Meta-Signal Processing with Data/Pilot Combining for Beidou B2 Signals

Abstract

Beidou Navigation Satellite System (BDS) third generation satellites currently broadcast Open Service (OS) signals into two closely spaced Radio Frequencies (RFs) in the B2 band. These are the B2a and B2b signal components, which form the current implementation of the Asymmetric Constant-Envelope Binary Offset Carrier (ACE-BOC) modulation. The B2a signal features both a data and a pilot channel, whereas the B2b component is data only with data symbols of 1 ms duration. The absence of a pilot channel and the fast data rate make the processing of the B2b component challenging. Tracking performance can, however, be improved by jointly processing the B2a and B2b components. In this respect, meta-signal approaches are investigated for jointly processing the B2a and B2b signals. Two meta-signal tracking architectures are proposed: the first considers the pilot channel of the B2a component and the data channel of the B2b signal. The second exploits all the power available and also implements data/pilot combining on the B2a channel. Both architectures allow the extension of the integration time beyond the data symbol duration using non-coherent approaches. Theoretical results are supported by simulations and real data analysis performed using a custom Software Defined Radio (SDR) receiver. Simulation and experimental results clearly show the benefits of the meta-signal approach, which can be effectively adopted for the processing of asymmetric modulations such as the current implementation of the ACE-BOC, which lacks a pilot channel on the B2b component.

1. Introduction

Modern Global Navigation Satellite Systems (GNSSs) broadcast several signal components on different frequencies. This is the case with Beidou Navigation Satellite System (BDS), whose third generation satellites broadcast signals on five frequencies [1]. Some of the components transmitted are separated by few MHz. This is the case of the B1I and B1C components, which are broadcast into the L1 frequency band and are separated by only $14.322$ MHz. In the B2 band, centred around the $1191.795$ MHz frequency, two signals are broadcast around adjacent frequencies [1,2,3]. These are the B2a [2] and B2b [3] signals, and represent the current implementation of the Asymmetric Constant-Envelope Binary Offset Carrier (ACE-BOC) [4,5]. The availability of signals from different frequencies can be exploited to design advanced receiver algorithms, which jointly process the different components. In particular, it is possible to treat different frequency components as a single entity, which is the so-called GNSS meta-signal [6,7]. A meta-signal features a Gabor bandwidth much larger than those of the individual sideband components. A large Gabor bandwidth is a pre-condition to obtain high-accuracy pseudorange measurements [8]. While meta-signals have the potential to provide improved receiver performance, specialized algorithms are required to avoid false code locks and ambiguous pseudorange measurements. A GNSS receiver produces code measurements by maximizing the correlation function of the received signals. When combining components from different frequencies, signals with multi-peaked correlation functions are obtained. It is important to avoid the receiver locking on to a secondary correlation peak, leading to biased code measurements. A possible approach to avoid false code locks is to use triple-loop tracking architectures [7,9] that project an ambiguous one-dimensional correlation function into a two-dimensional unambiguous domain. In this respect, several solutions have been investigated for the BI1/BIC meta-signal [10,11,12]. All of these solutions track an additional component, the subcarrier, which accounts for relative variations between sideband components. Less work has been conducted with respect to the meta-signal obtained by jointly considering the B2a and B2b components. A software receiver processing the B1C and B2a signals is detailed in [13,14]. A second Software Defined Radio (SDR) receiver implementation for different BDS signals is described in [15]. In both cases, it was not possible to consider the B2b component since its specifications were published only in 2020 [3].

Meta-signal processing can also be implemented at the measurement level, where the carrier and pseudoranges from the sideband components are combined [8]. This approach was used in [16] to test different BDS meta-signal combinations. The B2a/B2b case was, however, not considered, since the receiver used for the tests was able to process only the B2I component, a Beidou second generation signal, which is being gradually replaced by the B2b component.

In addition to signals transmitted on different frequencies, modern GNSSs also broadcast several components for a given frequency. A common configuration is given by the simultaneous transmission of a data and a pilot channel for a given frequency [17,18,19,20]. This is the case of several Galileo and BDS signals [2,21]. A pilot channel allows one to extend the integration time used to estimate the signal correlation function. Moreover, a four quadrant Phase Lock Loop (PLL) [22] can be adopted to further improve tracking performance. While both pilot and data components are available for the B1C signal, the B1I modulation is data only. Thus, asymmetric tracking techniques [11] are required for the processing of the resulting meta-signal. These techniques are also asymmetric because the two sideband components feature different spectral characteristics. Similar considerations apply for the signals broadcast into the B2 frequency band: while the full ACE-BOC modulation foresees the presence of a data and pilot channel on both sideband components [4,5], its current implementation adopts a data only channel for the B2b signal [3]. When both data and pilot channels are available, data/pilot combining strategies can be used to further improve receiver performance [17,20,23,24]. Data/pilot combining strategies allow the receiver to recover all the available power and improve performance. While the possibility of adopting data/pilot combining strategies for meta-signal processing is mentioned by [7], a complete analysis on the subject is, however, missing in the literature. This is the main focus of this paper, which develops two meta-signal tracking strategies for the processing of the composite B2a/B2b signal. The first considers a single channel for both components: the B2a pilot channel and the B2b data signal. The second exploits all of the signal power available and implements data/pilot combining for the B2a component. Both strategies implement non-coherent integrations that allow the extension of the integration time beyond the symbol duration of the data channels. This is particularly relevant for the B2b component, which features a data symbol duration of 1 ms. This relatively high data rate is required for the intended use of the B2b signal that broadcasts Precise Point Positioning (PPP) corrections for China and surrounding areas [25]. Non-coherent integrations are thus required to implement reliable signal tracking.

The proposed tracking strategies are analysed from a theoretical point of view, and analytical formulas are provided for the tracking jitter of the proposed meta-signal Delay Lock Loops (DLLs) and PLLs. Theoretical results are supported by semi-analytic simulations: the benefits of meta-signal processing and data/pilot combination clearly emerge.

The proposed tracking architectures have also been implemented in a custom SDR receiver developed in Python. The receiver was used to test the proposed solutions using In-Phase Quadrature (I/Q) data collected using a wideband front-end able to recover the full ACE-BOC modulation. Experimental results further support theoretical and simulation findings. Both meta-signal approaches enable a more reliable tracking of the B2b component that benefits from the information brought by the B2a signal. The architecture implementing data/pilot combining allows one to recover all the signal power available and further reduce the tracking jitter.

The remainder of this paper is organized as follows: the signal and system models adopted in the paper are described in Section 2. Meta-signal dual-frequency tracking architectures are detailed in Section 3, whereas the associated tracking jitter is derived in Section 4. Semi-analytic simulation results are provided in Section 5, and experimental findings are discussed in Section 6. Finally, conclusions are drawn in Section 7.

2. Signal and System Model

The Alternative Binary Offset Carrier (AltBOC) and ACE-BOC modulations feature two data and two pilot channels jointly broadcast on two close frequencies [5,26]. In this case, the GNSS signal at the receiver antenna can be modelled as [22,24]:

$$y\left(t\right)=y_{ls}\left(t\right)+y_{us}\left(t\right)+\eta \left(t\right),$$

(1)

where $y_{ls}\left(t\right)$ and $y_{us}\left(t\right)$ are the lower and upper sideband components, respectively. In the following, the subscript ‘ls’ is used to indicate quantities related to the lower sideband component, whereas ‘us’ refers to elements belonging to the upper sideband signal. t is the time variable, and $\eta \left(t\right)$ is an Additive White Gaussian Noise (AWGN) process. The two sideband components in (1) can be modelled as [22,24]:

$$\begin{array}{cc}\hfill y_{ls}\left(t\right)& =\sqrt{2C_{ls}{\alpha }_{ls}}{d}_{ls}\left(t-{\tau }_0\right)c_{d,ls}\left(t-{\tau }_0\right)cos\left(2\pi (f_{RF,ls}+f_{0,ls})t+{\phi }_{0,ls}+{\phi }_{d,ls}\right)+\hfill \\ \hfill & \sqrt{2C_{ls}}{c}_{p,ls}\left(t-{\tau }_0\right)cos\left(2\pi (f_{RF,ls}+f_{0,ls})t+{\phi }_{0,ls}\right)\hfill \end{array}$$

(2)

and

$$\begin{array}{cc}\hfill y_{us}\left(t\right)& =\sqrt{2C_{us}{\alpha }_{us}}{d}_{us}\left(t-{\tau }_0\right)c_{d,us}\left(t-{\tau }_0\right)cos\left(2\pi (f_{RF,us}+f_{0,us})t+{\phi }_{0,us}+{\phi }_{d,us}\right)+\hfill \\ \hfill & \sqrt{2C_{us}}{c}_{p,us}\left(t-{\tau }_0\right)cos\left(2\pi (f_{RF,us}+f_{0,us})t+{\phi }_{0,us}\right),\hfill \end{array}$$

(3)

where $C_{ls}$ and $C_{us}$ are the power levels received on the pilot components of the lower and upper sideband components, respectively. ${\alpha }_{ls}$ and ${\alpha }_{us}$ are the ratios between the power levels of the data and pilot signals of the two sideband components. $c_{d,ls}\left(t\right)$, $c_{p,ls}\left(t\right)$, $c_{d,us}\left(t\right)$ and $c_{p,us}\left(t\right)$ are the four ranging codes, with index d indicating quantities related to data channels and p referring to pilot components. $f_{RF,ls}$ and $f_{RF,us}$ are the Radio Frequencies (RFs) of the two sideband components, and $f_{0,ls}$ and $f_{0,us}$ the corresponding Doppler shifts introduced by the communication channel. ${\tau }_0$ is a delay term considered common to both components. Finally, ${\phi }_{0,ls}$ and ${\phi }_{0,us}$ are the carrier phase affecting the two sideband components. Note that data and pilot signals can be transmitted in phase, quadrature or opposition of phase. For this reason, phase terms, ${\phi }_{d,ls}$ and ${\phi }_{d,us}$, have been introduced in (2) and (3). These phase terms are, however, known at the receiver side and specified in the Signal-In-Space (SIS) ICD [2,3,21]. In the AltBOC and ACE-BOC, ${\phi }_{d,ls}={\phi }_{d,us}=\pi /2$. In both (2) and (3), the Doppler effect on the code components has been neglected. Moreover, a single satellite signal has been assumed: while several components are broadcast by different satellites, a receiver is able to effectively separate them through the orthogonality property of the ranging codes modulating the different components.

Signal (1) is filtered, downconverted and digitized. Different downconversion strategies can be assumed [7]: for instance, a single wide-band front-end able to capture both sideband components can be adopted. Alternatively, two separate synchronized narrow-band front-ends can be used. In the latter case, two synchronous data streams are obtained:

$$\begin{array}{cc}\hfill y_{ls}\left[n\right]=y_{ls}\left(n{T}_s\right)& =\left[\sqrt{C_{ls}{\alpha }_{ls}}{d}_{ls}\left(n{T}_s-{\tau }_0\right)c_{d,ls}\left(n{T}_s-{\tau }_0\right)e^{j{\phi }_{d,ls}}\right.\hfill \\ \hfill & \left.+\sqrt{C_{ls}}{c}_{p,ls}\left(n{T}_s-{\tau }_0\right)\right]e^{j2\pi f_{0,ls}n{T}_s+j{\phi }_{0,ls}}+{\eta }_{ls}\left[n\right]\hfill \end{array}$$

(4)

and

$$\begin{array}{cc}\hfill y_{us}\left[n\right]=y_{us}\left(n{T}_s\right)& =\left[\sqrt{C_{us}{\alpha }_{us}}{d}_{us}\left(n{T}_s-{\tau }_0\right)c_{d,us}\left(n{T}_s-{\tau }_0\right)e^{j{\phi }_{d,us}}\right.\hfill \\ \hfill & \left.+\sqrt{C_{us}}{c}_{p,us}\left(n{T}_s-{\tau }_0\right)\right]e^{j2\pi f_{0,us}n{T}_s+j{\phi }_{0,us}}+{\eta }_{us}\left[n\right],\hfill \end{array}$$

(5)

where $T_s$ is the sampling interval adopted to recover the two sideband components and n the time index. Note that both (4) and (5) include a noise term, ${\eta }_{ls}\left[n\right]$ and ${\eta }_{us}\left[n\right]$. These two processes derive from $\eta \left(t\right)$ in (1), and are complex circularly symmetric AWGNs. They are assumed independent, since they are obtained considering different frequency bands, and with the same variance. More specifically, the real and imaginary parts of ${\eta }_{ls}\left[n\right]$ and ${\eta }_{us}\left[n\right]$ are assumed independent each with variance, ${\sigma }_{\eta }^2$. While this variance depends on several factors, the following model is adopted:

$${\sigma }_{\eta }^2=N_0{B}_{Rx},$$

(6)

where $N_0$ is the Power Spectral Density (PSD) of the input noise, $\eta \left(t\right)$ and $B_{Rx}$ the front-end one-sided bandwidth. The same bandwidth is assumed for the synchronous front-ends used for the recovery of the two sideband components. The front-end bandwidth can be effectively approximated as:

$$B_{Rx}\approx \frac{f_s}{2},$$

(7)

where $f_s=\frac{1}{T_s}$ is the frequency adopted to sample the input analogue signal (1). The ratio between the received useful signal power and $N_0$ defines the Carrier-To-Noise Power Spectral Density Ratio ($C/N_0$), a key parameter used to assess receiver performance. In this case, several $C/N_0$ values can specified depending on the signal component considered. Since different signal combinations are analysed in the following, the exact $C/N_0$ definition is specified for each case. While (4) and (5) are obtained considering two separate front-ends, a similar model is found using a single wide-band device. These two options and their equivalence are discussed in [7].

The signal model defined by (4) and (5) can be rewritten in terms of common phase and frequency parameters. In particular, the following transformation can be adopted:

$$\left[\begin{array}{c}{f}_0\\ f_{0,sub}\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\left[\begin{array}{c}{f}_{0,us}\\ f_{0,ls}\end{array}\right]\mathrm{a}\mathrm{n}\mathrm{d}\left[\begin{array}{c}{\phi }_0\\ {\phi }_{0,sub}\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\left[\begin{array}{c}{\phi }_{0,us}\\ {\phi }_{0,ls}\end{array}\right].$$

(8)

Thus, (4) and (5) become

$$\begin{array}{cc}\hfill y_{ls}\left[n\right]=y_{ls}\left(n{T}_s\right)& =\left[\sqrt{C_{ls}{\alpha }_{ls}}{d}_{ls}\left(n{T}_s-{\tau }_0\right)c_{d,ls}\left(n{T}_s-{\tau }_0\right)e^{j{\phi }_{d,ls}}\right.\hfill \\ \hfill & \left.+\sqrt{C_{ls}}{c}_{p,ls}\left(n{T}_s-{\tau }_0\right)\right]e^{j2\pi \left(f_0-f_{0,sub}\right)n{T}_s+j\left({\phi }_0-{\phi }_{0,sub}\right)}+{\eta }_{ls}\left[n\right]\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill y_{us}\left[n\right]=y_{us}\left(n{T}_s\right)& =\left[\sqrt{C_{us}{\alpha }_{us}}{d}_{us}\left(n{T}_s-{\tau }_0\right)c_{d,us}\left(n{T}_s-{\tau }_0\right)e^{j{\phi }_{d,us}}\right.\hfill \\ \hfill & \left.+\sqrt{C_{us}}{c}_{p,us}\left(n{T}_s-{\tau }_0\right)\right]e^{j2\pi \left(f_0+f_{0,sub}\right)n{T}_s+j\left({\phi }_0+{\phi }_{0,sub}\right)}+{\eta }_{us}\left[n\right].\hfill \end{array}$$

(10)

The matrix appearing in Equation (8) defines a Hadamard transform [27] of order 2 and brings the sideband parameters, $f_{0,us}$, $f_{0,ls}$, ${\phi }_{0,us}$ and ${\phi }_{0,ls}$, into the meta-signal domain. In particular, the meta-signal carrier and subcarrier Doppler frequencies, $f_0$ and $f_{0,sub}$, are obtained. Frequency $f_0$ accounts for common variations affecting both components, whereas $f_{0,sub}$ accounts for differential variations between components. Similar considerations apply for the phase terms, ${\phi }_0$ and ${\phi }_{0,sub}$, which are the meta-signal carrier and subcarrier phases.

The receiver generates local replicas of the data/pilot codes and of the carrier terms in (9) and (10). These local replicas are generated according to the receiver best estimates of the code delay, ${\tau }_0$, and of the Doppler and phase unknowns. These replicas are correlated with the digital signals in (9) and (10). For instance, for the pilot lower sideband component, the following process is implemented:

$$P_{ls,p,i}=\frac{1}{N}\sum _{n=iN}^{(i+1)N-1}{y}_{ls}\left[n\right]c_{p,ls}\left(n{T}_s-\tau \right)e^{j2\pi \left(f_d-f_{d,sub}\right)n{T}_s+j\left(\phi -{\phi }_{sub}\right)},$$

(11)

where N is the number of samples used in the integration process and defines the coherent integration time, $T_c=N{T}_s$. N is limited by the duration of the navigation symbols on the data channels. In the following, a common coherent integration time is assumed for all the components. It is determined by the data symbol with the shortest duration. For instance, the Beidou B2a data channel is charactered by a data symbol duration of 5 ms [2], whereas the B2b component has a data duration equal to 1 ms [3]. In this case, $T_c=1$ ms. The integration time can be further extended either coherently, for instance on the pilot channels or on data components with longer symbol durations. Alternatively, non-coherent approaches can be adopted. For this reason, several correlators are computed considering consecutive blocks of input samples; this is accounted for by the index i introduced in (11). This index denotes correlators computed for different consecutive time epochs. $\tau$ is the best estimate of the signal code delay, ${\tau }_0$, and $\phi$ and ${\phi }_{sub}$ are the phases tested by the receiver. Similarly, $f_d$ and $f_{d,sub}$ are the meta-signal carrier and subcarrier Doppler frequencies estimated by the receiver. In the following, frequency lock conditions are assumed and residual Doppler effects are neglected. More specifically, it is assumed that $f_d\approx f_0$ and $f_{d,sub}\approx f_{0,sub}$. A similar process is adopted for the data channel and for the upper sideband components. In this way, a set of correlators is found:

$$\begin{array}{cc}\hfill P_{ls,p,i}& =A_{ls}{R}_{ls,p}\left(\mathsf{\Delta }\tau \right)e^{j(\mathsf{\Delta }\phi -\mathsf{\Delta }{\phi }_{sub})}+{\eta }_{ls,p,i}\hfill \\ \hfill P_{ls,d,i}& =k_{ls}{A}_{ls}{d}_{ls,i}{R}_{ls,d}\left(\mathsf{\Delta }\tau \right)e^{j(\mathsf{\Delta }\phi -\mathsf{\Delta }{\phi }_{sub})}+{\eta }_{ls,d,i}\hfill \\ \hfill P_{us,p,i}& =A_{us}{R}_{us,p}\left(\mathsf{\Delta }\tau \right)e^{j(\mathsf{\Delta }\phi +\mathsf{\Delta }{\phi }_{sub})}+{\eta }_{us,p,i}\hfill \\ \hfill P_{us,d,i}& =k_{us}{A}_{us}{d}_{us,i}{R}_{us,d}\left(\mathsf{\Delta }\tau \right)e^{j(\mathsf{\Delta }\phi +\mathsf{\Delta }{\phi }_{sub})}+{\eta }_{us,d,i},\hfill \end{array}$$

(12)

where $\mathsf{\Delta }\tau =\tau -{\tau }_0$ is the residual delay error. Similarly, $\mathsf{\Delta }\phi =\phi -{\phi }_0$ and $\mathsf{\Delta }{\phi }_{sub}={\phi }_{sub}-{\phi }_{sub}$ are the residual meta-signal carrier and subcarrier phases. As for the previous equations, indices $ls$ and $us$ denote quantities related to the lower and upper sideband components, respectively. Subscripts p and d are used to indicate elements related to pilot and data channels. Finally, i is the epoch index. Amplitude factors, $A_{ls}$ and $A_{us}$, are given by

$$A_{ls}=\sqrt{C_{ls}} \mathrm{a}\mathrm{n}\mathrm{d} A_{us}=\sqrt{C_{us}}$$

(13)

whereas the known constants, $k_{ls}$ and $k_{us}$, take into account phase and amplitude differences between the data and pilot correlators:

$$k_{ls}=\sqrt{{\alpha }_{ls}}{e}^{j{\phi }_{d,ls}}\mathrm{a}\mathrm{n}\mathrm{d}{k}_{us}=\sqrt{{\alpha }_{us}}{e}^{j{\phi }_{d,us}}.$$

(14)

In (12), $R_{ls,p}(·)$, $R_{ls,d}(·)$, $R_{us,p}(·)$ and $R_{us,p}(·)$ denote the autocorrelation functions of the different signal components. The effect of front-end filtering has been neglected. $d_{ls,i}=d_{ls}\left[iN{T}_s\right]$ and $d_{us,i}=d_{us}\left[iN{T}_s\right]$ are the navigation symbols transmitted on the lower and upper sideband components and considered constant over the coherent integration time.

Finally, ${\eta }_{ls,p,i}$, ${\eta }_{ls,d,i}$, ${\eta }_{us,p,i}$ and ${\eta }_{us,d,i}$ are four noise components obtained from ${\eta }_{ls}\left[n\right]$ and ${\eta }_{us}\left[n\right]$. They are zero mean, independent and identically distributed (i.i.d.), where independence derives from the fact that they were obtained from the correlation of ${\eta }_{ls}\left[n\right]$ and ${\eta }_{us}\left[n\right]$ with practically orthogonal codes and considering signals from different frequencies. The variance of the real and imaginary parts of these four noise terms is given by

$${\sigma }^2=\frac{{\sigma }_{\eta }^2}{N}=\frac{N_0{B}_{Rx}}{N}\approx \frac{N_0{f}_s}{2N}=\frac{N_0{f}_s}{2N}=\frac{N_0}{2N{T}_s}=\frac{N_0}{2T_c}.$$

(15)

A joint data/pilot correlator can be obtained considering the approach derived in [24], where symbols are removed from the data correlators using a soft decision based on the hyperbolic tangent function:

$$\begin{array}{cc}\hfill P_{ls,i}& =P_{ls,p,i}+tanh\left(\frac{{\tilde{A}}_{ls}}{{\tilde{\sigma }}^2}\Re \left\{P_{ls,d,i}{k}_{ls}^{\ast }\right\}\right)P_{ls,d,i}{k}_{ls}^{\ast }\hfill \\ \hfill P_{us,i}& =P_{us,p,i}+tanh\left(\frac{{\tilde{A}}_{us}}{{\tilde{\sigma }}^2}\Re \left\{P_{us,d,i}{k}_{us}^{\ast }\right\}\right)P_{us,d,i}{k}_{us}^{\ast },\hfill \end{array}$$

(16)

where ${\tilde{A}}_{ls}$ and ${\tilde{A}}_{us}$ are estimates of $A_{us}$ and $A_{ls}$, respectively. Similarly, ${\tilde{\sigma }}^2$ is an estimate of the variance of the noise term affecting the correlators. These estimates are continuously produced by the receiver, and can be obtained using standard approaches. The hyperbolic tangent terms in (16) can be considered as soft-estimates of the data symbols. Moreover, they account for the reliability of the information provided by the data channels: for low signal amplitudes, ${\tilde{A}}_{ls}$ and ${\tilde{A}}_{us}$ tend to zero and the contribution of the data components in (16) becomes negligible. Thus, the composite correlators converge to the pilot components only. After removing the impact of the data symbols as in (16), the integration time can be further extended and the following final correlators are found:

$$\begin{array}{cc}\hfill P_{ls}& =\sum _{i=0}^{K-1}{P}_{ls,i}=\sum _{i=0}^{K-1}{P}_{ls,p,i}+\sum _{i=0}^{K-1}tanh\left(\frac{{\tilde{A}}_{ls}}{{\tilde{\sigma }}^2}\Re \left\{P_{ls,d,i}{k}_{ls}^{\ast }\right\}\right)P_{ls,d,i}{k}_{ls}^{\ast }\hfill \\ \hfill P_{us}& =\sum _{i=0}^{K-1}{P}_{us,i}=\sum _{i=0}^{K-1}{P}_{us,p,i}+\sum _{i=0}^{K-1}tanh\left(\frac{{\tilde{A}}_{us}}{{\tilde{\sigma }}^2}\Re \left\{P_{us,d,i}{k}_{us}^{\ast }\right\}\right)P_{us,d,i}{k}_{us}^{\ast }.\hfill \end{array}$$

(17)

Expressions in (17) are general and assume the presence of data and pilot channels in both sideband components. However, some terms may be missing. This is the case, for instance of the B2b signal, which features a data channel only [3]. For this signal, the corresponding pilot components should be removed and (17) degenerates to:

$$\begin{array}{cc}\hfill P_{ls}& =\sum _{i=0}^{K-1}{P}_{ls,p,i}+\sum _{i=0}^{K-1}tanh\left(\frac{{\tilde{A}}_{ls}}{{\tilde{\sigma }}^2}\Re \left\{P_{ls,d,i}{k}_{ls}^{\ast }\right\}\right)P_{ls,d,i}{k}_{ls}^{\ast }\hfill \\ \hfill P_{us}& =\sum _{i=0}^{K-1}tanh\left(\frac{{\tilde{A}}_{us}}{{\tilde{\sigma }}^2}\Re \left\{P_{us,d,i}{k}_{us}^{\ast }\right\}\right)P_{us,d,i}{k}_{us}^{\ast }\hfill \end{array}$$

(18)

Finally, different combinations of coherent/non-coherent integrations can be adopted for the upper and lower sideband components. As already mentioned, this may depend for instance on different data symbol durations. The data correlator within the hyperbolic tangent function should be coherently integrated on the longest possible duration in order to improve the soft symbol estimation process. A common total integration time,

$$T_u=KN{T}_s$$

(19)

is, however, assumed for the different components. The correlators discussed are integrated in the tracking loops described in the next section, and $T_u$ is also the update rate of such loops.

Finally, in addition to the Prompt correlators detailed above, a GNSS receiver also computes Early and Late correlators that are used to estimate the code delay, ${\tau }_0$. These correlators are obtained by considering local code replicas delayed and advanced by a fixed amount, the Early-minus-Late spacing. The final Early and Late correlators are found as:

$$\begin{array}{cc}\hfill E_{ls}& =\sum _{i=0}^{K-1}{E}_{ls,p,i}+\sum _{i=0}^{K-1}tanh\left(\frac{{\tilde{A}}_{ls}}{{\tilde{\sigma }}^2}\Re \left\{P_{ls,d,i}{k}_{ls}^{\ast }\right\}\right)E_{ls,d,i}{k}_{ls}^{\ast }\hfill \\ \hfill E_{us}& =\sum _{i=0}^{K-1}{E}_{us,p,i}+\sum _{i=0}^{K-1}tanh\left(\frac{{\tilde{A}}_{us}}{{\tilde{\sigma }}^2}\Re \left\{P_{us,d,i}{k}_{us}^{\ast }\right\}\right)E_{us,d,i}{k}_{us}^{\ast }\hfill \\ \hfill L_{ls}& =\sum _{i=0}^{K-1}{L}_{ls,p,i}+\sum _{i=0}^{K-1}tanh\left(\frac{{\tilde{A}}_{ls}}{{\tilde{\sigma }}^2}\Re \left\{P_{ls,d,i}{k}_{ls}^{\ast }\right\}\right)L_{ls,d,i}{k}_{ls}^{\ast }\hfill \\ \hfill L_{us}& =\sum _{i=0}^{K-1}{L}_{us,p,i}+\sum _{i=0}^{K-1}tanh\left(\frac{{\tilde{A}}_{us}}{{\tilde{\sigma }}^2}\Re \left\{P_{us,d,i}{k}_{us}^{\ast }\right\}\right)L_{us,d,i}{k}_{us}^{\ast },\hfill \end{array}$$

(20)

where the computation of the single epoch data/pilot Early and Late correlators, $E_{ls,p,i}$, $L_{ls,p,i}$, $E_{us,p,i}$ and $L_{ls,p,i}$ is not detailed here, since it is similar to that of the Prompt correlators and can be found in standard textbooks on GNSS [22]. Note that in (20), soft symbol estimation and removal is performed using the information from the Prompt correlators. A joint code discriminator using Early and Late correlators is described in the next section.

3. Combined Dual-Frequency Tracking

The composite correlators derived in the previous section can be integrated in a dual-frequency meta-signal tracking loop, which jointly exploits information from the two sideband components. In [7], a general meta-signal tracking loop architecture was derived using bicomplex numbers. This architecture can be restated using standard components according to the schematic representation provided in Figure 1.

This representation is general, and can accommodate different correlation processes in addition to the ones discussed in Section 3. The upper sideband component, $y_{us}\left[n\right]$, enters a correlation block, and is used to produce the Early, Late and Prompt correlators. The upper sideband Prompt correlator, $P_{us}$, is passed to a phase loop discriminator that provides an estimate of the phase error on the upper sideband component. The phase discriminator output on the upper sideband component is denoted as $D_{\phi ,us}$, and provides an estimate of the residual error on ${\phi }_{0,us}$.

Similarly, the lower sideband component, $y_{ls}\left[n\right]$, is processed by a second correlation block, which produces a second set of correlators. Also in this case, the Prompt correlator, $P_{ls}$, is passed to a phase discriminator that provides the lower sideband phase discriminator output, $D_{\phi ,ls}$. Up to this point, the phase estimates on the two components have been kept separate as in (4) and (5). A Hadamard transform of order two is then applied to $D_{\phi ,us}$ and $D_{\phi ,ls}$:

$$\left[\begin{array}{c}{D}_{{\phi }_c}\\ D_{{\phi }_{sub}}\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\left[\begin{array}{c}{D}_{\phi ,us}\\ D_{\phi ,ls}\end{array}\right].$$

(21)

In this way, $D_{{\phi }_c}$ and $D_{{\phi }_{sub}}$ are obtained. $D_{{\phi }_c}$ is the discriminator output of the meta-signal carrier phase, and is obtained as the average of the upper and lower sideband phase discriminator outputs. Similarly, $D_{{\phi }_{sub}}$ is the discriminator output of the meta-signal subcarrier component, which accounts for differential variations between sideband components. $D_{{\phi }_{sub}}$ is the semi-difference of the upper and lower sideband phase discriminator outputs. While meta-signal carrier and subcarrier phase discriminator outputs have been obtained here through a Hadamard transform, carrier and subcarrier discriminators jointly processing sideband Prompt correlators can be used [7]. The actual carrier discriminator on the two sideband components depends on the type of correlators produced. If (17) or (18) are considered, four-quadrant arctangent discriminators can be adopted.

The carrier and subcarrier discriminator outputs, $D_{{\phi }_c}$ and $D_{{\phi }_{sub}}$, are then independently filtered and carrier and subcarrier Doppler estimates are produced. An inverse Hadamard transform is then used to find the Doppler estimates for the sideband components. These estimates are used to drive the carrier Numerically Controlled Oscillators (NCOs) of the two correlation blocks depicted in Figure 1.

The correlators from both components are jointly used for the estimation of the code delay. In this respect, a common code delay is assumed for both sideband components. The correlators enter a joint code discriminator block that produces a delay error that drives the DLL operations. The discriminator used here is a form of non-coherent Early-minus-Late envelope discriminator [7,22]:

$$D_{\tau }=\frac{|E_{ls}|+\gamma |E_{us}|-|L_{ls}|-\gamma |L_{us}|}{|E_{ls}|+\gamma |E_{us}|+|L_{ls}|+\gamma |L_{us}|},$$

(22)

where Early and Late correlators have been defined in (20), and $\gamma$ is the ratio between the amplitudes of two channel components:

$$\gamma =\frac{\sqrt{C_{us}}}{\sqrt{C_{ls}}}.$$

(23)

The code discriminator output is filtered and a code rate estimate is obtained. The code rate is then used to drive the two code NCOs integrated within the correlation blocks used to process the two sideband components.

The processing scheme detailed in Figure 1 provides a simple way for implementing a triple-loop architecture made of a DLL, a PLL and a Subcarrier Phase Lock Loop (SPLL). The sideband components are jointly used to estimate a common carrier phase term, which is affected by a reduced equivalent noise as compared to the individual sideband phases. Moreover, the subcarrier term is characterized by a nominal frequency, which is significantly lower than that of the individual sideband components. For this reason, the subcarrier loop filter can be designed adopting a small equivalent loop bandwidth. A single DLL benefits from the availability of correlators from two frequencies.

While the main focus of this paper are the correlators detailed in Section 2, different correlator blocks can be implemented within the architecture depicted in Figure 1. Three possible correlation blocks compatible with Figure 1 are shown in Figure 2. In Figure 2a, a pure pilot correlation block is depicted. It is obtained by neglecting the data channel and setting $k_{ls}$ or $k_{us}$ equal to zero.

After carrier and code wipe-off, the samples are coherently integrated and three correlators are produced. A data correlation block is described in Figure 2b. In this case, soft-bit estimation and removal is implemented. Bits are estimated from the Prompt correlators using the hyperbolic tangent function introduced in the previous section. Finally, Figure 2c shows the correlation block for the data/pilot architecture proposed in [24] and detailed in (17). After soft-bit removal, the integration time is extended on both data and pilot channels. Correlators from the two components are then combined to produce $P_{ls}$ or $P_{us}$.

In the following, two dual-frequency meta-signal tracking configurations are considered. As already discussed, the B2b signal is data only. Thus, the correlation block adopted for processing the B2b components is the one depicted in Figure 2b, where data symbols are estimated and removed. The B2a signal also features a pilot component. Thus, the following configurations are considered:

• Dual-frequency, single channel: pure-pilot processing is adopted for the B2a component according to Figure 2b. In this case, a single channel is considered for each sideband component: the pilot channel for the B2a signal and the data channel for the B2b component.
• Dual-frequency, dual channel: the data/pilot combination is adopted for the B2a component. In this case, all the energy available for the ACE-BOC modulation is recovered.

The performance of these two configurations is analysed in the following.

Other types of tracking channels can be used for processing the B2b components. For instance, a squaring PLL with non-coherent integrations can be adopted [28]. The solution with soft-bit estimation is however preferred here since the correlators generated by the blocks shown in Figure 2 can be directly used by the joint DLL discriminator.

4. Tracking Jitter Analysis

The tracking jitter measures the noise transferred from the input signals, $y_{us}\left[n\right]$ and $y_{ls}\left[n\right]$, to the final delay and phase estimates. When using data/pilot and meta-signal combining strategies, useful power is recovered from both channels and combined to reduce the variance of the final delay/phase estimates. The tracking jitter is a normalized standard deviation and is defined as [29]:

$${\sigma }_j=\frac{{\sigma }_{d,j}}{G_{d,j}}\sqrt{2B_{eq}{T}_u}.$$

(24)

Subscript j is used to indicate that the tracking jitter is either for the phase estimates from the PLL/SPLL or for the delay estimate from the DLL. In this respect, j is set to $\phi$ if the phase estimate is considered or to $\tau$ for the delay. ${\sigma }_{d,j}$ is the standard deviation of the phase/delay discriminator output, and $G_{d,j}$ is the corresponding discriminator gain [29]. A loop discriminator provides a control signal that is approximately proportional to the actual phase/delay error. $G_{d,j}$ is the proportionality constant approximately relating the discriminator output and the actual phase/delay error. It is computed as [29]:

$$G_{d,j}={\left.\frac{\partial E\left[D_j\right]}{\partial \epsilon }\right|}_{\epsilon =0}={\left.\frac{\partial S\left(\epsilon \right)}{\partial \epsilon }\right|}_{\epsilon =0},$$

(25)

where $D_j$ is the discriminator output and $\epsilon$ is the actual phase/delay error. $E[·]$ denotes the expected value and $S\left(\epsilon \right)$, often referred as S-function, is the noiseless response of the loop discriminator to an input phase/delay error, $\epsilon$.

Finally, $B_{eq}$ is the loop equivalent bandwidth and $T_u$ is the total integration time, (19), which corresponds to the loop update rate. In the following, the tracking jitter for the different PLL/SPLL and DLL configurations are analysed.

4.1. Carrier Phase Tracking Jitter

The tracking jitter is first derived for carrier phase estimated by the dual-frequency meta-signal tracking loop depicted in Figure 1. To determine the tracking jitter, it is first necessary to evaluate the variance of the meta-signal discriminator outputs, ${\sigma }_{d,\phi }^2$. In this respect, Equation (21) provides a way to determine ${\sigma }_{d,\phi }^2$:

$${\sigma }_{d,\phi }^2=\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{{\phi }_c}\right\}=\frac{1}{4}\left[\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\phi ,us}\right\}+\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\phi ,ls}\right\}\right],$$

(26)

where $\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\phi ,us}\right\}$ and $\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\phi ,ls}\right\}$ are the variances of the upper and lower sideband discriminator outputs. Note that (26) is valid for both meta-signal carrier and subcarrier phases. Indeed, the variance is a quadratic operator and the minus sign in the second row of the matrix transform in (21) is squared. Since (26) is valid for both carrier and subcarrier components, only the carrier tracking jitter is analysed here. The same results apply for the subcarrier.

When a pilot correlation block is considered along with a four-quadrant arctangent discriminator, the output variance is given by [22]:

$$\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\phi ,pilot}\right\}=\frac{1}{2\frac{C}{N_0}{T}_u}\left(1+\frac{1}{2\frac{C}{N_0}{T}_u}\right)$$

(27)

C is the power received from a single channel and can be set either to $C_{ls}$ or to $C_{us}$.

The discriminator variance of a data only channel with soft-bit removal can be effectively approximated with that of a standard Costas loop with extended integrations:

$$\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\phi ,data}\right\}\approx \frac{1}{2\frac{C}{N_0}{T}_u}\left(1+\frac{1}{2\frac{C}{N_0}{T}_u}\right).$$

(28)

Also in this case, C represents the power received from a single channel. In Section 2, the power of data channels was denoted as either $C_{ls}{\alpha }_{ls}$ or $C_{us}{\alpha }_{us}$. Finally, the variance of a single-frequency data/pilot combined channel was found in [24], and it is given by:

$$\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\phi ,combo}\right\}=\frac{1}{2C/N_0{T}_{u}g(C/N_0)}\left[1+\frac{1}{2C/N_{0}g(C/N_0)T_u}\right],$$

(29)

with $g(C/N_0)$ given by

$$g(C/N_0)=\frac{{\left[1+{|k|}^{2}tanh\left(2\frac{C}{N_0}{|k|}^2{T}_c\right)\right]}^2}{1+{|k|}^2{tanh}^2\left(2\frac{C}{N_0}{|k|}^2{T}_c\right)}.$$

(30)

In (30), the constant k is reported without any subscript. This is because it can refer either to the lower or to the upper sideband component. Thus, k can be equal to $k_{ls}$ or to $k_{us}$. A similar convention applies to C: in (30), this is the power received from the pilot channel, and can be equal to $C_{ls}$ or to $C_{us}$.

For low $C/N_0$, the hyperbolic tangent terms in (30) tend to zero and $g(C/N_0)\approx 1$. For high $C/N_0$ values, the hyperbolic tangent terms converges to unity and $g(C/N_0)=1+{|k|}^2$. Thus, for low $C/N_0$ values, the tracking jitter of the selected data/pilot combining strategy converges to that of pure pilot tracking. For high $C/N_0$ values, the power of the data and pilot channels is coherent summed and $g(C/N_0)>1$ leads to a tracking jitter reduction. For symmetric power splitting between data and pilot components, i.e., for $|k|=1$, $g(C/N_0)$ converges to 2.

Using these results, it is finally possible to determine the variance of the meta-signal phase discriminator outputs. In the following, results are provided only for the carrier component since similar findings are obtained for the subcarrier. In particular, for the dual-frequency single channel case, one obtains:

$$\begin{array}{cc}\hfill \mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{{\phi }_c}\right\}& =\frac{1}{4}\left[\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\left(1+\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\right)+\frac{1}{2\frac{C_{us}}{N_0}{T}_u}\left(1+\frac{1}{2\frac{C_{us}}{N_0}{T}_u}\right)\right]\hfill \\ \hfill & =\frac{1}{4}\left[\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\left(1+\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\right)+\frac{1}{2\frac{{\gamma }^2{C}_{ls}}{N_0}{T}_u}\left(1+\frac{1}{2\frac{{\gamma }^2{C}_{ls}}{N_0}{T}_u}\right)\right]\hfill \\ \hfill & =\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\frac{1+{\gamma }^2}{4{\gamma }^2}\left(1+\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\frac{1+{\gamma }^4}{{\gamma }^2(1+{\gamma }^2)}\right)\hfill \end{array}$$

(31)

where $\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{{\phi }_c}\right\}$ has been expressed as a function of the power of the lower sideband component, $C_{ls}$. $\gamma$ is the ratio between the amplitudes of two channel components introduced in (23). When the two sideband components have the same power, $\gamma =1$, and variance (31) is reduced by a factor $\frac{1}{2}$ with respect to single-frequency tracking.

The variance of the dual-frequency, dual channel PLL discriminator is obtained in a similar way:

$$\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{{\phi }_c}\right\}=\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\frac{g(C_{ls}/N_0)+{\gamma }^2}{4g(C_{ls}/N_0){\gamma }^2}\left[1+\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\frac{g^2(C_{ls}/N_0)+{\gamma }^4}{g(C_{ls}/N_0){\gamma }^2(g(C_{ls}/N_0)+{\gamma }^2)}\right].$$

(32)

Variance (32) refers to the B2 case where all the available energy has been recovered. In this case, the data and pilot components from the B2a channel and the data signal from the B2b component are used. Under good signal conditions, and for equal data/pilot power splitting, $g(C_{ls}/N_0)$ converges to 2. Moreover, assuming $\gamma =1$, one obtains that $\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{{\phi }_c}\right\}$ is reduced by about a factor $3/8$ ($4.26$ dB) as compared to the single-frequency single channel case.

While, at the moment, the B2b signal does not feature a pilot component, it is also possible to derive the variance of a carrier discriminator featuring four channels on two different frequencies. This is given by:

$$\begin{array}{cc}\hfill \mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{{\phi }_c}\right\}& =\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\frac{g(C_{ls}/N_0)+{\gamma }^{2}g(\gamma C_{ls}/N_0)}{4g(C_{ls}/N_0){\gamma }^{2}g(\gamma C_{ls}/N_0)}\hfill \\ \hfill & ·\left[1+\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\frac{g^2(C_{ls}/N_0)+{\gamma }^4{g}^2(\gamma C_{ls}/N_0)}{g(C_{ls}/N_0){\gamma }^{2}g(\gamma C_{ls}/N_0)[g(C_{ls}/N_0)+{\gamma }^{2}g(\gamma C_{ls}/N_0)]}\right].\hfill \end{array}$$

(33)

This variance is reported here for completeness, but it is not further analysed, since the main focus of the paper is the processing of the current BDS B2 signal.

All of the phase discriminators considered above have unit gain. Thus, the final tracking jitter for the different cases can be directly computed using (24). The phase tracking jitters obtained for the different cases are summarized in

Table 1. Carrier phase tracking jitter expressions for the different schemes considered in the paper.

Scheme	Tracking Jitter	Source
Standard, single-frequency	$\sqrt{\frac{B_{eq}}{C_{ls}/N_0}\left[1+\frac{1}{2C_{ls}/N_0{T}_c}\right]}$	[22]
Data/pilot, tanh, single-frequency	$\sqrt{\frac{B_{eq}}{C_{ls}/N_{0}g(C_{ls}/N_0)}\left[1+\frac{1}{2C_{ls}/N_{0}g(C_{ls}/N_0)T_u}\right]}$	[24]
Squaring PLL, single channel	$\sqrt{\frac{B_{eq}}{C_{us}/N_0}\left[1+\frac{1}{2C_{us}/N_0{T}_u}\right]}$	[28]
Meta-signal, dual-frequency	$\sqrt{\frac{B_{eq}}{C_{ls}/N_0}\frac{1+{\gamma }^2}{4{\gamma }^2}\left[1+\frac{1}{2C_{ls}/N_0{T}_u}\frac{1+{\gamma }^4}{{\gamma }^2(1+{\gamma }^2)}\right]}$
Meta-signal, data/pilot + data	$\sqrt{\frac{B_{eq}}{C_{ls}/N_0}\frac{g(C_{ls}/N_0)+{\gamma }^2}{4g(C_{ls}/N_0){\gamma }^2}\left[1+\frac{1}{2\frac{C_{ls}}{N_0}{T}_u}\frac{g^2(C_{ls}/N_0)+{\gamma }^4}{g(C_{ls}/N_0){\gamma }^2(g(C_{ls}/N_0)+{\gamma }^2)}\right]}$

. The table also includes the tracking jitter obtained in [28] for the squaring PLL. These results will be further analysed in Section 5.

4.2. Code Tracking Jitter

The code tracking jitter is obtained following a process analogous to that adopted in the previous section for the carrier component. In particular, it is first necessary to determine the normalized variance

$$\frac{\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\tau }\right\}}{G_{d,j}^2}.$$

(34)

A general expression for (34) is obtained in Appendix A for code discriminator (22). Moreover, different special cases are analysed. The results derived in Appendix A can be directly used for computing the tracking jitter.

In this section, only the case where the different components adopt a Binary Phase Shifiting Keying (BPSK) modulation with the same rate are analysed. Moreover, a common half Early-minus-Late correlator spacing, $\mathsf{\Delta }$, is considered. These conditions are denoted here as symmetric BPSK case and correspond to the current implementation of the ACE-BOC modulation. General results are presented in Appendix A and are not repeated here.

Using the results obtained in Appendix A, it is possible to show that the normalized variance (34) for the symmetric BPSK case is given by:

$$\frac{\mathrm{V}\mathrm{a}\mathrm{r}\left\{D_{\tau }\right\}}{G_{d,\tau }^2}=\frac{\left[1+|k_{ls}{|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)+{\gamma }^2{|k_{us}|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]\mathsf{\Delta }}{2\frac{C_{ls}}{N_0}{T}_u{\left[1+|k_{ls}{|}^{2}tanh\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)+{\gamma }^2{|k_{us}|}^{2}tanh\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]}^2},$$

(35)

where the data/pilot combination has been adopted for the lower sideband components, i.e., the B2a signal, and data-only processing has been considered for the upper sideband channel, i.e., the B2b component. Combining (35) with (24), the code tracking jitter is found:

$${\sigma }_{\tau }=\sqrt{\frac{B_{eq}\left[1+|k_{ls}{|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)+{\gamma }^2{|k_{us}|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]\mathsf{\Delta }}{\frac{C_{ls}}{N_0}{\left[1+|k_{ls}{|}^{2}tanh\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)+{\gamma }^2{|k_{us}|}^{2}tanh\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]}^2}}.$$

(36)

This expression can be written in a compact form by introducing the function:

$$g_c(\gamma ,|k_{ls}|,|k_{us}|)=\frac{{\left[1+|k_{ls}{|}^{2}tanh\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)+{\gamma }^2{|k_{us}|}^{2}tanh\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]}^2}{1+|k_{ls}{|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)+{\gamma }^2{|k_{us}|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)},$$

(37)

which represents the gain, in terms of $C/N_0$, provided by the combining strategy used to jointly track the common signal delay. Using (37), (36) is written in a more compact form as

$${\sigma }_{\tau }=\sqrt{\frac{B_{eq}\mathsf{\Delta }}{\frac{C_{ls}}{N_0}{g}_c(\gamma ,|k_{ls}|,|k_{us}|)}}.$$

(38)

The single-frequency case with the processing of the pilot component alone is found for $\gamma =0$ and $k_{ls}=0$. Under such conditions, there is no combining gain and $g_c(0,0,|k_{us}|)=1$. The dual-frequency case, with a pilot component on the lower sideband channel, is found for $k_{ls}$ = 0.

The single-frequency case with data/pilot combination is found with similar consideration in Appendix A. A summary of the code tracking jitters considered in this section, that is for the BPSK symmetric case, is provided in

Table 2. Code delay tracking jitter expressions for the different schemes obtained under the symmetric BPSK case. Non-coherent Early-minus-Late envelope discriminator.

Scheme	Tracking Jitter
Standard, single-frequency	$\sqrt{\frac{B_{eq}\mathsf{\Delta }}{\frac{C_{ls}}{N_0}}}$
Data/pilot, tanh, single-frequency	$\sqrt{\frac{B_{eq}\left[1+|k_{ls}{|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)\right]\mathsf{\Delta }}{\frac{C_{ls}}{N_0}{\left[1+|k_{ls}{|}^{2}tanh\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)\right]}^2}}$
Meta-signal, pilot + data	$\sqrt{\frac{B_{eq}\left[1+{\gamma }^2{|k_{us}|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]\mathsf{\Delta }}{\frac{C_{ls}}{N_0}{\left[1+{\gamma }^2{|k_{us}|}^{2}tanh\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]}^2}}$
Meta-signal, data/pilot + data	$\sqrt{\frac{B_{eq}\left[1+|k_{ls}{|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)+{\gamma }^2{|k_{us}|}^2{tanh}^2\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]\mathsf{\Delta }}{\frac{C_{ls}}{N_0}{\left[1+|k_{ls}{|}^{2}tanh\left(\frac{A_{ls}^2{|k_{ls}|}^2}{{\sigma }^2}\right)+{\gamma }^2{|k_{us}|}^{2}tanh\left(\frac{A_{ls}^2{|k_{us}|}^2}{{\sigma }^2}\right)\right]}^2}}$

.

5. Semi-Analytic Simulations

Theoretical results discussed in the previous section have been supported by semi-analytic simulations [30]. This type of approach adopts analytical results to reduce the computational burden that full Monte Carlo simulations would required. In this case, analytic results are used for the computation of the correlator outputs, whereas all the other elements of the tracking loops are fully simulated. The semi-analytic framework developed in [31] has been ported in Python and extended to support the different tracking options considered in this paper. For all the simulations, the parameters reported in

Table 3. Parameters used for the semi-analytic simulations.

Parameter	Value
Sampling frequency, $f_s$	50 MHz
Simulation runs	$5\times {10}^4$
Coherent integration time, $T_c$	1 ms
Data/pilot power ratio	1

have been adopted along with an equal power distribution, that is $\gamma = |k_{ls}| = |k_{us}| = 1$.

The sampling frequency, $f_s$, and the coherent integration time, $T_c$, have been selected taking into account the characteristics of the B2a and B2b signals, which form the ACE-BOC modulation. A sampling frequency equal to 50 MHz allows one to jointly sample the two B2 components, whereas $T_c=1$ ms corresponds to the duration of the B2b primary code. Different combinations of K and equivalent bandwidth, $B_{eq}$, are considered in the following to analyse different trade-offs between sensitivity and ability to track signal dynamics. Equivalent bandwidths and loop update rates have been selected to ensure loop stability.

Results obtained for a PLL tracking the meta-signal carrier phase are presented first. The analysis performed with respect to the DLL component is then discussed. Since the SPLL is characterized by similar properties to the PLL, this component is not analysed in the following.

5.1. PLL Semi-Analytic Simulation Results

Semi-analytic simulation results are briefly presented in this section for the PLL and its tracking jitter. The analysis focuses on the two meta-signal tracking configurations introduced at the end of Section 3, which are suitable for the processing of the current implementation of the ACE-BOC modulation. In this respect, Figure 3 provides a comparison between the tracking jitter obtained by simulations and theoretical Formula (31). In this case, a dual-frequency PLL considering the pilot channel on the lower sideband component and a data channel on the upper sideband component is analysed.

The tracking jitter is provided as a function of the $C/N_0$ of the lower sideband component. Moreover, different loop equivalent bandwidths, $B_{eq}$, are considered. From the figure, a good agreement between theoretical and simulation results emerges. As expected, the tracking jitter decreases with the reduction in the loop bandwidth. For low $C/N_0$ values, theoretical and simulation curves start diverging. This is due to the non-linear loop behaviour that is neglected by (24). This behaviour is captured by the semi-analytic simulations. More specifically, the simulation curves shown in Figure 3 are characterized by a vertical trend in the region between 20 and 25 dB-Hz. This trend indicates that the PLL has lost lock and the phase estimates are diverging. This differences at low $C/N_0$ are, however, expected. For moderate/high $C/N_0$ values, theoretical results derived in the previous section properly capture the behaviour of the meta-signal single channel PLL.

The meta-signal PLL with data/pilot combination on the lower sideband component and data only processing on the upper sideband channel is considered in Figure 4. Also in this case, a good agreement between theoretical and simulation results is obtained for moderate and high $C/N_0$ values. Semi-analytic simulations allows one to analyse the non-linear behaviour of the meta-signal PLL that also in this case loses lock in the 20–25 dB-Hz region depending on the loop parameters.

In both Figure 3 and Figure 4, $K=5$ non-coherent integrations are considered. The tracking jitter results from these figures are compared in Figure 5. From the figure, the advantage of recovering useful power from all the available components clearly emerges. For all the three $B_{eq}$ values considered, the PLL with data/pilot combination on the lower sideband component achieves lower tracking jitters.

As the $C/N_0$ decreases, the tracking jitters from the two PLLs converges. This is the effect of the hyperbolic tangent used in (18) to estimate the data symbols. At low $C/N_0$ values, the symbol estimation process becomes unreliable and the information from the data channels should be discarded.

The impact of K is investigated in Figure 6, which further compares the two meta-signal PLLs. As for the previous configurations, a good agreement between theoretical and simulation results is found for moderate to high $C/N_0$ values. Divergence between theoretical and simulation results are observed at low $C/N_0$ values when the PLLs are loosing lock.

Also in this case, the advantages of the PLL using data/pilot combing on the lower sideband component clearly emerge. Lower tracking jitters are obtained for all the configurations investigated in Figure 6.

Finally, Figure 7 compares the tracking jitter of meta-signal PLLs with the performance of single-frequency tracking loops. Only simulation results are presented.

As expected, meta-signal PLLs achieve better tracking jitter for medium-to-high $C/N_0$ values. This is expected, since these PLLs are able to recover more power than their single-frequency counterpart. This effect is particularly evident when considering the results obtained for single-frequency pure pilot processing, which is characterized by the highest tracking jitter. For medium-to-high $C/N_0$ values, the PLLs using a data and a pilot channel, either from one or two frequencies, achieve the similar tracking jitter. This is expected since, in the configuration considered here, they are able to recover the same amount of useful power. The meta-signal PLL using three components has the lowest tracking jitter. At low $C/N_0$ values, non-linear effects start becoming relevant and the single-frequency PLLs have slightly better performance, most of all in terms of loss of lock, with respect to their meta-signal counterpart. This is due to the fact that the upper sideband component features a data-only channel. In this case, the de-weighting performed by the hyperbolic tangent is not effective, since there is no combining with a corresponding pilot component. Information from the upper sideband component is however needed to estimate the common meta-signal phase and cannot be further de-weighted. This degradation is limited and the tracking threshold is reduced by approximately 2 dBs. This result suggests that for low $C/N_0$ values a switching mechanism, between single and dual-frequency processing, should be implemented. Despite being aided by the lower sideband component, the upper channel becomes unrecoverable and should be discarded.

The results from Figure 7 are better analysed in

Table 4. Comparison of PLL tracking jitters for the different processing strategies and $C/N_0$ values. $K=5$, $B_{eq}$ = 10 Hz.

	C/No	25 dB-Hz	30 dB-Hz	35 dB-Hz
Strategy
Single-frequency, pure pilot		$0.22$ rad	$0.11$ rad	$0.06$ m
Pilot + Data		$0.17$ rad	$0.08$ rad	$0.04$ rad
Single-frequency, Data/pilot		$0.16$ rad	$0.08$ rad	$0.04$ rad
Data/pilot + Data		$0.16$ rad	$0.065$ rad	$0.034$ m

, which explicitly compares the tracking jitters obtained for the different strategies and for different $C/N_0$ values. The tracking jitters reported in the table confirm the benefits of signal combining for phase processing most of all at medium/high $C/N_0$ values. For low $C/N_0$, the information from data channels becomes unreliable and should be discarded.

The results obtained using semi-analytic simulations support the theoretical findings discussed in Section 4.1 and confirm the benefits of effectively combining power from all the sources available.

5.2. DLL Semi-Analytic Simulation Results

Sample results obtained using semi-analytic simulation for the assessment of meta-signal DLLs performance are presented in this section. Figure 8 provides a comparison between the theoretical tracking jitters and simulation results obtained considering a meta-signal dual-frequency DLL with pilot only processing on the lower sideband component and data only processing on the upper sideband component. The symmetric BPSK case is considered, and different Early-minus-Late spacings are tested.

Theoretical and simulation results agree well, most of all for medium to high $C/N_0$ values. As for the PLL case, the $C/N_0$ here is the one of the pilot channel on the lower sideband component. While some slight differences are present, these occurs at low $C/N_0$ values where the approximations performed in Appendix A are less valid. Despite this fact, a good agreement is found and simulation results support theoretical findings.

A similar analysis is performed in Figure 9 for a meta-signal DLL using data/pilot combination on the lower sideband component and data only processing on the upper sideband component.

Also in this case, a good agreement between simulation and theoretical results is observed. Also, in the DLL case, the advantages of using all the power available arise. This fact clearly emerges by comparing the tracking jitters shown in Figure 8 and Figure 9: lower tracking jitters are found for the meta-signal DLL with data/pilot combination.

Different DLL configurations are compared in Figure 10 for a single Early-minus-Late spacing. As expected, a clear tracking jitter reduction is found when moving from the single-frequency, single channel case to the dual-frequency configuration using data/pilot combination.

Table 5. Comparison of DLL tracking jitters for the different processing strategies and $C/N_0$ values. $K=10$, $B_{eq}$ = 2 Hz, $\mathsf{\Delta }=0.25$.

	C/No	25 dB-Hz	30 dB-Hz	35 dB-Hz
Strategy
Single-frequency, pure pilot		$13.5$ m	$6.8$ m	$2.8$ m
Pilot + Data		$9.5$ m	$4.8$ m	$2.5$ m
Data/pilot + Data		$7.6$ m	$4.1$ m	$2.1$ m

explicitly compares the tracking jitter values obtained for the different DLL architectures considered in Figure 10 for three $C/N_0$ values. The results provided in the table confirm the benefits of recovering all the available power. At low $C/N_0$ values, the benefits are more evident and for $C/N_0$ = 25 dB-Hz the tracking jitter is almost halved when passing from the single-frequency, pure pilot case to meta-signal approach with data/pilot combining.

The single-frequency DLL with data/pilot combination is not considered in Figure 10, since it achieves the same performance of the meta-signal DLL using data and pilot components from different frequencies. In both cases, the same signal power is recovered.

While additional configurations have been considered, the related results are not reported here to avoid the repetition of similar findings. Simulation results support the validity of theory developed in Section 4.2.

6. Experimental Analysis

Results obtained considering real BDS B2 signals are briefly discussed in this section. The experimental setup adopted is described first.

6.1. Experimental Setup

Signals were collected using a National Instruments (NI) Universal Software Radio Peripheral (USRP)-2944R wideband front-end connected to a multi-frequency Trimble Zephyr 2 geodetic antenna placed under open-sky conditions. The NI front-end was configured according to the parameters reported in

Table 6. Parameters used for configuring the USRP-2944R front-end used for the data collection.

Parameter	Value
Sampling frequency, $f_s$	50 MHz
Centre frequency, $f_c$	$1191.795$ MHz
No. of bits	8 bits
Sampling type	Complex, IQ

.

In this way, it was possible to jointly capture the B2a and B2b components. In order to test the different tracking algorithms under noisy conditions, the approach schematically represented in Figure 11 was adopted. Synthetic AWGN was added to the real I/Q signals collected using the USRP front-end. The I/Q data from the USRP front-end were first used to estimate the variance of the noise already present in the input signal. This variance was then scaled and used to determine the variance of the additional AWGN generated through simulations.

The additional AWGN was generated in order to obtain a $C/N_0$ ramp: the variance of the noise affecting the useful GNSS components was increased by 1 dB each 10 s. The full experiment lasted 300 s with a final noise degradation of 30 dB.

The resulting data were processed using a custom SDR receiver implemented in Python. The software implements both acquisition and tracking. The meta-signal acquisition strategy described in [7] was implemented and used to obtain code and Doppler frequency estimates for initializing the tracking loops. The receiver also implements several stages such as initial Frequency Lock Loop (FLL) processing and secondary code removal on the B2a component. While a complete description of the software receiver is outside the scope of this paper, details on the implementation of the data/pilot combining strategy implemented can be found in [24]. The parameters used for tracking are reported in

Table 7. Software receiver parameters used for processing the ACE-BOC signals.

Parameter	Value
PLL order	3rd
PLL bandwidth	15 Hz
SPLL order	2nd
SPLL bandwidth	2 Hz
DLL order	2nd
DLL bandwidth	2 Hz
Coherent integration time	1 ms
Non-coherent integrations, K	5

.

6.2. Experimental Results

Sample results obtained processing the I/Q data collected according to the experimental setup described in the previous section are discussed here. Figure 12 compares the total $C/N_0$ estimated by the software receiver for the different processing configuration. The receiver continuously estimates the $C/N_0$ from correlator outputs (17), which include components from different frequencies and from different channels. Thus, in this case, the $C/N_0$ includes the power recovered from all the components processed by the different tracking strategies. The adjective ‘total’ is used here to differentiate the $C/N_0$ estimated by the receiver from the one considered in Section 5, which was relative to the pilot component of the lower sideband signal. The total $C/N_0$ is used here to quantify the ability of the different tracking strategies to recover useful signal power and as loss of lock indicator [29]. Five tracking strategies have been considered: three single-frequency and two dual-frequencies. Single-frequency architectures include standard pilot processing and data/pilot combination [24] for the B2a signal, and the squaring tracking loop [28] for the B2b component. The two dual-frequencies architectures are the two meta-signal approaches analysed in the previous sections.

From Figure 12, it emerges that the meta-signal architecture using data/pilot combination on B2a signal is the most effective in recovering all the available power. In this respect, the total $C/N_0$ recovered by this strategy is always above that of the other architectures. The two strategies employing data and pilot components, either from the same or from two different frequencies, have similar performance with very close $C/N_0$ values. Single-frequency, single component tracking strategies recover power from a single signal and the corresponding $C/N_0$ is the lowest. With respect to the loss of lock, the results shown in Figure 12 are in agreement with the simulation findings discussed in Section 5: loss of lock occurs for single channel $C/N_0$ values in the [20–25] dB-Hz range.

Figure 13 compares the standard deviations of the Doppler estimates obtained using different BDS B2 tracking algorithms. The Doppler standard deviations are provided as a function of time for the $C/N_0$ ramp experiment. Standard deviations have been estimated from the PLL loop filter outputs and can be used as receiver performance indicators.

As for the total $C/N_0$ case, the strategy recovering all the power available is the most effective and lower Doppler standard deviations are achieved. The results in Figure 13 confirm the findings already discussed for the total $C/N_0$.

Experimental results further support the validity of the findings discussed in the previous sections and confirm the feasibility of implementing meta-signal approaches for the processing of the ACE-BOC modulation, even when a pilot component is missing, in this case, for the upper sideband component.

7. Conclusions

In this paper, different strategies for jointly tracking BDS B2 signals have been analysed. The processing of the B2b signal is particularly challenging, since it does not include a pilot channel and features a short code duration of 1 ms. Particular focus was devoted to the meta-signal approach and two meta-signal processing strategies were proposed and analysed. The first one used the B2a pilot signal and the B2b data channel. The second strategy was designed to recover all the available power and adopted the data/pilot combination for the B2a component. The proposed strategies were compared with standard approaches from the literature, such a single-frequency pilot processing and single-frequency data/pilot combining.

The different strategies have been analysed theoretically, through semi-analytic simulations and by processing real BDS B2 signals collected using a wideband RF front-end. The different strategies also enable non-coherent integrations that allow one to extend the integration process beyond the symbol duration of the data components. The analysis shows the flexibility of the meta-signal approach that can be implemented considering asymmetric modulations and can incorporate data/pilot combining. When these approaches are adopted, all of the signal power available is effectively exploited and a clear improvement in terms of tracking jitter is observed. Pseudoranges and carrier phase observations are derived from the tracking loop outputs. Thus, lower tracking jitters imply GNSS measurements with lower variances. In this respect, the proposed approaches can reduce the variance of GNSS measurements and thus of the final position solution.

The proposed approach is effective for the processing of the current implementation of the ACE-BOC, which lacks a pilot channel on the upper sideband component.

Future research directions include better data/pilot combining strategies under low $C/N_0$ conditions: in weak signal scenarios, the contribution of the B2b signal becomes unreliable. While this is accounted for by the soft-bit estimation process considered in the current work, additional strategies can be considered. Kalman filter-based approaches exploiting the relationships between the Doppler frequencies of the different components could, for instance, be investigated.