lobe of the ACF is about 0.7 chips, thus a delay error higher, in absolute value, than 0.35 chips would mean a loss of lock). The second comparison criterion is the Mean Time to Loose Lock (MTLL). The statistics have been computed over 9000 random realizations of channel parameters. In Fig. 7, the 4 compared modulations are: TMBOC with xpow = 0.5 (i.e., M = 9, N = 11, and w1 = 0.9045), TMBOC with xpow = 0.75 p (i.e., M = 29, N = 33, and w1 = 0.9374), CBOC with w1 = 10/11 = 0.9535, and SinBOC(1,1) (it is to be noticed that the MBOC model given by eq (10) holds also for SinBOC(1,1) signal, with w1 = 1). In Fig. 8, TMBOC with xpow = 0.9 (i.e., M = 89, N = 99, and w1 = 0.9482) was included instead of xpow = 0.5. Clearly, NEML performance is reverse proportional with w1 index, i.e., the higher the weight of SinBOC(1,1) part, the lower tracking performance at moderate to good Carrier to Noise ratios (CNR) (and the highest the MTLL). However, the performances with HRC discriminator and various modulation types (Fig. 8) are very similar (slightly better RMSE values for SinBOC(1,1) and slightly worse MTLL values). The simulation results of Figs. 7 and 8 are in conformity with the semi-analytical findings of Section IV. VI. C ONCLUSIONS MBOC modulation has been proposed for future satellite signals in order to enhance the delay tracking in multipath scenarios. Here, an analytical framework has been derived which covers several MBOC modulation classes and allows a fast computation of the spectral and time-domain correlation shapes of MBOC signal. The theoretical formulas were validated first via a semi-analytical Multipath Error Envelope model and, then, via simulation-based model. It has been noticed that the wellpraised advantage of MBOC modulation over SinBOC(1,1)
Narrow correlator 55
TMBOC, w =0.9045 (x 1
=0.5)
pow
TMBOC, w1=0.9374 (xpow=0.75)
50
CBOC, w1=0.9535 SinBOC(1,1) (w =1)
45
1
RMSE [m]
40 35 30
modulation does not hold for all discriminator types. For example, it holds for narrow correlator, but MBOC modulation does not bring a tracking benefit when used with HRC discriminator. However, the HRC performance with MBOC is normal, since this correlator has a design matched for the GPS C/A code. Further studies are needed in order to understand the advantage of MBOC modulation with other, more advanced, delay tracking architectures, such as the above-mentioned Sidelobe cancellation methods or Multipath Estimating Delay Locked Loops.
25
R EFERENCES
20 15 20
[1] 25
30
35 CNR [dB−Hz]
40
45
50
Narrow correlator
[2]
60 55 50
[3]
MTLL [s]
45 40 TMBOC, w =0.9045 (x 1
35
=0.5)
[4]
pow
TMBOC, w =0.9374 (x 1
=0.75)
pow
CBOC, w1=0.9535
30
SinBOC(1,1) (w =1)
[5]
1
25 20 20
[6] 25
30
35 CNR [dB−Hz]
40
45
50
Fig. 7. RMSE (upper plot) and MTLL (lower plot) for SinBOC(1,1) and MBOC signals in 4-path Nakagami fading channels, for Narrow Correlator, ∆EL = 0.1 chips.
[7]
[8] High resolution correlator 55
TMBOC, w1=0.9374 (xpow=0.75) TMBOC, w =0.9482 (x
50
1
=0.9)
pow
[9]
CBOC, w1=0.9535
45
[10]
SinBOC(1,1) (w1=1)
RMSE [m]
40 35
[11]
30
[12]
25
[13]
20 15 10 20
[14] 25
30
35 CNR [dB−Hz]
40
45
50
[15]
High resolution correlator 60 55
[16]
50
[17]
MTLL [s]
45 40 TMBOC, w =0.9374 (x 1
35
=0.75)
[18]
pow
TMBOC, w =0.9482 (x 1
=0.9)
pow
CBOC, w1=0.9535
30
SinBOC(1,1) (w1=1) 25 20 20
25
30
35 CNR [dB−Hz]
40
45
50
Fig. 8. RMSE (upper plot) and MTLL (lower plot) for SinBOC(1,1) and MBOC signals in 4-path Nakagami fading channels, for High Resolution Correlator, ∆EL = 0.1 chips.
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