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Research ArticleRegular Papers
Open Access

GNSS Meta-Signal Tracking Using a Bicomplex Kalman Filter

Daniele Borio and Melania Susi
NAVIGATION: Journal of the Institute of Navigation December 2024, 71 (4) navi.674; DOI: https://doi.org/10.33012/navi.674
Daniele Borio
1European Commission, Joint Research Centre, Italy
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Melania Susi
2Topcon Positioning Systems, Inc., Italy
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Abstract

Global navigation satellite system (GNSS) signals from different frequencies can be effectively treated as a single entity, characterized by common delays and carrier phases, leading to so-called GNSS meta-signals. A convenient approach for deriving meta-signal acquisition and tracking algorithms has been recently introduced based on bicomplex numbers, which are a bidimensional extension of complex numbers. Bicomplex numbers allow one to represent two signals from different frequencies as a single quantity, providing a compact notation for algorithm development. In this work, an error-state Kalman filter (KF) is developed, and two signals from different frequencies are tracked simultaneously using the bicomplex number paradigm. A triple-loop architecture, in which loop filters are replaced by a single KF, is developed, implemented, and tested using real Galileo alternative binary offset carrier and BeiDou B1I/B1C meta-signals. This analysis clearly shows the advantages of KF tracking for processing GNSS meta-signals with components from different frequencies.

Keywords
  • dual-frequency
  • GNSS
  • Kalman filter
  • meta-signals
  • triple-loop

1 INTRODUCTION

Modern global navigation satellite systems (GNSSs) broadcast several signals on different frequencies, providing interesting opportunities for improving receiver performance and for advanced scientific applications, such as the study of ionospheric conditions (Vilà-Valls et al., 2020). While dual-frequency receivers were once used only for military and high-end surveying applications, several mass-market receivers now have this capability, including smartphones (Crosta et al., 2018). The optimal tracking of dual-frequency signals remains an active area of research. At the same time, new signal options have been considered. In the context of next-generation GNSS signals, an option under consideration is the introduction of a new component into the Galileo L1/E1 frequency band, denoted as E1D (Schwalm et al., 2020). While the E1D components will be modulated into a radio frequency (RF) different from the L1 center frequency, at 1575.42 MHz, the selection of this frequency will depend on several factors, including the ability of a receiver to fully exploit the benefits of signals from different frequencies (Schwalm et al., 2020; Sénant et al., 2018). Introducing signals in frequency gaps close to existing components provides interesting opportunities to test advanced signal processing algorithms (Beck et al., 2023). In this respect, the BeiDou navigation satellite system (BDS)-3 has maintained the B1I component together with the B1C signal. The two components are separated by a few MHz and can be processed as a single signal (Gao et al., 2020).

A promising approach to simultaneously deal with signals from different frequencies is based on the so-called meta-signal concept (Issler et al., 2010), in which components from different frequencies are treated as a single entity characterized by common code delays and carrier phases. Several algorithms have been developed to process GNSS meta-signals (Gao et al., 2020; Moriana et al., 2023; Paonni et al., 2014; Tian et al., 2022). While effective, these algorithms often require complicated notations and derivations, which mask the underlying nature of the approaches developed. A convenient approach for deriving meta-signal acquisition and tracking algorithms has recently been introduced by Borio (2023). This approach is based on bicomplex numbers (Alpay et al., 2014), which are an extension of complex numbers with four real components. Bicomplex numbers allow one to represent two signals from different frequencies as a single quantity, providing a compact notation for algorithm development (Borio, 2023). Bicomplex numbers, different from quaternions, are characterized by a commutative product and allow the representation of two modulated signals through the multiplication of three terms: a bicomplex code, a carrier, and a subcarrier. This representation extends the single-frequency case, where a modulated signal is represented as the real part of the product of a complex baseband code and a carrier component used to bring the signal to RF. This type of product representation has important implications for GNSS receiver design, where code and carrier components are tracked separately using dedicated tracking loops. Moreover, sideband components are preserved by bicomplex numbers through their orthogonal representation. A bicomplex number can always be rewritten as the sum of two orthogonal components multiplying two idempotent units, which are two bicomplex numbers with special properties. More specifically, their product is zero, and their powers always assume the same value.

In the work by Borio (2023), a triple-loop tracking architecture was also derived, in which the code, carrier, and subcarrier components are tracked separately by a bicomplex delay lock loop (DLL), phase lock loop (PLL), and subcarrier phase lock loop (SPLL), respectively. The proposed triple-loop architecture follows directly from the bicomplex signal representation in which the code, carrier, and subcarrier components are isolated. This type of tracking architecture was originally designed for binary offset carrier (BOC) signals (Borio, 2014; Hodgart & Blunt, 2007) and was then extended to dual-frequency components, including the BeiDou B1I/B1C combination (Gao et al., 2020; Wang et al., 2017; Zhang et al., 2019). For these architectures, it is possible to show that the loop filter outputs, opportunely normalized by the corresponding nominal frequencies, assume similar values, enabling carrier aiding for the code and subcarrier components (Borio, 2023). This result opens the possibility to adopt advanced tracking algorithms, for example, based on the Kalman filter (KF), where a joint dynamic model is used for the errors tracked by the DLL, PLL, and SPLL. The use of KF-based tracking algorithms has become popular over the past few years to enhance the robustness of GNSS tracking loops (O’Driscoll et al., 2011; Susi & Borio, 2020; Vila-Valls et al., 2017). Owing of its adaptive nature, a KF allows one to deal with variable and challenging conditions, for example, in the presence of weak signals or under dynamic scenarios. Moreover, the KF represents the optimum linear filter with respect to the minimum mean square error under the assumption of additive Gaussian noise.

Two main approaches are usually adopted to design a KF-based tracking algorithm: the error-state Kalman filter (ESKF) (O’Driscoll et al., 2011; Susi et al., 2017; Tang et al., 2017) and the direct-state Kalman filter (DSKF) (Cortès et al., 2023; Won et al., 2012). The ESKF replaces the loop filters with a single KF, whereas the DSKF utilizes an extended KF in place of the standard PLL and DLL (Vila-Valls et al., 2017).

In this paper, we extend the ESKF to the meta-signal case in which two signals from different frequencies are tracked simultaneously. The bicomplex number paradigm and the triple-loop architecture introduced by Borio (2023) are modified, and the three separate loop filters are replaced by a single KF. The KF state vector consists of five terms, including the code delay and the carrier and subcarrier phase errors. In addition, a residual Doppler frequency shift and a Doppler rate correction term are included in the model. In the system dynamic model, which describes the time evolution of the state vector, these last two terms contribute to the evolution of the code delay, carrier, and subcarrier phases and are used to effectively implement carrier aiding. In this respect, the dynamic model adopted herein is similar to that introduced by Luo et al. (2018) for tracking BOC signals. The measurement model is based on the discriminator outputs of the DLL, PLL, and SPLL. The variances of the three discriminator outputs are determined as a function of the input carrier-to-noise power spectral density ratio (C/N0), which is continuously estimated. These variances are used to build the measurement noise covariance matrix.

Bicomplex KF tracking has been implemented in a custom software-defined radio (SDR) GNSS receiver developed in Python. The tracking architecture implemented within the receiver features several stages. After initial frequency and phase lock are achieved using standard loops, KF-based tracking is enabled, and code, carrier, and subcarrier parameters are jointly estimated. The developed receiver has been used to process Galileo alternative binary offset carrier (AltBOC) and BeiDou B1I/B1C signals collected using SDR front-ends. Both modulations feature signal components from nearby frequencies and can be considered meta-signals. Different experiments were conducted in both static and dynamic modes. The dynamic experiment focused on a B1I/BIC meta-signal consisting of two asymmetric sideband components, representing a more general case with respect to AltBOC modulation. Experiments were conducted with a HackRF One front-end (Ossmann, 2023), which was mounted on a car and used to collect in-phase quadrature (I/Q) data. These data, in turn, were processed via the Python SDR receiver implementing KF-based tracking. From the analysis, the benefits of KF tracking for processing GNSS meta-signals clearly emerge. The use of a KF allows one to obtain smooth code and subcarrier estimates. The use of a model, embedded in the KF, improves tracking performance with respect to dynamic conditions.

This paper is an extended version of a previous conference paper (Borio & Susi, 2023), where additional experimental results have been included.

The remainder of this paper is organized as follows: Section 2 briefly summarizes the signal model and the GNSS bicomplex signal representation. Section 3 describes a triple-loop architecture in which two GNSS sideband components are jointly processed as a single meta-signal, whereas Section 4 illustrates an extension of the architecture introducing the KF tracking loop. The experimental setup and related results are described in Section 5 and Section 6, respectively. Finally, Section 7 concludes this paper.

2 SIGNAL MODEL AND BICOMPLEX REPRESENTATION

The set of bicomplex numbers, denoted as Bℂ or ℂ2 (Alpay et al., 2014; Price, 1991), is a bidimensional extension of complex numbers. Bicomplex numbers are characterized by a real component and three imaginary terms, which are multiplied by three imaginary units. Two of these units, denoted as i and j, square to −1, whereas the third unit, k, squares to 1 and is called the hyperbolic unit. Borio (2023) showed that bicomplex numbers can be effectively used to provide a joint baseband representation of two RF signals from different frequencies.

When a single GNSS signal on a single frequency is considered, a general baseband signal model is given as follows:

x(t)=Ad,xdx(t)cd,x(t)+jAp,xcp,x(t)1

where the subscript d indicates quantities related to the data component and the index p indicates terms related to the pilot signal. Ad,x and Ap,x are the amplitudes of the data and pilot components, respectively. dx(t) models the data navigation message, and cd,x(t) and cy,d(t) are the two ranging codes.

Signal x(t) is brought to RF and transmitted on frequency, fb. After passing through the communication channel, the signal is finally recovered by the receiver front-end. A common model adopted for received GNSS signals considers a communication channel introducing a delay on the code components and a Doppler shift and a phase variation on the carrier terms (Misra & Enge, 2006, p. 432):

xRX(t)=βℜ{x(t−τ0)ej2π(fb+fb,d)t+jφb}+ηRF,b(t)=βAd,xdx(t−τ0)cd,x(t−τ0)cos(2π(fb+fb,d)t+φb)−βAp,xcp,x(t−τ0)sin(2π(fb+fb,d)t+φb)+ηRF,b(t)2

where β models the attenuation introduced by the channel and ηRF,b is additive white Gaussian noise (AWGN). τ0 denotes the code delay, fb,d represents the Doppler frequency, and φb denotes the carrier phase. The Doppler effect on the code component is neglected here.

At the receiver side, the use of an I/Q downconversion front-end (Tsui, 2004) is assumed. In this way, a digital baseband representation of xRX(t) is recovered:

xbb[n]=βx(nTs−τ0)ej2πfb,dnTs+jφb+ηb(nTs)3

xbb[n] is a digital sequence obtained by sampling the received signal at a frequency fs. In Equation (3), n denotes the time index, and Ts=1fs is the sampling interval.

When two signals from different frequencies are considered, a second component is introduced:

ybb[n]=αy(nTs−τ0)ej2πfa,dnTs+jφa+ηa(nTs)4

where:

y(t)=Ay,ddy(t)cy,d(t)+jAy,pcy,p(t)5

models the baseband transmitted signal with data and pilot components. Similar to Equation (1), Ay,d and Ay,p are the amplitudes of the data and pilot components of the second signals, respectively. dy(t) models the data navigation message broadcast by y(t), and cy,d(t) and cy,d(t) are the data and pilot ranging codes. In Equation (4), α models the attenuation introduced by the communication channel, whereas fa,d and φa are the Doppler frequency and carrier phase introduced on y(t). ηa(nTs) is a second AWGN independent from ηRF,b.

Using bicomplex numbers, it is possible to build a composite signal:

z[n]=2ybb[n]e1+2xbb[n]e26

where e1 and e2 are the two idempotent orthogonal elements of Bℂ:

e1=1+k2; e2=1−k27

The composite signal, z[n], can be expressed as follows (Borio, 2023):

z[n]=2αy(nTs−τ0)ej2πfa,dnTs+jφae1+2ηa(nTs)e1+2βx(nTs−τ0)ej2πfb,dnTs+jφbe2+2ηb(nTs)e2=2[αy(nTs−τ0)e−j2πfsub,0nTs−jφsub,0e1+βx(nTs−τ0)ej2πfsub,0nTs+jφsub,0e2]ej2πfd,0nTs+jφ0+η[n]=zBB(nTs−τ0)ei2πfsub,0nTs+iφsub,0ej2πfd,0nTs+jφ0+η[n]8

where:

η[n]=2ηa(nTs)e1+2ηb(nTs)e29

and:

zBB(nTs)=2αy(nTs)e1+2βx(nTs)e210

The last expression in Equation (8) is obtained by exploiting the properties of bicomplex numbers and the representation of a pure i-exponential, as derived by Borio (2023, Appendix C). Equation (8) shows that the useful signal component, obtained as a combination of the two sideband terms, can be written as the product of a bicomplex code times a complex carrier and a subcarrier: fd,0 and fsub,0 are the residual Doppler frequencies of the meta-signal carrier and subcarrier components, respectively. Similarly, φ0 and φsub,0 are the meta-signal carrier and subcarrier phases, defined as follows:

fd,0=fa,d+fb,d2, fsub,0=fb,d−fa,d2, φ0=φa+φb2, φsub,0=φb−φa211

The product representation in Equation (8) suggests that a triple-loop architecture can be used to jointly track Equations (3) and (4). This architecture is briefly reviewed in Section 3.

η[n] is a zero-mean bicomplex AWGN with four independent and identically distributed components assumed to be uncorrelated with the same variance, ση2. This variance depends on several factors related to the antenna and front-end implementations. A commonly adopted model for ση2 is given as follows:

ση2=N0BRx12

where BRx is the front-end one-sided bandwidth and N0 is the power spectral density (PSD) of the input noises. The same PSD and front-end bandwidth are assumed for the two sideband components. A good approximation for BRx is half the sampling frequency:

BRx≈fs213

2.1 Bicomplex Codes

To acquire and track the signal in Equation (8), the receiver must generate local codes, which are correlated with z[n]. As discussed by Borio (2023), these codes should mimic the original transmitted signal or at least be strongly correlated with it. Ideally, we would have the following:

c˜[n]=2αy(nTs)e1+2βx(nTs)e2=2α(Ad,ydy(nTs)cd,y(nTs)+jAp,ycp,y(nTs))e1+2β(Ad,xdx(nTs)cd,x(nTs)+jAp,xcp,x(nTs))e214

Note that α and β, the attenuations introduced by the communication channel, are present in Equation (14): when generating the local codes, it is necessary to give more weight to the received component with the largest amplitude. Because the navigation messages, dy (nTs) and dx (nTs), of the data channels may be unknown, several options are available (Borio, 2023). In the remainder of this paper, we consider pilot-only and mixed data–pilot processing. In the first case, only the pilot channels are considered, and the data components are neglected. Equation (14) is then simplified as follows:

c˜[n]=2αjAp,ycp,y(nTs)e1+2βjAp,xcp,x(nTs)e215

Note that the j units in Equation (15) can be removed. With the removal of these terms, the different phases recovered by baseband processing will be expressed with respect to the pilot channel phases. Moreover, a common normalization can be adopted. In this way, we have the following:

c˜[n]=cp,y(nTs)e1+γcp,x(nTs)e216

where γ=βAp,xαAp,y is the relative amplitude between the two sideband pilot components. For signals such as the AltBOC, where symmetric sideband components are adopted, γ = 1. In the mixed data–pilot processing case, one of the sideband components consists of only a data channel; this is the case of the BeiDou B1I/B1C meta-signal, where the B1I sideband component features a data-only channel. One possibility is to neglect the data channel of the sideband signal that also contains a pilot component, yielding the following local code:

c˜[n]=cd,y(nTs)e1+jγcp,x(nTs)e217

where, in this case, γ=βAp,xαAd,y. The unknown data stream dy (nTs) is neglected in Equation (17), and phase discriminators insensitive to data bits must be adopted. This point is discussed in more detail in Section 3.2.

In general, c˜[n]=c˜(nTs) can be expressed as follows:

c˜[n]=c˜(nTs)=y˜(nTs)e1+γx˜(nTs)e218

where y˜(nTs) and x˜(nTs) contain one of the possible local code options discussed above.

3 TRIPLE-LOOP TRACKING

The triple-loop architecture with bicomplex signals and correlators introduced by Borio (2023) is depicted in Figure 1. While the architecture shown in Figure 1 is equivalent to that of other algorithms described in the literature (Gao et al., 2020; Tian et al., 2022), bicomplex numbers provide a compact and effective representation of the different signal and correlator components, allowing one to highlight the main tracking loop operations. The architecture in Figure 1 implements a DLL, PLL, and SPLL. The bicomplex carrier and subcarrier are first removed through multiplication by the local components generated by carrier and subcarrier numerically controlled oscillators (NCOs). Then, the bicomplex baseband signal is multiplied by early, prompt, and late replicas of the bicomplex local code obtained according to the process summarized in Section 2.1. The early, prompt, and late correlators are then obtained through an integration process. In particular, N samples are integrated, leading to the three bicomplex correlator outputs:

P=1N∑n=0N−1z[n]c˜*(nTs−τ^)e−2π(jf^d+if^sub)nTs−(jφ^+iφ^sub)E=1N∑n=0N−1z[n]c˜*(nTs−τ^−ds/2)e−2π(jf^d+if^sub)nTs−(jφ^+iφ^sub)L=1N∑n=0N−1z[n]c˜*(nTs−τ^+ds/2)e−2π(jf^d+if^sub)nTs−(jφ^+iφ^sub)19

where:

  • c˜*(nTs)=c˜*[n] is the complex conjugate of the local code introduced in Section 2.1,

  • τ^ is the delay estimated by the DLL during the previous processing epoch,

  • f^d and f^sub are the Doppler frequencies on the carrier and subcarrier components estimated by the PLL and SPLL during the previous processing epoch,

  • φ^ and φ^sub are the residual carrier and subcarrier phases determined by the PLL and SPLL,

  • ds is the early–minus–late correlator spacing.

FIGURE 1
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FIGURE 1

Triple-loop architecture using three separate loop filters

The input signal, z[n], is bicomplex and combines two sideband components from two different frequencies. Signals and correlators in the upper box are bicomplex.

N and Ts, the sampling interval, define the coherent integration time, Tc = NTs. Bicomplex numbers admit several conjugations. In Equation (19), (·)* is the *-conjugation (Alpay et al., 2014; Borio, 2023), which is obtained by negating the sign of the components multiplying pure imaginary units. Finally, note that in Equation (19), a single correlator spacing is assumed for all of the signal components forming c˜*(nTs). While this option has been adopted here to simplify derivations, different correlator spacings can be adopted for the upper and lower sideband components. This more general option is considered in Section 3.1 for the analysis of the triple-loop code discriminator.

Using Equation (8), Equation (19) can be expressed as follows:

P=1N∑n=0N−1zBB(nTs−τ0)c˜*(nTs−τ^)e2π(jΔfd+iΔfsub)nTs+(jΔφ+iΔφsub)+ηPE=1N∑n=0N−1zBB(nTs−τ0)c˜*(nTs−τ^−ds/2)e2π(jΔfd+iΔfsub)nTs+(jΔφ+iΔφsub)+ηEL=1N∑n=0N−1zBB(nTs−τ0)c˜*(nTs−τ^+ds/2)e2π(jΔfd+iΔfsub)nTs+(jΔφ+iΔφsub)+ηL20

where Δfd=fd,0−f^d and Δfsub=fsub,0−f^sub are the residual carrier and subcarrier Doppler frequencies, respectively. Similarly, Δφ=φ0−φ^ and Δφsub=φsub,0−φ^sub are the residual carrier and subcarrier phases. ηP, ηE, and ηL are three bicomplex noise terms obtained by correlating η[n] with the local code, carrier, and subcarrier components.

The three bicomplex correlators can be expressed in terms of complex sideband correlators by exploiting orthogonal decomposition (Equation (6)). In the following, the prompt correlator is considered, but similar results are obtained for the early and late terms. Moreover, to simplify the derivations, it is assumed that frequency lock has been achieved, that is, Δfd ≈ 0 and Δfsub ≈ 0. In this way, we have the following:

P=2N∑n=0N−1[αy(nTs−τ0)e1+βx(nTs−τ0)e2]c˜*(nTs−τ^)ejΔφ+iΔφsub+ηP21

The exponential term in Equation (21) can be expressed in terms of orthogonal components, as described by Borio (2023, Appendix C):

e−(jΔφ+iΔφsub)=e−j(Δφ−Δφsub)e1+e−j(Δφ+Δφsub)e222

Moreover, the local code can be generically expressed as Equation (18). In this way, Equation (21) has the following form:

P=2N∑n=0N−1[αy(nTs−τ0)e1+βx(nTs−τ0)e2][y˜*(nTs−τ^)e1+γx˜*(nTs−τ^)e2]⋅[ej(Δφ−Δφsub)e1+ej(Δφ+Δφsub)e2]+ηP23

Given the orthogonality of the idempotent units, e1 and e2, the operations in Equation (23) can be performed component-wise. In this way, we obtain the following for Equation (23):

P=2αN∑n=0N−1y(nTs−τ0)y˜*(nTs−τ^)ej(Δφ−Δφsub)e1+ηP,1e1+2βγN∑n=0N−1x(nTs−τ0)x˜*(nTs−τ^)ej(Δφ+Δφsub)e2+ηP,2e2=2αR˜y(Δτ)ej(Δφ−Δφsub)e1+ηP,1e1+2βγR˜x(Δτ)ej(Δφ+Δφsub)e2+ηP,2e2=2Pye1+2γPxe224

where Py and Px are the sideband prompt correlators obtained as follows:

Py=αN∑n=0N−1y(nTs−τ0)y˜*(nTs−τ^)ej(Δφ−Δφsub)+ηP,1=1N∑n=0N−1ybb[n]y˜*(nTs−τ^)e−j(φ^−φsub)Px=βN∑n=0N−1x(nTs−τ0)x˜*(nTs−τ^)ej(Δφ+Δφsub)+ηP,2=1N∑n=0N−1xbb[n]x˜*(nTs−τ^)e−j(φ^+φsub)25

These terms are computed from the sideband samples, xbb[n] and ybb[n].

In Equation (24), the noise component ηp has been decomposed into its orthogonal components, ηP,1 and ηP,2, which derive from the noise processes affecting the upper and lower sideband components, respectively. R˜y(Δτ) and R˜x(Δτ) are the cross-correlations between the sideband signals and the corresponding local codes. Their actual values depend on the choice of the local code, as discussed in the previous section. If a data channel is considered as in Equation (17), the data bit present in the received signal and not compensated in the local code will affect the correlator computation. Δτ is the residual delay error, Δτ=τ0−τ^. Models similar to Equation (24) can be obtained for the early and late correlators. These correlators are used by the different loop discriminators to estimate the code, carrier, and subcarrier residual errors. The three discriminators and their statistical properties are more fully discussed in the following subsections. The variances of the three discriminators will be used in Section 4 to construct the KF measurement noise matrix.

The discriminator outputs are independently filtered using three loop filters. These are standard components, and their description can be found in textbooks from the GNSS literature (Kaplan & Hegarty, 2017). The output of the loop filters is finally used to drive the three loop NCOs that generate new code, carrier, and subcarrier local replicas. In this way, the three loops are closed. This architecture is the starting point used in this paper for the design of a bicomplex KF tracking loop.

3.1 Code Discriminator

A form of normalized non-coherent early–minus–late envelope discriminator was derived by Borio (2023) as follows:

Dc=1Gdℜ{|E|k}−ℜ{|L|k}ℜ{|E|k}+ℜ{|L|k}=1Gd|Ey|+γ|Ex|−|Ly|−γ|Lx||Ey|+γ|Ex|+|Ly|+γ|Lx|26

where Dc is the discriminator output, computed as a function of the bicomplex early and late correlators, E and L, defined as follows:

E=2Eye1+2γExe2L=2Lye1+2γLxe227

where Ex, Lx, Ey, and Ly are the complex correlators obtained from the sideband components, xbb[n] and ybb[n]. Equation (27) can be obtained in a manner similar to Equation (24). γ is the ratio between the amplitudes of the received signal sideband components introduced in Section 2. |·|k is the hyperbolic modulus (Alpay et al., 2014, pp. 12), which can be expressed with respect to the orthogonal components as follows:

|E|k=2|Ey|e1+2γ|Ex|e2|L|k=2|Ly|e1+2γ|Lx|e228

The absolute values in Equation (28) render the code discriminator, Dc, insensitive to both phase and data bits. Finally, Gd is a normalization constant used to make the discriminator gain equal to 1:

E{∂Dc∂τ}|τ=0=129

In the work by Borio (2023, Appendix A), the following is shown:

Gd=sy+γ2sx1−syds,y/2+γ2(1−sxds,x/2)30

where sy and sx are the slopes of the correlator functions of the two sideband components. ds,y and ds,x are the early–minus–late spacings used for the two sideband components. In the previous section, a single early–minus–late spacing was adopted to simplify the notation. However, it is possible to use two different spacings by applying different delays to the sideband components of the local codes. For symmetric modulations, such as the AltBOC, γ = 1 and sy = sx. In this way, for ds,y = ds,x, Equation (30) simplifies as follows:

Gd=sx1−sxds,c/231

The variance of Equation (26) was not derived in the work by Borio (2023) and is evaluated in this paper in Appendix A. The code discriminator variance is used in the following to build the measurement noise correlation matrix:

σc2=Var{Dc}=14CyN0Tcsyds,y+γ2sxds,x[sy+γ2sx]232

where CyN0 is the C/N0 value of the lower sideband component, ybb[n]. In particular, Cy is the power received on the ybb[n] component:

Cy=(αAp,y)233

The combined use of the sideband components in the meta-signal approach provides a reduction in the code discriminator variance with respect to the single-frequency case. The single-frequency code discriminator variance can be obtained by setting γ equal to zero in Equation (32):

σc,sf2=ds,y4CyN0Tcsy34

where the subscript “sf” is used to indicate results for the single-frequency case. The advantages of the meta-signal approach in terms of code discriminator variance reduction are particularly evident for the symmetric case, where both sideband components have the same characteristics. In this case, we obtain the following for Equation (32):

σc,sy2=ds,y8CyN0Tcsy35

where the subscript “sy” indicates the symmetric meta-signal case. By comparing Equation (34) with Equation (35), a 3-dB improvement is observed when moving from the single-frequency case to the symmetric meta-signal case. In the meta-signal approach, the power from both sideband components is effectively combined, and the code discriminator variance is consequently reduced.

3.2 Carrier Discriminator

The carrier discriminator is found as follows (Borio, 2023):

Dφ=12∠|P|j2=12arctan2(ℑ{P·P¯},ℜ{P·P¯})36

where P is the bicomplex prompt correlator evaluated in Equation (24). Here, ⋅¯ and |·|j denote the bar-conjugation and the j-modulus defined by Borio (2023, Appendix A).

As discussed by Borio (2023), the discriminator in Equation (36) can be expressed in terms of the sideband components as follows:

Dφ=12∠PyPx=12arctan2(ℑ{PyPx},ℜ{PyPx})37

where Px and Py are the prompt correlators from the sideband components. Equation (37) is the average phase of the prompt correlators evaluated from the sideband components. The discriminator in Equation (37) is insensitive to bit changes common to the two sideband prompt correlators. To obtain a discriminator that is insensitive to independent bit changes on the two sideband components, it is sufficient to replace the arctan2(·) function with the standard arctangent:

Dφ=12arctan(J{PyPx}ℜ{PyPx})38

Consider, for instance, the case in which the lower sideband component is a data channel and local code (Equation (16)) is used. Because the data bit is not accounted for in the local code, Px is modulated by dx, the value assumed by dx(t – τ0) in Equation (2) during the correlator integration time. dx can be equal to either 1 or –1 and commutes with the real and imaginary parts in Equation (38). Because dx is present in both the numerator and denominator of Equation (38), this term cancels out and has no effect on the final discriminator output. For this reason, Equation (38) is bit-insensitive.

A pure pilot discriminator, which can be used when both sideband components are pilot signals, was also suggested by Borio (2023). All of these discriminators have a unit gain, and their variance is determined in Appendix B. In particular, we have the following:

σφ2=Var{Dφ}=18γ2CyN0Tc[(1+γ2)+2γ2+(1+γ2)2γ2CyN0Tc]39

The variance σφ2 is expressed in squared radians.

It is possible to compare the variance in Equation (39) with that of a standard single-frequency arctangent discriminator. Note that, in this case, the single-frequency carrier discriminator variance cannot be obtained by simply setting γ = 0, because two different quantities are estimated in the two cases. In the single-frequency case, the phase of a single signal carrier is estimated. In the meta-signal case, the average phase of the two sideband carriers is tracked. By setting γ = 0, one implies that the upper sideband component is not available, and thus, the average phase of the two sideband components becomes unobservable. The carrier discriminator variance for the single-frequency case is given by the following (Borio, 2011):

σφ,sf2=12CyN0Tc[1+12CyN0Tc]40

This case also results in a reduction in terms of discriminator noise variance. This reduction is particularly evident for the symmetry meta-signal case, which is characterized by the following carrier discriminator variance:

σφ,sy2=14CyN0Tc[1+1CyN0Tc]41

A gain of approximately 3 dB is also observed in this case.

3.3 Subcarrier Discriminator

The subcarrier phase discriminator is obtained as follows (Borio, 2023):

Dφsub=12∠|P|i2=12arctan2(ℑ{P·P†},ℜ{P·P†})42

where (·)† denotes the †-conjugation and |·|i is the i-modulus defined by Borio (2023, Appendix A). Moreover, Equation (42) can be expressed in terms of sideband components as follows:

Dφsub=12∠PxPy*=12arctan2(J{PxPy*},ℜ{PxPy*})43

In this case, the subcarrier phase discriminator is the half-difference of the phases of the sideband prompt correlators. Different variants of this discriminator, sensitive or not to data bit changes, can be obtained by using a process similar to that used for the carrier phase discriminator, as discussed by Borio (2023).

The discriminator in Equation (43) has a unit gain and variance:

σφsub2=Var{Dφsub}=18γ2CyN0Tc[(1+γ2)+2γ2+(1+γ2)2γ2CyN0Tc]44

The variance of the subcarrier discriminator output can be derived by using an approach similar to that adopted in Appendix B for the carrier component. When expressed in squared radians, Equations (39) and (44) assume the same values. This condition follows from the fact that the two discriminator outputs only differ for a conjugation operation. Appropriate scaling must be adopted in Equation (44) to consider the unit of measurement utilized in the actual processing of the subcarrier component.

4 TRIPLE-LOOP KALMAN FILTER

The triple-loop architecture discussed in the previous section has been modified in order to take advantage of the relationships between code, carrier, and subcarrier components. This type of approach has been used in the literature for standard dual-loop architectures (O’Driscoll et al., 2011) and for sine-BOC triple-loop tracking (Luo et al., 2018). In such cases, the loop filters in Figure 1 are replaced by a single KF, which uses a dynamic model to improve tracking performance. A schematic representation of the triple-loop KF architecture adopted in this work is shown in Figure 2. A single KF replaces the three independent loop filters present in Figure 1.

FIGURE 2
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FIGURE 2

Schematic representation of a triple-loop KF combining two GNSS signals from different frequencies

The two signals are considered as a single bicomplex component, z[n]. Signals and correlators in the upper box are bicomplex.

The KF uses a dynamic model to predict and estimate the time evolution of a state vector that represents the parameters of interest (O’Driscoll et al., 2011; Susi & Borio, 2020). In this work, the five-dimensional state vector introduced by Luo et al. (2018) for BOC signals is considered:

xh=[Δτ,Δφsub,Δφ,Δf,Δa]h45

where Δτ, Δφ, and Δφsub are the delay, carrier, and subcarrier phase errors, respectively. Δf is the residual Doppler frequency shift error, and Δa is the Doppler rate correction. h is the time index. Note that xh in Equation (45) is similar to the state vector considered in standard dual-loop KF tracking (O’Driscoll et al., 2011; Susi & Borio, 2020). In this case, the state vector has been extended by adding the subcarrier phase error to xh.

The system dynamic model, which describes the state vector evolution over time, is defined as follows:

xh+1=Axh+wh46

where A is the state transition matrix defined as follows (Luo et al., 2018; Susi & Borio, 2020):

A=[100βcodeTc12βcodeTc2010βsubTc12βsubTc20012πTcπTc20001Tc00001]47

Here, βcode is the ratio between the nominal code rate and the carrier RF. Similarly, βsub is the ratio between the nominal subcarrier rate and the carrier RF.

In Equation (46), wh denotes an additive zero-mean Gaussian noise process that is uncorrelated over time:

wh∼N(0,Q)48

where Q is the covariance matrix modeling the uncertainties due to oscillator instability, user dynamics, and code–carrier and subcarrier–carrier divergence.

The approach adopted by O’Driscoll et al. (2011) and Luo et al. (2018) has been used for setting Q. In particular, we have the following:

Q=qcode[Tc000000000000000000000000]+qsub[000000Tc000000000000000000]+(2πf0c)2qa[Tc520βcode2Tc520βcodeβsubTc520βcodeTc48βcodeTc36βcodeTc520βcodeβsubTc520βsub2Tc520βsubTc48βsubTc36βsubTc520βcodeTc520βsubTc520Tc48Tc36Tc48βcodeTc48βsubTc48Tc33Tc22Tc36βcodeTc36βsubTc36Tc22Tc]+(2πf0)2qd[Tc33βcode2Tc33βcodeβsubTc33βcodeTc22βcode0Tc33βcodeβsubTc33βsub2Tc33βsubTc22βsub0Tc33βcodeTc33βsubTc33Tc220Tc22βcodeTc22βsubTc22Tc000000]+(2πf0)2qb[Tcβcode2TcβcodeβsubTcβcode00TcβcodeβsubTcβsub2Tcβsub00TcβcodeTcβsubTc000000000000]49

The first two blocks in Equation (49) model the code–carrier and subcarrier–carrier divergence. Possible code–subcarrier divergences are neglected, and for this reason, no corresponding terms are present in Equation (49). qcode and qsub are the associated PSDs and have been found empirically.

The third block in the second line of Equation (49) models the effects of the line-of-sight acceleration (O’Driscoll et al., 2011) and has also been considered by Luo et al. (2018) for a BOC KF tracking loop. qa is the PSD of the random process driving the line-of-sight acceleration. Finally, the fourth and fifth blocks represent the effects of carrier frequency and phase noises due to the local oscillator. qd and qb are the associated PSDs and are found from the oscillator h-parameters. In this work, standard values from the literature have been used (see work by O’Driscoll et al. (2011, Table 1)).

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TABLE 1

Characteristics of the Digital Baseband Galileo AltBOC and BeiDou B1 Signals Used for the Static Tests

The KF uses Equation (46) to predict the state vector at the next time epoch, h + 1. Moreover, measurements are used to correct the predicted states. While a full review of the KF is outside the scope of this paper, the measurement model is briefly reviewed here (see work by Brown and Hwang (2012) for additional details on KF principles). In particular, the measurement model can be expressed as follows:

zh=Hxh+vh50

where:

zh=[DcDφsubDφ]h51

is the vector with the code, subcarrier, and carrier discriminator outputs at epoch h. H is the observation matrix mapping the state vector into the measurements and is defined as follows (Luo et al., 2018; Susi & Borio, 2020):

H=[100−βcodeTc2βcodeTc26010−βsubTc2βsubTc26001−πTcπTc23]52

vh is a noise vector affecting the measurements. This vector is assumed to have a zero mean with uncorrelated components:

vh∼N(0,Rh)53

where Rh is obtained as follows:

Rh=[σc2000σφsub2000σφ2]54

The diagonal components of Equation (54) are the discriminator output variances derived in Section 3. Note that the variances of the discriminator outputs are a function of the signal C/N0, which is continuously estimated.

Using the elements described in this section, it is finally possible to implement the KF tracking loop.

5 EXPERIMENTAL SETUP

Several experiments using baseband I/Q data were performed to test the triple-loop KF tracking architecture. Both static and dynamic conditions were considered. These experiments are briefly described below.

5.1 Static Scenarios: AltBOC and BeiDou B1I/B1C Signals

A first set of tests was conducted by applying the data used by Borio (2023) to demonstrate bicomplex tracking. The two static data sets, which were considered by Borio (2023) for AltBOC and BeiDou B1I/B1C signals, have been reprocessed here using KF tracking. AltBOC signals were collected using a National Instruments universal software radio peripheral 2944R front-end, whereas the BeiDou B1 signals were taken from open data shared by Gao et al. (2020). A summary of the signal characteristics is provided in Table 1. The interested reader is referred to the work by Borio (2023) for additional details on the two static data sets.

5.2 Dynamic Scenario: BeiDou B1I/B1C

Two sets of tests were conducted under dynamic conditions targeting the BeiDou B1I/B1C meta-signal. For both cases, which considered different environmental conditions, a HackRF One platform (Ossmann, 2023) was used for data collection. A view of the equipment used for data collection is shown in Figure 3(a). A multi-frequency patch antenna was mounted on the rooftop of the car used for the experiments. The antenna was connected through an RF splitter to a u-blox ZED-F9R module (u-blox, 2023) and to the HackRF One platform. The u-blox ZED-F9R module was used to collect reference data. Views of the first experiment, which was conducted in an industrial zone in the suburbs of Turin, Italy, are provided in Figures 3(b) and (c). The tests were conducted on a straight road with buildings on both sides. For the tests, the driver performed several loops on the road shown in Figure 3(c), performing u-turns at each end of the road. The second set of tests was performed in the environment shown in Figures 3(d) and (e), in the proximity of Lake Maggiore, Ispra, Italy. This environment is considered challenging, not only because of the dynamic conditions but also because of the presence of tall trees that impaired signal reception. The signals were collected using the parameters reported in Table 2. In the table, the nominal center frequency denotes the parameter used to configure the HackRF One platform. However, as reported by O’Driscoll and Curran (2018), in such low-cost SDR devices, slightly different frequencies are adopted despite the parameters specified by the user. For this reason, the adjective “nominal” is used in the table. Because the device adopts a center frequency different from the one set in the device configuration, a residual intermediate frequency (IF) is present. This residual IF must be estimated and accounted for in the implementation of the KF tracking loop. A residual IF of approximately –24.735 Hz was estimated via the procedure described by Borio and Susi (2023). The collected data were processed using a custom SDR receiver implemented in Python (Borio, 2023). The receiver implements different stages. After acquiring the different signals, tracking is started. The adopted tracking strategy features different states and includes initial fine frequency estimation via a frequency lock loop (FLL) followed by a standard PLL. When phase lock conditions are achieved, KF tracking can finally start. The receiver also recovers the secondary codes on the different signal components and implements extended coherent integrations.

FIGURE 3
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FIGURE 3

Views of the experimental setup adopted for collecting BeiDou B1I/B1C baseband I/Q data under dynamic conditions: (a) internal view of the car used for data collection with the different devices used in the experiment, (b) external view of the car and location of the patch antenna used for the first set of tests (industrial zone in Turin, Italy), (c) view of the road/scenario where the first set of tests was conducted, (d) external view of the car and location of the patch antenna used for the second set of tests (tall-trees scenario, Ispra, Italy), and (e) view of the road/ scenario where the second set of tests was conducted

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TABLE 2

Characteristics of the Digital Baseband BeiDou B1 Signals Used for the Dynamic Tests

The triple-loop architecture discussed by Borio (2023) has also been considered for comparison purposes.

6 EXPERIMENTAL RESULTS

The experimental results obtained using the KF tracking loop are briefly discussed in this section.

6.1 Static AltBOC Processing

Here, the findings obtained via processing of static AltBOC data are discussed. Static tests were conducted primarily to verify proper functioning of the proposed KF architecture and to confirm that no significant improvements are expected under such conditions. The actual benefits from KF tracking are expected under dynamic conditions.

The Doppler estimates obtained for the code, subcarrier, and carrier components for the static AltBOC test are compared in Figure 4 for the signal received from the satellite with pseudorandom number (PRN) 2. Each component has been normalized by its nominal rate. In the left panel of the figure, the results obtained via a triple-loop architecture with separate filters are provided. For the test, a third-order PLL with a 15-Hz bandwidth was considered along with second-order DLL and SPLL with 2- and 3-Hz bandwidths, respectively. An integration time of 5 ms is adopted after secondary code synchronization. Under static conditions, the standard triple-loop architecture can effectively track the different signal components. The Doppler estimates in the left panel of Figure 4 overlap, justifying the adoption of a common state in the KF implementation. The right panel in Figure 4 shows the Doppler estimates obtained via KF tracking. After an initial transient, where standard FLL/PLL is used to achieve frequency and phase lock conditions, KF operations are started. The KF provides a single estimate for the three Doppler components, as clearly shown in the right panel in Figure 4: after approximately 2 s from the start of the test, the three Doppler frequency estimates overlap, and a single line is visible.

FIGURE 4
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FIGURE 4

Comparison between Doppler estimates for the code, subcarrier, and carrier components for the static AltBOC test (signal with PRN 2): (left) standard triple-loop architecture; (right) KF tracking

The use of a single state for the Doppler estimates in Equation (45) does not affect lock conditions. This fact is more fully investigated in Figure 5, which provides a comparison between overall C/N0 values obtained using a standard triple-loop architecture and KF tracking. In this paper, the term “overall C/N0” is used to indicate the total C/N0 obtained by jointly considering the signals and correlators from the two sidebands. The C/N0 estimates are used as lock indicators to verify that the SPLL is effectively able to align in phase the two sideband prompt correlators and to confirm that tracking is effectively performed. The comparison in Figure 5 shows that the two architectures achieve similar performance under static conditions. KF tracking is able to properly align the different signal components, obtaining the same C/N0 values found via the standard triple-loop architecture. As already mentioned, the real advantages of a KF tracking loop are expected under dynamic conditions, when the adaptive nature of this tracking loop can be fully exploited.

FIGURE 5
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FIGURE 5

Comparison between overall C/N0 values obtained using a standard triple-loop architecture and KF tracking (static AltBOC experiment, PRN 2)

6.2 Static BeiDou B1 Processing

Results similar to those described in Section 6.1 were obtained for processing of the static BeiDou B1 signals. This fact is confirmed by the sample results presented in Figure 6, which compares the normalized Doppler estimates obtained via the triple-loop architecture and KF tracking. In this case, the two architectures achieve similar results. When the receiver is operating with the KF, a single Doppler estimate is found, effectively tracking the different signal components. As observed for the previous case, no performance degradations are observed when the KF architecture is used.

FIGURE 6
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FIGURE 6

Comparison between Doppler estimates for the code, subcarrier, and carrier components for the static BeiDou B1 test (PRN 45): (left) standard triple-loop architecture; (right) KF tracking

6.3 Dynamic BeiDou B1 Processing

Sample results obtained for the dynamic BeiDou experiment are analyzed in this section. As discussed in Section 5.2, the HackRF One is a low-cost SDR platform, which features a low-quality temperature-compensated crystal oscillator. These characteristics present further challenges in processing the collected data, which are affected by the apparent dynamics and non-idealities introduced by the local oscillator. For all of the tests discussed in the following, an integration time equal to 10 ms was adopted. This corresponds to the duration of a single B1C primary code period.

Sample results obtained for the industrial zone tests are presented first. In this scenario, six satellites broadcasting both B1I and B1C components were available. All of the satellite signals were acquired and tracked by the Python software receiver implementing KF tracking. Figure 7 provides the C/N0 values obtained using the triple-loop KF during one of the tests conducted in the industrial zone scenario. The C/N0 time series were low-pass-filtered in order to improve the quality of the representation. During the test, the car made several loops, periodically passing the same location: this periodic behavior is reflected in the C/N0 time series, which are characterized by periodic patterns. Three signals (PRN 34, 42, and 43) were received with high and stable C/N0 values that were only marginally affected by the buildings on the two sides of the road. The remaining signals (PRN 22, 23, and 25) were affected by deep fades that compromised the estimated C/N0 values. Despite this fact, the KF triple-loop architecture was able to maintain lock on all of the signal components. This fact clearly emerges from the last part of the C/N0 time series shown in Figure 7, where stable C/N0 values are obtained for all signals. After completing two full loops, the car remained static for several seconds: because a lock was maintained for all signals, stable C/N0 values were obtained. This was not the case for the standard triple-loop architecture, which is discussed more fully below.

FIGURE 7
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FIGURE 7

C/N0 values estimated for the different signals during one of the dynamic experiments conducted in the industrial zone scenario

Six signals were successfully tracked via the KF architecture. Oscillations in the time series reflect the different loops performed during the test.

In particular, the carrier Doppler variations of the different signals tracked during the dynamic experiment considered in Figure 7 are provided in Figure 8 for both the KF approach and the standard triple-loop architecture. Carrier Doppler variations were obtained by removing the Doppler median values, observed during the whole duration of the test, from the corresponding carrier Doppler frequencies. Doppler variations were considered instead of the actual Doppler values, to improve the representation clarity. When plotted together, the large and different Doppler values estimated for the different signals would have hidden the variations caused by the actual motion of the car. Carrier Doppler variations estimated using the KF approach are presented in the left panel of Figure 8. The implemented KF tracking is able to correctly recover the different Doppler components, including the component estimated from signals with PRN 22, 23, and 25. This fact was verified by considering other metrics such as the correlator outputs: at the end of the test, when the car used for the experiment was static, all of the signal energy was concentrated in the in-phase branch of the correlators. The oscillations visible on all of the estimated carrier Doppler variations have the same period and are consistent between different loops. Results for the triple-loop architecture with standard filters are provided in right panel of Figure 8. To process the different signals, different loop bandwidth/integration time combinations were considered. The results provided in the right panel of Figure 8 were obtained with a third-order PLL and second-order DLL and SPLL. The PLL and SPLL bandwidths were set to 20 and 8 Hz, respectively. This increase with respect to the static case was introduced to better follow the signal dynamics. The integration time was maintained at 10 ms. These parameters, which were selected empirically, provide a good compromise in term of noise reduction and tracking of signal dynamics. Using these loop parameters, we obtained the results shown in the right panel of Figure 8, where only signals with PRN 25 and 23 were affected by problems.

FIGURE 8
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FIGURE 8

Carrier Doppler variations of the different signals tracked during the dynamic experiment considered in Figure 7 for the industrial zone scenario: (left) results obtained using the meta-signal KF approach; (right) results obtained using the standard triple-loop architecture

In particular, phase lock was lost on the signal with PRN 25, whereas a false frequency lock occurred for the signal with PRN 23. The false frequency lock was determined by analyzing the corresponding correlator outputs: in the standard case, energy was split between the in-phase and quadrature components. These problems do not occur for the KF architecture. Moreover, from the comparison in Figure 8, it is clear that smoother Doppler estimates are obtained by using KF tracking.

Note that the meta-signal KF approach also has inherent advantages over the single-frequency KF tracking architecture (O’Driscoll et al., 2011). In the meta-signal approach, the power from both sideband components is effectively recovered. This fact has been discussed in Section 4, where a 3-dB gain was shown for the meta-signal symmetric case with respect to single-frequency processing. This power gain is reflected in the tracking performance. Figure 9 compares the carrier Doppler variations obtained via the meta-signal KF approach (left panel) with those estimated by a single-frequency KF tracking architecture operating on the B1C component alone. For single-channel processing, the architecture described by O’Driscoll et al. (2011) has been implemented. The same parameters (oscillator h-parameters, dynamic constants, and code–carrier divergence) have been used for the two KF implementations. The comparison in Figure 9 confirms the benefits of the meta-signal approach. Single-channel KF tracking is able to maintain lock only on the strongest B1C signals.

FIGURE 9
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FIGURE 9

Carrier Doppler variations of the different signals tracked during the dynamic experiment considered in Figure 7 for the industrial zone scenario: (left) results obtained using the meta-signal KF approach; (right) results obtained using a single-frequency KF tracking architecture operating on the B1C component alone

Similar conclusions can be drawn from the tests conducted in the second scenario, where dynamic experiments were performed below tall trees. Figure 10 shows the C/N0 values estimated for the different signals during one of the tests performed under the second scenario. In addition, in this case, six signals with both B1I and B1C components were acquired and tracked. Oscillations in the time series are clearly visible; the car repeated the same loop several times, periodically returning to the same location with similar reception conditions. The oscillations correspond to the different loops performed by the car. KF tracking is able to maintain lock on the different signals, even the signal with PRN 39, which was characterized by low C/N0 values. The ability of the KF tracking to maintain lock is clearly shown in Figure 11, which displays the carrier Doppler variations of the different signals tracked during the dynamic experiment introduced in Figure 10. Figure 11 confirms that the implemented KF tracking is able to correctly recover the different Doppler components, including the component estimated from the signal with PRN 39. In this test, standard loop filters are not effective, and the receiver is not able to maintain lock on the weakest signal components. This finding is analyzed in Figure 12, which compares the overall C/N0 values obtained via the standard triple-loop architecture and KF tracking. Here, the specific case of the signal with PRN 39 is analyzed. For the triple-loop architecture, the same loop parameters considered for the industrial zone test were adopted. During the first 20 s of the test, the car was static. During this portion of the test, the triple-loop architecture with standard loop filters is able to maintain lock. However, once the car starts moving, the standard triple-loop architecture loses lock.

FIGURE 10
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FIGURE 10

C/N0 values estimated for the different signals tracked during one of the tests conducted under tall trees

Six signals were successfully tracked. Oscillations in the time series reflect the different loops performed during the test.

FIGURE 11
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FIGURE 11

Carrier Doppler variations of the different signals tracked during the dynamic experiment considered in Figure 10 for the tall-trees scenario

FIGURE 12
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FIGURE 12

Comparison between overall C/N0 values obtained via a standard triple-loop architecture and KF tracking (tall-tree experiment, BeiDou B1I/B1C PRN 39)

These experimental results confirm the benefits of KF filter tracking for meta-signal processing under dynamic scenarios.

7 CONCLUSIONS

In this paper, an error-state KF loop architecture was designed for simultaneously tracking two GNSS signals from different frequencies. The KF tracking loop was designed using the bicomplex number paradigm, which allows one to represent two GNSS signals from different frequencies as the product of a code, carrier, and subcarrier component. In this respect, the KF extends previous triple-loop architectures, where the code, carrier, and subcarrier components are processed via independent loop filters. The proposed architecture has been implemented and tested using real Galileo AltBOC and BeiDou B1I/B1C meta-signals, under both static and dynamic conditions. The collected data were processed by a custom Python software receiver implementing KF tracking. The analysis clearly shows the advantages of KF tracking for processing GNSS meta-signals with components from different frequencies. In particular, tests conducted under dynamic conditions show the ability of the KF architecture to continuously track weak GNSS signals received from a moving vehicle. These results are not achievable when using standard loops, which require large bandwidths to handle dynamic conditions.

HOW TO CITE THIS ARTICLE

Borio, D., & Susi, M. (2024). GNSS meta-signal tracking using a bicomplex Kalman filter. NAVIGATION, 71(4). https://doi.org/10.33012/navi.674

CONFLICT OF INTEREST

The authors declare no potential conflicts of interest.

APPENDIX

A VARIANCE OF THE CODE DISCRIMINATOR OUTPUT

In this appendix, the variance of the code discriminator in Equation (26) is found using the Delta method (Casella & Berger, 2001, pp. 240). The Delta method allows one to approximate the variance of a function of several random variables as follows:

Var{f(θ)}≈∑i=0I−1∑j=0I−1∂f(θ)∂θi|θ=θ0∂f(θ)∂θj|θ=θ0σi,j55

where θ is a vector of I random variables, θi. θ0 is a vector with the average values of θ, and σi,j are the variances/covariances of the components of θ.

The code discriminator in Equation (26) is a function of the real and imaginary parts of the sideband early and late correlators:

Ey,I=ℜ{Ey}∼N(αAyRy(ds,y2)cosφ,σ2)Ey,Q=ℑ{Ey}∼N(αAyRy(ds,y2)sinφ,σ2)Ly,I=ℜ{Ly}∼N(αAyRy(ds,y2)cosφ,σ2)Ly,Q=J{Ly}∼N(αAyRy(ds,y2)sinφ,σ2)Ex,I=ℜ{Ex}∼N(βAxRx(ds,x2)cosφ,σ2)Ex,Q=ℑ{Ex}∼N(βAxRx(ds,x2)sinφ,σ2)Lx,I=ℜ{Lx}∼N(βAxRx(ds,x2)cosφ,σ2)Lx,Q=ℑ{Lx}∼N(βAxRx(ds,x2)sinφ,σ2)56

where σ2 is the correlator variance and φ is the residual phase error not recovered by the PLL/SPLL. The correlator variance is found by propagating the input signal noise variance through the correlation process: σ2=σn2N.

Note that the code discriminator in Equation (26) is phase-independent. Thus, the resulting variance will also be phase-independent.

Note that components from different frequencies are statistically independent. Moreover, in-phase and quadrature components are also independent. However, early and late components from the same channel are correlated, and we have the following:

Cov(Ey,I,Ly,I)=Cov(Ey,Q,Ly,Q)=σ2Ry(ds,y)Cov(Ex,I,Lx,I)=Cov(Ex,Q,Lx,Q)=σ2Rx(ds,x)57

To compute the variance of Equation (26), let us first consider the unnormalized discriminator:

D¯c=|Ey|+γ|Ex|−|Ly|−γ|Lx||Ey|+γ|Ex|+|Ly|+γ|Lx|58

Its partial derivatives with respect to the different components are given by the following:

∂D¯c∂Ey,I=2(|Ly|+γ|Lx|)(|Ey|+γ|Ex|+|Ly|+γ|Lx|)2Ey,I|Ey|∂D¯c∂Ey,Q=2(|Ly|+γ|Lx|)(|Ey|+γ|Ex|+|Ly|+γ|Lx|)2Ey,Q|Ey|∂D¯c∂Ly,I=−2(|Ey|+γ|Ex|)(|Ey|+γ|Ex|+|Ly|+γ|Lx|)2Ly,I|Ly|∂D¯c∂Ly,Q=−2(|Ey|+γ|Ex|)(|Ey|+γ|Ex|+|Ly|+γ|Lx|)2Ly,Q|Ly|∂D¯c∂Ex,I=2γ(|Ly|+γ|Lx|)(|Ey|+γ|Ex|+|Ly|+γ|Lx|)2Ex,I|Ex|∂D¯c∂Ex,Q=2γ(|Ly|+γ|Lx|)(|Ey|+γ|Ex|+|Ly|+γ|Lx|)2Ex,Q|Ex|∂D¯c∂Lx,I=−2γ(|Ey|+γ|Ex|)(|Ey|+γ|Ex|+|Ly|+γ|Lx|)2Lx,I|Lx|∂D¯c∂Lx,Q=−2γ(|Ey|+γ|Ex|)(|Ey|+γ|Ex|+|Ly|+γ|Lx|)2Lx,Q|Lx|59

When computed for the average values of the early and late correlators, the partial derivatives in Equation (59) are as follows:

∂D¯c∂Ey,I|θ=θ0=cosφ2αAy[Ry(ds,y2)+γ2Rx(ds,x2)]∂D¯c∂Ey,Q|θ=θ0=sinφ2αAy[Ry(ds,y2)+γ2Rx(ds,x2)]∂D¯c∂Ly,I|θ=θ0=−cosφ2αAy[Ry(ds,y2)+γ2Rx(ds,x2)]∂D¯c∂Ly,Q|θ=θ0=−sinφ2αAy[Ry(ds,y2)+γ2Rx(ds,x2)]∂D¯c∂Ex,I|θ=θ0=cosφ2αAy[Ry(ds,y2)+γ2Rx(ds,x2)]∂D¯c∂Ex,Q|θ=θ0=sinφ2αAy[Ry(ds,y2)+γ2Rx(ds,x2)]∂D¯c∂Lx,I|θ=θ0=−γcosφ2αAy[Ry(ds,y2)+γ2Rx(ds,x2)]∂D¯c∂Lx,Q|θ=θ0=−γsinφ2αAy[Ry(ds,y2)+γ2Rx(ds,x2)]60

From these results and Equation (55), the variance of Equation (58) can be approximated as follows:

Var{D¯c}≈σ2[1−Ry(ds,y)+γ2(1−Rx(ds,x))]2α2Ay2[Ry(ds,y2)+γ2Rx(ds,x2)]261

The variance of Dc is given by the following:

Var{Dc}=1Gd2Var{D¯c}62

The gain, Gd, has been reported by Borio (2023) and can be expressed as follows:

Gd=−R˙y(ds,y2)+γ2R˙x(ds,x2)Ry(ds,y2)+γ2Rx(ds,x2)63

where Ṙy(·) and Ṙx(·) are the derivatives of the correlation functions of the two sideband components.

Using these results, we obtain the following for Equation (62):

Var{Dc}=σ2[1−Ry(ds,y)+γ2(1−Rx(ds,x))]2α2Ay2[R˙y(ds,y2)+γ2R˙x(ds,x2)]264

The ratio α2Ay2σ2 is the post-coherent signal-to-noise ratio (SNR) and can be expressed as follows:

SNRp=α2Ay2σ2=2CyN0Tc65

In this way, Equation (64) takes the following form:

Var{Dc}=14CyN0Tc[1−Ry(ds,y)+γ2(1−Rx(ds,x))][R˙y(ds,y2)+γ2R˙x(ds,x2)]266

For piece-linear correlation functions, such as in the case of binary phase shifting keying and BOC modulations, the main correlation peaks can be expressed as Ry(τ) = 1 – sy |τ| and Rx(τ) = 1 – sx |τ|. In this way, we obtain the following:

Var{Dc}=14CyN0Tcsyds,y+γ2sxds,x[sy+γ2sx]267

B VARIANCE OF THE CARRIER DISCRIMINATOR OUTPUT

The variance of the carrier discriminator in Equation (36) is found using an approach similar to that developed by Borio (2011). In particular, Borio (2011) showed that the variance of discriminators based on the arctangent function approximately depends on the inverse of the SNR at the input of the arctangent function. In the case of Equation (36), we have the following:

Var{Dφ}≈141Rs(1+1Rs)68

where:

Rs=|E{PyPx}|212Var{PyPx}69

Rs can be easily computed using the properties of the sideband correlators, Px and Py:

|E{PyPx}|2=|E{Py}|2|E{Px}|2=α2Ay2β2Ax2=γ2α4Ay4=γ2Cy270

The variance of PyPx can be computed as follows:

Var{PyPx}=(Var{Px}+|E{Px}|2)(Var{Py}+|E{Py}|2)−|E{Py}|2|E{Px}|2=2σ2(2σ2+β2Ax2+α2Ay2)=2σ2(2σ2+Cy(1+γ2))71

Using these results, we obtain the following for the inverse of Rs:

1Rs=1γ2Cyσ2[(1+γ2)+2Cyσ2]72

From Equation (65), we obtain the following:

1Rs=12γ2CyN0Tc[(1+γ2)+1CyN0Tc]73

Finally, it is possible to compute Var{Dφ}:

Var{Dφ}=18γ2CyN0Tc[(1+γ2)+2γ2+(1+γ2)2γ2CyN0Tc+1+γ2(γCyN0Tc)2+1(2γCyN0Tc)3]74

Equation (74) can be effectively truncated, retaining only the first two terms in the square brackets, leading to the following:

Var{Dφ}=18γ2CyN0Tc[(1+γ2)+2γ2+(1+γ2)2γ2CyN0Tc]75

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

REFERENCES

  1. ↵
    1. Alpay, D.,
    2. Luna-Elizarrarás, M. E.,
    3. Shapiro, M., &
    4. Struppa, D. C.
    (2014). Basics of functional analysis with bicomplex scalars, and bicomplex Schur analysis. Springer, Cham. https://doi.org/10.1007/978-3-319-05110-9
  2. ↵
    1. Beck, F. C.,
    2. Enneking, C.,
    3. Thölert, S., &
    4. Meurer, M.
    (2023). Impact of payload distortions on BeiDou B1I-B1C meta-signal ranging performance. Proc. of the 36th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2023), Denver, CO, 329–339. https://doi.org/10.33012/2023.19395
  3. ↵
    1. Borio, D.
    (2011). Squaring and cross-correlation codeless tracking: Analysis and generalisation. IET Radar, Sonar & Navigation, 5(9), 958–969. https://doi.org/10.1049/iet-rsn.2011.0235
  4. ↵
    1. Borio, D.
    (2014). Double phase estimator: New unambiguous binary offset carrier tracking algorithm. IET Radar, Sonar & Navigation, 8(7), 729–741. https://doi.org/10.1049/iet-rsn.2013.0306
  5. ↵
    1. Borio, D.
    (2023). Bicomplex representation and processing of GNSS signals. NAVIGATION, 70(4), 1–33. https://doi.org/10.33012/navi.621
  6. ↵
    1. Borio, D., &
    2. Susi, M.
    (2023). Bicomplex Kalman filter tracking for GNSS meta-signals. Proc. of the 36th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2023), Denver, CO, 3353–3373. https://doi.org/10.33012/2023.19233
  7. ↵
    1. Brown, R. G., &
    2. Hwang, P. Y. C.
    (2012). Introduction to random signals and applied Kalman filtering (4th ed.). Wiley. https://www.wiley.com/en-us/Introduction+to+Random+Signals+and+Applied+Kalman+Filtering+with+Matlab+Exercises%2C+4th+Edition-p-9780470609699
  8. ↵
    1. Casella, G., &
    2. Berger, R.
    (2001). Statistical inference. Duxbury Resource Center. https://books.google.com/books/about/Statistical_Inference.html?id=FAUVEAAAQBAJ
  9. ↵
    1. Cortés, I.,
    2. van der Merwe, J. R.,
    3. Lohan, E. S.,
    4. Nurmi, J., &
    5. Felber, W.
    (2023). Evaluation of low-complexity adaptive full direct-state Kalman filter for robust GNSS tracking. Sensors, 23(7), 3658. https://doi.org/10.3390/s23073658
  10. ↵
    1. Crosta, P.,
    2. Zoccarato, P.,
    3. Lucas, R., &
    4. De Pasquale, G.
    (2018). Dual frequency mass-market chips: Test results and ways to optimize PVT. Proc. of the 31st International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2018), Miami, FL, 323–333. https://doi.org/10.33012/2018.15882
  11. ↵
    1. Gao, Y.,
    2. Yao, Z., &
    3. Lu, M.
    (2020). High-precision unambiguous tracking technique for BDS B1 wideband composite signal. NAVIGATION, 67(3), 633–650. https://doi.org/10.1002/navi.377
  12. ↵
    1. Hodgart, M., &
    2. Blunt, P.
    (2007). Dual estimate receiver of binary offset carrier modulated signals for global navigation satellite systems. Electronics Letters, 43, 877–878. http://doi.org/10.1049/el:20071101
  13. ↵
    1. Issler, J.-L.,
    2. Paonni, M., &
    3. Eissfeller, B.
    (2010). Toward centimetric positioning thanks to L- and S-band GNSS and to meta-GNSS signals. Proc. of the ESA Workshop on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (NAVITEC), Noordwijk, Netherlands, 1–8. https://doi.org/10.1109/NAVITEC.2010.5708075
  14. ↵
    1. Kaplan, E. D., &
    2. Hegarty, C.
    (Eds.). (2017). Understanding GPS/GNSS: Principles and applications (3rd ed.). Artech House Publishers. https://us.artechhouse.com/Understanding-GPSGNSS-Principles-and-Applications-Third-Edition-P1871.aspx
  15. ↵
    1. Luo, Y.,
    2. Zhang, L., &
    3. El-Sheimy, N.
    (2018). An improved DE-KFL for BOC signal tracking assisted by FRFT in a highly dynamic environment. Proc. of the IEEE/ION Position, Location and Navigation Symposium (PLANS), Monterey, CA, 1525–1534. https://doi.org/10.1109/PLANS.2018.8373547
  16. ↵
    1. Misra, P., &
    2. Enge, P.
    (2006). Global Positioning System: Signals, measurements, and performance (2nd ed.). Ganga-Jamuna Press. https://books.google.com/books/about/Global_Positioning_System.html?id=5WJOywAACAAJ
  17. ↵
    1. Moriana, C.,
    2. Ortas, G.,
    3. Garbin, E.,
    4. Benedetti, E.,
    5. Boreham, N.,
    6. Boto, P.,
    7. Míguez, J.,
    8. García-Molina, J. A., &
    9. Melman, F.
    (2023). Evaluating performance of meta-signal exploitation in end user. Proc. of the 36th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2023), Denver, CO, 340–354. https://doi.org/10.33012/2023.19399
  18. ↵
    1. O’Driscoll, C., &
    2. Curran, J. T.
    (2018). Carrier phase tracking considerations for commodity SDR hardware. Proc. of the 31st International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2018), Miami, FL, 4182–4196. https://doi.org/10.33012/2018.16117
  19. ↵
    1. O’Driscoll, C.,
    2. Petovello, M., &
    3. Lachapelle, G.
    (2011). Choosing the coherent integration time for Kalman filter-based carrier-phase tracking of GNSS signals. GPS Solutions, 15, 345–356. https://doi.org/10.1007/s10291-010-0194-4
  20. ↵
    1. Ossmann, M.
    (2023). HackRF GitHub page [Online; last accessed 20 December 2023]. https://github.com/greatscottgadgets/hackrf
  21. ↵
    1. Paonni, M.,
    2. Curran, J.,
    3. Bavaro, M., &
    4. Fortuny-Guasch, J.
    (2014). GNSS meta signals: Coherently composite processing of multiple GNSS signals. Proc. of the 27th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2014), Tampa, FL, 2592–2601. https://www.ion.org/publications/abstract.cfm?articleID=12322
  22. ↵
    1. Price, G. B.
    (1991). An introduction to multicomplex spaces and functions (1st ed.). CRC Press. https://doi.org/10.1201/9781315137278
  23. ↵
    1. Schwalm, C.,
    2. Enneking, C., &
    3. Thoelert, S.
    (2020). Ziv-Zakai bound and multicorrelator compression for a Galileo E1 meta-signal. Proc. of the European Navigation Conference (ENC), Dresden, Germany, 1–9. https://doi.org/10.23919/ENC48637.2020.9317389
  24. ↵
    1. Sénant, E.,
    2. Gadat, B.,
    3. Charbonieras, C.,
    4. Roche, S.,
    5. Aubault, M., &
    6. Marmet, F.-X.
    (2018). Tentative new signals and services in upper L1 and S bands for Galileo evolutions. Proc. of the 31st International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2018), Miami, FL, 913–942. https://doi.org/10.33012/2018.15889
  25. ↵
    1. Susi, M.,
    2. Andreotti, M.,
    3. Aquino, M., &
    4. Dodson, A.
    (2017). Tuning a Kalman filter carrier tracking algorithm in the presence of ionospheric scintillation. GPS Solutions, 21, 1149–1160. https://doi.org/10.1007/s10291-016-0597-y
  26. ↵
    1. Susi, M., &
    2. Borio, D.
    (2020). Kalman filtering with noncoherent integrations for Galileo E6-B tracking. NAVIGATION, 67(3), 601–618. https://doi.org/10.1002/navi.380
  27. ↵
    1. Tang, X.,
    2. Falco, G.,
    3. Falletti, E., &
    4. Lo Presti, L.
    (2017). Complexity reduction of the Kalman filter-based tracking loops in GNSS receivers. GPS Solutions, 21, 685–699. https://doi.org/10.1007/s10291-016-0557-6
  28. ↵
    1. Tian, Z.,
    2. Cui, X.,
    3. Liu, G., &
    4. Lu, M.
    (2022). LPRA-DBT: Low-processing-rate asymmetrical dual-band tracking method for BDS-3 B1I and B1C composite signal. Proc. of the International Technical Meeting of the Institute of Navigation, Long Beach, CA, 1027–1038. https://doi.org/10.33012/2022.18243
  29. ↵
    1. Tsui, J. B.-Y.
    (2004). Fundamentals of Global Positioning System receivers: A software approach (2nd ed.). Wiley-Interscience. https://doi.org/10.1002/0471712582
  30. ↵
    1. u-blox
    . (2023). ZED-F9R module: Product summary [Online; last accessed: 17 May 2023]. https://content.u-blox.com/sites/default/files/ZED-F9R_ProductSummary_UBX-19048775.pdf
  31. ↵
    1. Vila-Valls, J.,
    2. Closas, P.,
    3. Navarro, M., &
    4. Fernandez-Prades, C.
    (2017). Are PLLs dead? A tutorial on Kalman filter-based techniques for digital carrier synchronization. IEEE Aerospace and Electronic Systems Magazine, 32(7), 28–45. https://doi.org/10.1109/MAES.2017.150260
  32. ↵
    1. Vilà-Valls, J.,
    2. Linty, N.,
    3. Closas, P.,
    4. Dovis, F., &
    5. Curran, J. T.
    (2020). Survey on signal processing for GNSS under ionospheric scintillation: Detection, monitoring, and mitigation. NAVIGATION, 67(3), 511–536. https://doi.org/10.1002/navi.379
  33. ↵
    1. Wang, C.,
    2. Cui, X.,
    3. Ma, T.,
    4. Zhao, S., &
    5. Lu, M.
    (2017). Asymmetric dual-band tracking technique for optimal joint processing of BDS B1I and B1C signals. Sensors, 17(10), 1–16. https://doi.org/10.3390/s17102360
  34. ↵
    1. Won, J.-H.,
    2. Pany, T., &
    3. Eissfeller, B.
    (2012). Characteristics of Kalman filters for GNSS signal tracking loop. IEEE Transactions on Aerospace and Electronic Systems, 48(4), 3671–3681. https://doi.org/10.1109/TAES.2012.6324756
  35. ↵
    1. Zhang, W.,
    2. Yao, Z., &
    3. Lu, M.
    (2019). WHAT: wideband high-accuracy joint tracking technique for BDS B1 composite signal. Proc. of the IEEE 9th International Conference on Electronics Information and Emergency Communication (ICEIEC), Beijing, China, 178–182. https://doi.org/10.1109/ICEIEC.2019.8784630
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NAVIGATION: Journal of the Institute of Navigation: 71 (4)
NAVIGATION: Journal of the Institute of Navigation
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GNSS Meta-Signal Tracking Using a Bicomplex Kalman Filter
Daniele Borio, Melania Susi
NAVIGATION: Journal of the Institute of Navigation Dec 2024, 71 (4) navi.674; DOI: 10.33012/navi.674

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GNSS Meta-Signal Tracking Using a Bicomplex Kalman Filter
Daniele Borio, Melania Susi
NAVIGATION: Journal of the Institute of Navigation Dec 2024, 71 (4) navi.674; DOI: 10.33012/navi.674
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  • Article
    • Abstract
    • 1 INTRODUCTION
    • 2 SIGNAL MODEL AND BICOMPLEX REPRESENTATION
    • 3 TRIPLE-LOOP TRACKING
    • 4 TRIPLE-LOOP KALMAN FILTER
    • 5 EXPERIMENTAL SETUP
    • 6 EXPERIMENTAL RESULTS
    • 7 CONCLUSIONS
    • HOW TO CITE THIS ARTICLE
    • CONFLICT OF INTEREST
    • A VARIANCE OF THE CODE DISCRIMINATOR OUTPUT
    • B VARIANCE OF THE CARRIER DISCRIMINATOR OUTPUT
    • REFERENCES
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Keywords

  • dual-frequency
  • GNSS
  • Kalman filter
  • meta-signals
  • triple-loop

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