Quad-Frequency Wide-Lane, Narrow-Lane and Hatch–Melbourne–Wübbena Combinations: The Beidou Case

Abstract

The pseudoranges of a Global Navigation Satellite System (GNSS) meta-signal can be reconstructed from the observations of its side-band components. More specifically, the Hatch–Melbourne–Wübbena (HMW) code-carrier combination is used to solve the ambiguity associated to the wide-lane carrier phase combination of the side-band components, obtaining a high-accuracy pseudorange. The HMW and the wide-lane combinations thus play a key role in constructing meta-signal measurements. The theory of GNSS meta-signals was recently extended to the case with a number of components equal to a power of two. This theory can be used to generalize HMW and wide-lane combinations to the quad-frequency case. This is carried out through a Hadamard matrix of order four, which defines a narrow-lane and three wide-lane combinations. This paper characterizes meta-signal-inspired quad-frequency HMW and wide-lane measurements combinations using Beidou Navigation Satellite System (BDS) observations. Two professional Septentrio PolarRx5S multi-frequency, multi-constellation receivers were set up in a zero-baseline configuration and used to collect observables from all the BDS open frequencies. These measurements are used to characterize different quad-frequency HMW and wide-lane carrier combinations. Some of the combinations analyzed have large equivalent wavelengths and have the potential to enable single-epoch ambiguity resolution in scenarios where short convergence times are required.

1. Introduction

The availability of Global Navigation Satellite System (GNSS) signals from several Radio Frequencies (RFs) is a key enabler for decimeter-level positioning [1]. Multi-frequency measurements allow one to estimate and remove the impact of the ionosphere [2,3], generate measurement combinations with large wavelengths for reliable carrier phase ambiguity resolution [4,5,6] and implement algorithms with reduced or even instantaneous convergence times [7,8,9,10,11]. Galileo and Beidou Navigation Satellite System (BDS) currently provide open signals from four and five frequencies, respectively [12,13]. This has stimulated significant research in the measurement and position domain, where advanced algorithms have been designed to take advantage of the modern multi-frequency GNSS signal landscape. In this respect, the systematic analysis of triple-frequency measurement combinations performed by [14] has been extended to a higher number of components, also considering Galileo and BDS [15]. For instance, ref. [16] analyzed BDS triple-frequency combinations and their application to Three Carrier Ambiguity Resolution (TCAR). BDS quad-frequency combinations considering the B1C, B1I, B3I, and B2a signals were studied in [9]. Two Four Carrier Ambiguity Resolution (FCAR) methods based on the construction of dual- and triple-frequency carrier phase combinations, whose integer ambiguities are solved using a cascaded approach, were also proposed. The authors of [6,8] considered the same BDS signals and analyzed integer ambiguity resolution strategies based on dual- and triple-frequency combinations. The focus of the analyses was mainly on double differences and long baselines. The processing of triple-frequency BDS measurements and their application to the single-epoch ambiguity resolution problem was considered in [17]. Finally, the authors of [7] provided a general model for the combination of measurements from five frequencies. The analysis, however, mainly focused on combinations obtained with a lower number of components.

Despite the intense research work on multi-frequency measurement combinations and their application to Real Time Kinematic (RTK) and Precise Point Positioning (PPP) solutions, the complexity of the subject and the wide range of possibilities opened by the availability of up to five signals from different GNSSs continue to call for further investigations to improve accuracy, reliability and convergence time. A possibility is to consider solutions developed at the signal processing level and analyze their implications in the measurement domain [18,19,20]. Multi-frequency GNSS receivers implement strategies to optimally combine signal components from different frequencies [21,22,23] and exploit their common properties. These include joint signal tracking and aiding between different frequency components. In this context, the meta-signal approach [24] is receiving significant attention by the research community. A GNSS meta-signal is obtained by considering and processing as a single entity components from different frequencies [25]. A sensitivity gain is obtained by coherently combining signal power recovered from the different frequencies. The meta-signal concept has its counterpart in the measurement domain, and meta-signal observations have been related to the dual-frequency narrow- and wide-lane measurement combinations [18]. Meta-signal high accuracy pseudoranges can also be reconstructed from side-band observations through the Hatch–Melbourne–Wübbena (HMW) combination [26,27,28]. This combination is used to instantaneously solve for the integer ambiguities of the wide-lane carrier phase combination of the meta-signal side-band components.

The meta-signal paradigm was recently extended to the quad-frequency case [20,29]. Signals from four different frequencies are jointly processed through the introduction of three Subcarrier Phase Lock Loops (SPLLs) that align in phase the components recovered from different frequencies. Phase variations between the four RF signals are modeled through subcarrier terms that are tracked instead of the original signal carriers. These concepts found their counterpart in the measurement domain [20], and generalized quad-frequency narrow- and wide-lane combinations are found. These measurements are obtained through a Hadamard matrix [30] whose coefficients all equal $\pm 1$.

This paper analyzes meta-signal-inspired quad-frequency narrow- and wide-lane combinations obtained considering the BDS signal landscape. While these combinations were introduced by the conference paper in [20], their complete characterization was missing in the literature. Moreover, only Galileo signals were considered. This work extends previous results from the authors [20] and provides a full characterization of meta-signal-inspired quad-frequency measurement combinations. Properties such as the equivalent wavelength are derived and discussed. Moreover, an experimental characterization of the different BDS quad-frequency combinations is performed and, for each case, figures of merit, such as the standard deviation and fractional bias of the resulting combination, are provided. Since five BDS open signals are available, several quad-frequency configurations are possible. This paper mainly focuses on two configurations: the first considers the B1C, BI1, B2b, and B2a signals, whereas the second adopts the B3I component instead of the B1I signal. These quad-frequency combinations have not been considered before [6,8,9] and have the potential to enable single-epoch ambiguity resolution. The fact that all combination coefficients are equal to $\pm 1$ leads to low noise amplification factors. The paper also analyzes quad-frequency HMW combinations that provide estimates of the integer ambiguities of the corresponding wide-lane combinations. In this respect, this paper extends the work of [31] that provided a systematic analysis of dual-frequency HMW combinations in the BDS case.

Quad-frequency narrow-lane, wide-lane, and HMW combinations were characterized using measurements from two professional Septentrio PolarRx5S multi-frequency, multi-constellation receivers set up in a zero-baseline configuration [32]. In this way, observables from all the BDS open frequencies were collected. Combinations were characterized considering undifferenced, single-difference, and double-difference modes. In the undifferenced case, the impact of satellite biases was assessed using the precise products provided by the PPP-wizard project [33,34]. Moreover, fractional receiver biases were compensated using an approach similar to that developed by [35]. These elements were not considered in the previous analysis conducted by the authors [20], which focused on Galileo signals.

The analysis shows the benefits of meta-signal-inspired quad-frequency combinations, which, depending on the configuration, are characterized by large wavelengths and thus have the potential for reliable single-epoch ambiguity resolution. In the undifferenced case, the main limitation of this approach is represented by the presence of residual biases, which are not completely removed through the use of PPP corrections. The potential use of meta-signal-inspired quad-frequency combinations in conjunction with dual-frequency observations is also discussed.

The remainder of this paper is organized as follows: Section 2 provides a summary of the synthetic meta-signal measurement reconstruction paradigm, which is extended to the quad-frequency case in Section 3. Quad-frequency HMW combinations are introduced in Section 4. The BDS signal landscape and the signal configurations considered in this paper are described in Section 5, whereas the materials and methods used for the experimental analysis are detailed in Section 6. Results are provided in Section 7, and Section 8 discusses the benefits, limitations, and potential applications of the quad-frequency combinations analyzed. Finally, Section 9 concludes this paper.

2. Synthetic Meta-Signal Measurement Reconstruction

In the dual-frequency case, the synthetic meta-signal approach allows one to reconstruct the measurements of a wide-band modulation, such as the Alternative Binary Offset Carrier (AltBOC) [18,36] and the Asymmetric Constant-Envelope Binary Offset Carrier (ACE-BOC) [19,37], from its side-band observations. This process, which is strictly related to the wide/narrow-lane carrier phase and HMW combinations is briefly summarized in this section.

Meta-signal carrier phase observations in cycles are obtained as

$${\phi }_0={\displaystyle \frac{1}{2}}({\phi }_1+{\phi }_2)+{\displaystyle \frac{1}{2}}H={\displaystyle \frac{1}{2}}{\phi }_{nl}+{\displaystyle \frac{1}{2}}H,$$

(1)

where ${\phi }_0$, ${\phi }_1$, and ${\phi }_2$ are the carrier phase observations from the meta-signal and side-band components, respectively. ${\phi }_{nl}$ is the narrow-lane combination of ${\phi }_1$ and ${\phi }_2$, and H is a binary variable used to correct for possible half-cycle ambiguities [38]. H is computed from the HMW code-carrier combination as

$$H=\left\{\begin{array}{cc}0\hfill & \hfill \mathrm{if\; round}\left({\displaystyle \frac{C_{HMW}}{{\lambda }_{wl}}}\right) \mathrm{is\; even}\\ 1\hfill & \hfill \mathrm{if\; round}\left({\displaystyle \frac{C_{HMW}}{{\lambda }_{wl}}}\right) \mathrm{is\; odd}\end{array},\right.$$

(2)

where $C_{HMW}$ is the HMW code-carrier combination of the side-band observations [26,27,28]:

$$C_{HMW}={\Phi }_{wl}-{\rho }_{nl}={\lambda }_{wl}({\phi }_1-{\phi }_2)-{\displaystyle \frac{f_1{\rho }_1+f_2{\rho }_2}{f_1+f_2}}.$$

(3)

${\lambda }_{wl}$ is the wavelength of the wide-lane combination of the side-band carrier phase observations:

$${\lambda }_{wl}={\displaystyle \frac{c}{f_1-f_2}},$$

(4)

where c is the speed of light, and $f_1$ and $f_2$ arethe center frequencies of the meta-signal side-band components. In (3), ${\rho }_{nl}={\displaystyle \frac{f_1{\rho }_1+f_2{\rho }_2}{f_1+f_2}}$ is the narrow-lane combination of the side-band pseudoranges, ${\rho }_1$ and ${\rho }_2$. Pseudoranges are expressed in meters. ${\Phi }_{wl}={\lambda }_{wl}({\phi }_1-{\phi }_2)$ is the wide-lane combination of the side-band carrier phase observations expressed in meters.

Meta-signal pseudoranges can be reconstructed as

$${\rho }^{+}={\Phi }_{wl}-\mathrm{round}\left({\displaystyle \frac{C_{HMW}}{{\lambda }_{wl}}}\right){\lambda }_{wl},$$

(5)

where ${\rho }^{+}$ is the final high-accuracy meta-signal pseudorange obtained by solving the integer cycle ambiguity associated to ${\Phi }_{wl}$, the wide-lane carrier phase combination of the side-band components. This is carried out through rounding the HMW combination. Note that in the original synthetic meta-signal approach, a different code-carrier combination was used for solving the cycle ambiguity [18]. The HMW is preferred here since it provides an unbiased estimator of the wide-lane carrier integer ambiguities.

The summary provided above clearly shows that the reconstruction of meta-signal carrier phase and pseudorange measurements strictly depends on the narrow/wide-lane carrier phase combinations. When considered jointly, the following transformation is obtained:

$$\left[\begin{array}{c}{\phi }_{nl}\\ {\phi }_{wl}\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\left[\begin{array}{c}{\phi }_1\\ {\phi }_2\end{array}\right]={\mathbf{H}}_2\left[\begin{array}{c}{\phi }_1\\ {\phi }_2\end{array}\right],$$

(6)

where ${\mathbf{H}}_2$ is a Hadamard matrix of order 2.

3. Quad-Frequency Narrow and Wide-Lane Combinations

In the signal processing domain, quad-component meta-signals are processed by introducing a Hadamard transform of order four [29]. In this way, the original carrier components are projected into a transformed domain where a common carrier and three subcarriers are identified. In analogy to the signal processing domain, ref. [20] introduced quad-frequency narrow- and wide-lane combinations by using a Hadamard transform of order four:

$$\left[\begin{array}{c}{\phi }_{nl,4}\\ {\phi }_{wl1,4}\\ {\phi }_{wl2,4}\\ {\phi }_{wl3,4}\end{array}\right]=\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& -1& 1& -1\\ 1& 1& -1& -1\\ 1& -1& -1& 1\end{array}\right]\left[\begin{array}{c}{\phi }_1\\ {\phi }_2\\ {\phi }_3\\ {\phi }_4\end{array}\right]={\mathbf{H}}_4\left[\begin{array}{c}{\phi }_1\\ {\phi }_2\\ {\phi }_3\\ {\phi }_4\end{array}\right],$$

(7)

where ${\phi }_1$, ${\phi }_2$, ${\phi }_3$, and ${\phi }_4$ are carrier phase measurements expressed in cycles from four different frequencies. ${\mathbf{H}}_4$ is a Hadamard matrix of order 4 and ${\phi }_{nl,4}$ is the quad-frequency narrow-lane combination obtained as the sum of the four original carrier phase measurements. Three quad-frequency wide-lane combinations are also found. These correspond to the last three elements of (7) and are obtained by iterating the wide-laning process twice. For instance,

$${\phi }_{wl1,4}={\phi }_1-{\phi }_2+{\phi }_3-{\phi }_4=({\phi }_1-{\phi }_2)-({\phi }_4-{\phi }_3).$$

(8)

The wavelengths associated with the different quad-frequency carrier combinations can be evaluated as [8]

$${\lambda }_{(i,j,k,h)}={\displaystyle \frac{c}{i·f_1+j·f_2+k·f_3+h·f_4}},$$

(9)

where i, j, k, and h are the coefficients from the rows of ${\mathbf{H}}_4$, and $f_1$, $f_2$, $f_3$, and $f_4$ are the RF of the different carrier phase measurements. In the case of (8),

$${\lambda }_{(1,-1,1,-1)}={\displaystyle \frac{c}{f_1-f_2+f_3-f_4}}.$$

(10)

BDS currently features open signals from five frequencies [13]: the different quad-frequency combinations that can be obtained using these signals and the associated wavelengths are discussed in Section 5.

4. Hatch–Melbourne–Wübbena Combinations

The main contribution of this paper is to derive measurement combinations, which can be used to reconstruct high-accuracy pseudoranges related to quad-frequency meta-signals. In order to do so, it is necessary to solve for the integer ambiguities related to the quad-frequency combinations derived in Section 3. In the dual-frequency case, this is performed through a rounding operation on the HMW combinations that provide unbiased estimates on the wide-lane ambiguities [31]. Following the approach discussed in [20], quad-frequency HMW combinations can be obtained by considering a general code-carrier combination:

$$C_{wlm,4}={\lambda }_{(i,j,k,h)}{\phi }_{wlm,4}-({\alpha }_1{\rho }_1+{\alpha }_1{\rho }_2+{\alpha }_1{\rho }_3+{\alpha }_1{\rho }_4)$$

(11)

where ${\rho }_1$, ${\rho }_2$, ${\rho }_3$, and ${\rho }_4$ are pseudorange measurements from the different frequencies. The index m is used to denote one of three quad-frequency wide-lane combinations defined in (7) and $(i,j,k,h)$ are the coefficients defined by the corresponding row of ${\mathbf{H}}_4$. ${\alpha }_l$ with $l=1,2,3,4$ are the coefficients to be determined to obtain generalized HMW combinations. Note that in order to preserve the geometric components of the term derived from pseudorange measurements, the following condition needs to be satisfied:

$$\sum _{l=1}^4{\alpha }_l=1.$$

(12)

$C_{wlm,4}$ needs to preserve the properties of dual-frequency HMW combinations and provide a possibly unbiased estimate of the integer ambiguities of ${\phi }_{wlm,4}$. This can be achieved by rearranging (11) in terms of dual-frequency HMW combinations. Note that the last three rows of ${\mathbf{H}}_4$ all contain two 1 s and two $-1$ s. To obtain a general expression for the three quad-frequency HMW combinations defined in (7), four indexes, a, b, c, and d, are introduced. a and c are related to the elements of $(i,j,k,h)$ equal to one, whereas b and d refer to elements equal to $-1$. More specifically, the convention summarized in

Table 1. Convention with the index values for selecting the different measurements as a function of the different quad-frequency wide-lane combinations.

Combination	Weights $(\mathit{i},\mathit{j},\mathit{k},\mathit{h})$	a	b	c	d
${\phi }_{wl1,4}$	$(1,-1,1,-1)$	1	2	3	4
${\phi }_{wl2,4}$	$(1,1,-1,-1)$	1	3	2	4
${\phi }_{wl3,4}$	$(1,-1,-1,1)$	1	2	4	3

is adopted.

Using this convention, (11) can be written as

$$\begin{array}{cc}\hfill C_{wlm,4}& ={\lambda }_{(i,j,k,h)}{\phi }_{wlm,4}-({\alpha }_a{\rho }_a+{\alpha }_b{\rho }_b+{\alpha }_c{\rho }_c+{\alpha }_d{\rho }_d)\hfill \\ \hfill & ={\lambda }_{(i,j,k,h)}({\phi }_a-{\phi }_b+{\phi }_c-{\phi }_d)-({\alpha }_a{\rho }_a+{\alpha }_b{\rho }_b+{\alpha }_c{\rho }_c+{\alpha }_d{\rho }_d)\hfill \\ \hfill & ={\lambda }_{(i,j,k,h)}({\phi }_a-{\phi }_b)-({\alpha }_a{\rho }_a+{\alpha }_b{\rho }_b)\hfill \\ \hfill & +{\lambda }_{(i,j,k,h)}({\phi }_c-{\phi }_d)-({\alpha }_c{\rho }_c+{\alpha }_b{\rho }_b)\hfill \\ \hfill & ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{ab}}}\left[{\lambda }_{ab}({\phi }_a-{\phi }_b)-{\displaystyle \frac{{\lambda }_{ab}}{{\lambda }_{(i,j,k,h)}}}({\alpha }_a{\rho }_a+{\alpha }_b{\rho }_b)\right]\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{cd}}}\left[{\lambda }_{cd}({\phi }_c-{\phi }_d)-{\displaystyle \frac{{\lambda }_{cd}}{{\lambda }_{(i,j,k,h)}}}({\alpha }_c{\rho }_c+{\alpha }_d{\rho }_d)\right],\hfill \end{array}$$

(13)

where ${\lambda }_{ab}$ and ${\lambda }_{cd}$ are the wavelengths of the wide-lane combinations obtained using frequency combinations, $(a,b)$ and $(c,d)$, respectively. These wavelengths are computed analogously to (4). The ${\alpha }_l$ coefficients in square brackets in the last two rows of (13) have to correspond to the pseudorange coefficients of dual-frequency HMW combinations, and the following conditions should be met:

$$\begin{array}{cc}\hfill {\alpha }_a^{\left(1\right)}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{ab}}}{\displaystyle \frac{f_a}{f_a+f_b}}={\displaystyle \frac{f_a(f_a-f_b)}{(f_a-f_b+f_c-f_d)(f_a+f_b)}}\hfill \\ \hfill {\alpha }_b^{\left(1\right)}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{ab}}}{\displaystyle \frac{f_b}{f_a+f_b}}={\displaystyle \frac{f_b(f_a-f_b)}{(f_a-f_b+f_c-f_d)(f_a+f_b)}}\hfill \\ \hfill {\alpha }_c^{\left(1\right)}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{cd}}}{\displaystyle \frac{f_c}{f_c+f_d}}={\displaystyle \frac{f_c(f_c-f_d)}{(f_a-f_b+f_c-f_d)(f_c+f_d)}}\hfill \\ \hfill {\alpha }_d^{\left(1\right)}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{cd}}}{\displaystyle \frac{f_d}{f_c+f_d}}={\displaystyle \frac{f_d(f_c-f_d)}{(f_a-f_b+f_c-f_d)(f_c+f_d)}},\hfill \end{array}$$

(14)

where superscript $\left(1\right)$ was added to indicate that (14) is a first solution satisfying the requirements to build a quad-frequency HMW combination. Using coefficients (14), $C_{wlm,4}$ becomes a linear combination of dual-frequency HMW code-carrier combinations. Thus, $C_{wlm,4}$ is geometry and iono-free. Moreover, it is possible to show that in the absence of satellite and receiver biases and residual noise, $C_{wlm,4}$ is equal to ${\lambda }_{(i,j,k,h)}$ times the integer ambiguities affecting ${\phi }_{wlm,4}$. Following the derivation in [20], a second solution for the ${\alpha }_l$ coefficients can be found by choosing a different measurement order. $C_{wlm,4}$ can also be written as:

$$\begin{array}{cc}\hfill C_{wlm,4}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{ad}}}\left[{\lambda }_{ad}({\phi }_a-{\phi }_d)-{\displaystyle \frac{{\lambda }_{ad}}{{\lambda }_{(i,j,k,h)}}}({\alpha }_a{\rho }_a+{\alpha }_d{\rho }_d)\right]\hfill \\ \hfill & +{\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{cb}}}\left[{\lambda }_{cb}({\phi }_c-{\phi }_b)-{\displaystyle \frac{{\lambda }_{cb}}{{\lambda }_{(i,j,k,h)}}}({\alpha }_c{\rho }_c+{\alpha }_b{\rho }_b)\right]\hfill \end{array}$$

(15)

where ${\lambda }_{ad}$ and ${\lambda }_{cb}$ are the wavelengths of the wide-lane combinations obtained using frequency combinations, $(a,d)$ and $(c,b)$, respectively. The representation in (15) leads to a second solution for the ${\alpha }_l$ coefficients:

$$\begin{array}{cc}\hfill {\alpha }_a^{\left(2\right)}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{ad}}}{\displaystyle \frac{f_a}{f_a+f_d}}={\displaystyle \frac{f_a(f_a-f_d)}{(f_a-f_b+f_c-f_d)(f_a+f_d)}}\hfill \\ \hfill {\alpha }_b^{\left(2\right)}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{cb}}}{\displaystyle \frac{f_b}{f_c+f_b}}={\displaystyle \frac{f_b(f_c-f_b)}{(f_a-f_b+f_c-f_d)(f_c+f_b)}}\hfill \\ \hfill {\alpha }_c^{\left(2\right)}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{cb}}}{\displaystyle \frac{f_c}{f_c+f_b}}={\displaystyle \frac{f_c(f_c-f_b)}{(f_a-f_b+f_c-f_d)(f_c+f_b)}}\hfill \\ \hfill {\alpha }_d^{\left(2\right)}& ={\displaystyle \frac{{\lambda }_{(i,j,k,h)}}{{\lambda }_{ad}}}{\displaystyle \frac{f_d}{f_a+f_d}}={\displaystyle \frac{f_d(f_a-f_d)}{(f_a-f_b+f_c-f_d)(f_a+f_d)}},\hfill \end{array}$$

(16)

where superscript $\left(2\right)$ is used to indicate that (16) is a second solution for obtaining the range coefficients. Equations (14) and (16) are independent and can be combined to obtain a general solution for ${\alpha }_l$ coefficients:

$$\begin{array}{cc}\hfill {\alpha }_a& ={\alpha }_a^{\left(1\right)}\beta +{\alpha }_a^{\left(2\right)}(1-\beta )={\displaystyle \frac{f_a}{f_a-f_b+f_c-f_d}}\left[\beta {\displaystyle \frac{f_a-f_b}{f_a+f_b}}+(1-\beta ){\displaystyle \frac{f_a-f_d}{f_a+f_d}}\right]\hfill \\ \hfill {\alpha }_b& ={\alpha }_b^{\left(1\right)}\beta +{\alpha }_b^{\left(2\right)}(1-\beta )={\displaystyle \frac{f_b}{f_a-f_b+f_c-f_d}}\left[\beta {\displaystyle \frac{f_a-f_b}{f_a+f_b}}+(1-\beta ){\displaystyle \frac{f_c-f_b}{f_c+f_b}}\right]\hfill \\ \hfill {\alpha }_c& ={\alpha }_c^{\left(1\right)}\beta +{\alpha }_c^{\left(2\right)}(1-\beta )={\displaystyle \frac{f_c}{f_a-f_b+f_c-f_d}}\left[\beta {\displaystyle \frac{f_c-f_d}{f_c+f_d}}+(1-\beta ){\displaystyle \frac{f_c-f_b}{f_c+f_b}}\right]\hfill \\ \hfill {\alpha }_d& ={\alpha }_d^{\left(1\right)}\beta +{\alpha }_d^{\left(2\right)}(1-\beta )={\displaystyle \frac{f_d}{f_a-f_b+f_c-f_d}}\left[\beta {\displaystyle \frac{f_c-f_d}{f_c+f_d}}+(1-\beta ){\displaystyle \frac{f_a-f_d}{f_a+f_d}}\right]\hfill \end{array}$$

(17)

where $\beta$ is a free parameter, which can be selected according to several optimization criteria such as the minimization of the noise of the resulting quad-frequency HMW combination. The reduction in the noise is indeed of paramount importance for achieving faster solution convergence. A possibility is to minimize the variance of the pseudorange component of (11):

$$\mathcal{L}\left(\beta \right)=\sum _{i=1}^4{\sigma }_i^2{\alpha }_i^2=\sum _{i=1}^4{\sigma }_i^2{\left[{\alpha }_i^{\left(1\right)}\beta +{\alpha }_i^{\left(2\right)}(1-\beta )\right]}^2,$$

(18)

where ${\sigma }_i^2$ are the variances of the single-frequency pseudoranges and can be determined from the Carrier-To-Noise Power Spectral Density Ratio ($C/N_0$) values estimated by the receiver. In this paper, ${\sigma }_i^2$ is set proportional to the inverse of the $C/N_0$ expressed in linear units:

$${\sigma }_i^2\propto {\displaystyle \frac{1}{{\left.\frac{C_i}{N_0}\right|}_{lin}}},$$

(19)

where ${\left.{\displaystyle \frac{C_i}{N_0}}\right|}_{lin}$ is the $C/N_0$ of the ith signal component expressed in linear units. Note that the proportionality constant for the determination of ${\sigma }_i^2$ only scales $\mathcal{L}\left(\beta \right)$ without influencing the location of its minimum. The cost function $\mathcal{L}\left(\beta \right)$ is quadratic in $\beta$ and its minimum is attained for

$${\beta }^{*}={\displaystyle \frac{{\sum }_{i=0}^3{\sigma }_i^2{\alpha }_i^{\left(2\right)}\left({\alpha }_i^{\left(2\right)}-{\alpha }_i^{\left(1\right)}\right)}{{\sum }_{i=0}^3{\sigma }_i^2{\left({\alpha }_i^{\left(2\right)}-{\alpha }_i^{\left(1\right)}\right)}^2}}.$$

(20)

The reminder of the paper mostly considers the case where $\beta ={\beta }^{*}$, which minimizes the noise of the HMW combination. Equation (17) is general and applies to all the combinations defined by the last three rows of ${\mathbf{H}}_4$. The different cases can be found by using the parameters reported in Table 1.

5. The Beidou Case

Since BDS currently offers open signals on five different frequencies [13], five quad-frequency configurations exist where, at each time, one of the BDS signals is excluded.

Table 2. Frequencies and wavelengths of BDS open signals and of some wide-lane combinations.

Signal/Combination	Frequency (MHz)	Wavelength (Meters)
B1C	1575.42	0.19
B1I	1561.098	0.192
B3I	1268.52	0.236
B2b	1207.14	0.248
B2a	1176.45	0.255
B1C − B1I	14.322	20.932
B2b − B2a	30.69	9.768
B3I − B2b	61.38	4.884
B3I − B2a	92.07	3.256
B1I − B3I	292.578	1.025
B1C − B3I	306.9	0.977
Conf. 1, B3I excluded
B1C − B1I − B2b + B2a	16.368	18.316
B1C − B1I + B2b − B2a	45.012	6.66
B1C + B1I − B2b − B2a	752.928	0.398
Conf. 2, B1I excluded
B1C − B3I − B2b + B2a	276.21	1.085
B1C − B3I + B2b − B2a	337.59	0.888
B1C + B3I − B2b − B2a	460.35	0.651
Conf. 3, B1C excluded
B1I − B3I − B2b + B2a	261.888	1.145
B1I − B3I + B2b − B2a	323.268	0.927
B1I + B3I − B2b − B2a	446.028	0.672
Conf. 4, B2b excluded
B1C − B1I − B3I + B2a	77.748	3.856
B1C − B1I + B3I − B2a	106.392	2.819
B1C + B1I − B3I − B2a	691.548	0.434
Conf. 5, B2a excluded
B1C − B1I − B3I + B2b	47.058	6.371
B1C − B1I + B3I − B2b	75.702	3.960
B1C + B1I − B3I − B2b	660.858	0.454

provides the frequencies and wavelengths of the BDS open signals and of some wide-lane combinations obtainable considering combination coefficients equal to $\pm 1$. All five quad-frequency configurations are considered. The B1I and B1C components are closely spaced, and their center frequencies differ by only $14.322$ MHz, resulting in a wide-lane combination with a wavelength of about 21 m. The B2a and B2b components form the ACE-BOC modulation and are separated by only $30.69$ MHz, resulting in a wide-lane combination with a wavelength of about 10 MHz. The integer ambiguities of such combinations can be easily solved directly using the corresponding HMW combination [19]. Combinations with wavelengths on the order of a few meters are also found when considering the B3I, B2a, and B2b components.

In the following, two quad-frequency signal configurations are considered. In the first configuration, which is illustrated in the upper part of Figure 1, the B3I component is excluded. In this case, the ACE-BOC components are combined together with the B1I and B1C signals, which have often been considered jointly as a single meta-signal [22,39,40]. The frequencies and wavelengths of the three quad-frequency wide-lane combinations of this configuration are provided in the block denoted as “Conf. 1” in Table 2. The first two quad-frequency combinations of this configuration are characterized by wavelengths of $18.316$ and $6.66$ m, respectively. This configuration was selected for the large wavelengths, which are expected to lead to a reliable integer ambiguity resolution. Moreover, other GNSSs, such as Galileo, are considering placing a new signal component close to the $1575.42$ MHz frequency [41]. Thus, the study of this configuration can provide useful insights on the performance achievable by combining two dual-frequency meta-signals spaced by about 500 MHz. The third combination in this configuration is characterized by a wavelength of approximately 40 cm.

The second signal configuration is illustrated in the bottom part of Figure 1, and the related properties can be found in block “Conf. 2” of Table 2. In this case, the B1I signal is excluded, and a configuration similar to that obtained for the Galileo signal landscape is found. The B3I component plays a similar role to the Galileo E6 signal.

The properties of the remaining configurations are illustrated in the last three blocks of Table 2. The third configuration is obtained by removing the B1C components and has properties similar to those of the second configuration. This is due to the fact that the B1C and B1I signals are close in frequency and thus play similar roles for the second and third configurations, respectively. The last two configurations are obtained by excluding the B2b and B2a components, respectively. These configurations have characteristics similar to those of the first configuration. In the following, only configurations number one and two are analyzed in detail to avoid repetition of similar findings.

6. Material and Methods

This section describes the experiments and the methods used to analyze the quad-frequency wide-lane combinations obtained considering open BDS measurements.

The observations used in this study were recorded using two high-precision GNSS receivers configured in a zero-baseline setup. In the experimental configuration, two PolaRx5S receivers from Septentrio, Leuven, Belgium were connected through a splitter to a shared choke-ring antenna mounted on the rooftop of a building of the Warsaw University of Technology (WUT). A view of the rooftop antenna is provided in the right part of Figure 2, whereas a summary of the key characteristics of the different equipment involved in the experiment can be found in

Table 3. Key information of the zero-baseline experimental configuration.

Component	Specification
Receiver	Septentrio PolaRx5S, (Septentrio, Leuven, Belgium)
Antenna	Leica AT504, (Leica Geosystems, Heerbrugg, Switzerland)
GNSS Spiter	GPS source (S14)
Antenna location	52°13′15″ N, 21°00′37″ E

. The PolarRx5S receivers support multiple constellations and frequencies and are equipped with an ultra-low-noise Oven Controlled Crystal Oscillator (OCXO), offering several key advantages, including enhanced measurement precision and robust signal tracking. During the measurements, the receivers operated in static mode and were configured to track the full GNSS spectrum at a 1 Hz frequency. Since two receivers were used, the collected measurements could be combined and differentiated to eliminate the most common error sources. A view of the two receivers can be found in the left part of Figure 2. Employing a zero-baseline configuration in the experimental setup is essential to isolate receiver-specific errors and eliminate all spatially correlated errors [32,42]. This approach enables a precise evaluation of the fundamental quality of GNSS measurements and of the derived combinations.

The measurements collected according to the experimental setup detailed above were analyzed in undifferenced mode, considering the quad-frequency HMW combinations described in Section 4 and in double-difference mode for the analysis of wide-lane carrier phase combinations. For the quad-frequency HMW combinations, the processing scheme detailed in Figure 3 was adopted.

Measurements from the BDS signal configurations described in Section 5 were first corrected for satellite biases using the post-processed products from the PPP-Wizard project maintained by Centre National d’Études Spatiales (CNES) [33] (products retrieved from http://www.ppp-wizard.net/products/POST_PROCESSED/ (accessed on 15 January 2025)). Code and carrier biases are provided as BIA files and formatted according to the Solution INdependent EXchange (SINEX) standard [43]. Quad-frequency HMW combinations are then formed. Using an approach similar to that of [35], fractional receiver biases are also estimated and removed from the HMW combinations. In [35], the fractional receiver bias is estimated as the value that maximizes the number of floating ambiguities estimated though the HMW combinations approaching an integer value. Here, the fractional receiver bias is estimated as a weighted mean of the HMW fractional parts, where the weights are obtained as the inverse of the variances of HMW combinations. The final combinations are then analyzed, showing the benefits of these types of combinations.

For the analysis of carrier phase wide-lane combinations, dual-differences between receivers and with respect to a pivot satellite were computed. This type of processing allows one to remove most error sources and to analyze only the residual noise on carrier phase combinations.

7. Results

Sample results obtained by processing multi-frequency BDS data are provided in this section. Quad-frequency HMW combinations are analyzed first. Carrier phase wide-lane combinations are then considered.

7.1. Quad-Frequency HMW Combinations

In the absence of residual satellite and receiver biases, HMW combinations provide unbiased estimates of the integer ambiguities of the corresponding wide-lane carrier phase combinations. They also play the same role in the quad-frequency case, where the success of the ambiguity resolution process depends on several factors, including the combination wavelength. In the next figures, quad-frequency HMW combinations are analyzed for decreasing wavelengths and for different satellites. Configuration 1, where B3I signals are excluded, is considered at first. Figure 4 shows the undifferenced quad-frequency HMW combinations obtained for a wavelength equal to $18.32$ m and a combination signature equal to $(1,-1,-1,1)$ with frequencies ordered in a decreasing way, ($1575.42$ MHz, $1561.098$ MHz, $1207.14$ MHz, $1176.45$ MHz).

The figure also provides decision areas, which are depicted in gray. The ultimate goal of the HMW combinations is to estimate the ambiguities associated with the corresponding wide-lane combinations. The gray bands shown in Figure 4 map the HMW combinations to a single integer value and help understanding the quality of the combination. Measurements well within the decision areas lead to reliable ambiguity resolution.

The quad-frequency case shown in Figure 4 is obtained as the difference between the ACE-BOC and B1X combination. Differencing measurements from the B1C and B1I signals leads to this latter combination. These dual-frequency combinations are characterized by very large wavelengths, and the resulting quad-frequency case inherits this property. All the combinations shown in Figure 4 are well within the decision region that univocally maps them to the same integer value. The HMW combinations are also well aligned along integer values, confirming the effectiveness of the bias removal process described in Section 6. Figure 4 also shows the case of satellite with Pseudo Random Number (PRN) 22. This is a very low-elevation satellite tracked by the receiver for a very limited period of time. While the positioning engine will exclude the measurements from this satellite, the corresponding HMW combinations are aligned around an integer value, and the large wavelength allows for a reliable ambiguity resolution. As expected, the variations in the HMW combinations depend on the signal quality, which is determined by the satellite elevation being the receiver static and under open-sky conditions. Combinations originating from satellites with high elevations are less noisy.

The results shown in Figure 4 were obtained for $\beta ={\beta }^{*}$, which minimizes the noise variance of the code component of the HMW combination according to cost function (18). The impact of $\beta$ is studied in Figure 5, which shows the standard deviation of the quad-frequency HMW combination with $\lambda =18.32$ m as a function of the optimization parameter $\beta$. Standard deviations were estimated using the measurement combination from satellite C19. As $\beta$ varies, the standard deviation of the quad-frequency HMW combination changes from a few meters to about 30 cm.

The theoretical minimum is found using (20) and is equal to ${\beta }^{*}=0.00425$. Although the standard deviations shown in Figure 5 were empirically estimated from the actual measurement combinations, a good agreement between theoretical and empirical results for ${\beta }^{*}$ is observed. Note that in Figure 5, standard deviations were computed for a discrete set of $\beta$ values, which were varied with a step equal to $0.1$. In the remainder of this paper, only findings for $\beta ={\beta }^{*}$ are presented.

Results obtained for the case with a wavelength equal to $6.66$ m and a signature equal to $(1,-1,1,-1)$ are considered in Figure 6, which shows the related undifferenced HMW combinations for eight satellites.

Also, in this case, all the HMW combinations are well within a decision region, leading to the same integer ambiguity. Moreover, all combinations align along an integer value. This case, which is obtained as the sum of the ACE-BOC and B1X combinations, allows for reliable ambiguity resolution using undifferenced measurements. While the HMW combinations shown in the previous figures were computed by removing the satellite biases provided by the PPP-wizard portal, their large wavelengths allow for reliable ambiguity resolution even in the absence of such bias corrections. This fact is highlighted in Figure 7, which shows the residual fractional biases of the HMW measurement combinations as a function of time. These biases were computed using the precise products from the CNES PPP-wizard project and are all within a few hundredths of a cycle.

For example, for the combination with a wavelength equal to $18.32$ m considered in the top left box of Figure 7, all biases differ by less than $0.06$ cycles. Similar conclusions can be drawn for the other combinations. The biases in the bottom row of Figure 7 are related to the ACE-BOC and B1X combinations: they will play a marginal role also in these cases. The results shown in Figure 7 indicate that combinations obtained from the ACE-BOC and B1X can lead to reliable ambiguity resolution even without applying satellite bias corrections.

This is not the case for other quad-frequency combinations that are characterized by significantly smaller wavelengths. The last quad-frequency wide-lane combination obtained for configuration 1 is analyzed in Figure 8, which shows the related HMW combinations. In this case, the resulting wavelength is equal to only 40 cm, and the HMW combinations can lead to successful ambiguity resolution only for measurements from high-elevation satellites. In Figure 8, only satellite with PRN 20 leads to HMW combinations always within the same decision region corresponding to the same integer ambiguity. For all the other satellites, noisy measurements are obtained. In this case, it is not possible, in general, to reliably solve for the integer ambiguities instantaneously. The combinations will need to be filtered for several epochs to remove the impact of noise and other errors. This fact is confirmed by the results reported in

Table 4. Residual fractional biases and standard deviations of the quad-frequency HMW combinations obtained for configuration 1.

PRN	Fractional Mean (Cycles)	Standard Dev. (Cycles)
	$\lambda =18.32$ m
19	$-0.014$	$0.02$
20	$0.0$	$0.01$
22	$0.044$	$0.163$
23	$0.021$	$0.021$
29	$-0.028$	$0.049$
32	$-0.02$	$0.023$
35	$-0.045$	$0.037$
37	$-0.03$	$0.017$
	$\lambda =6.66$ m
19	$9.89\times {10}^{-3}$	$0.02$
20	$-1.78\times {10}^{-15}$	$0.008$
22	$1.15\times {10}^{-1}$	$0.158$
23	$-4.06\times {10}^{-3}$	$0.021$
29	$2.92\times {10}^{-2}$	$0.041$
32	$7.25\times {10}^{-3}$	$0.023$
35	$3.27\times {10}^{-2}$	$0.030$
37	$8.72\times {10}^{-3}$	$0.014$
	$\lambda =0.40$ m
19	$-0.185$	$0.282$
20	$-7.105\times {10}^{-15}$	$0.135$
22	$0.269$	$5.004$
23	$-0.433$	$0.4$
29	$-0.06$	$0.858$
32	$0.037$	$0.38$
35	$-0.319$	$0.573$
37	$0.067$	$0.263$

, which provides the residual fractional biases and standard deviations of the quad-frequency HMW combinations obtained for configuration 1.

The statistics reported in Table 4 are expressed in the cycle, and the HMW combinations obtained for the case with wavelength equal to 40 cm are characterized by large standard deviations that prevent the ambiguity resolution process. It is also noted that, in this case, the bias removal process is not completely successful, and three combinations are characterized by large fractional biases above $0.2$ cycles.

While the cause of such residual biases requires additional investigations, it is probably related to the impossibility of correcting several error sources when dealing with undifferenced measurements. This fact was analyzed by considering between receiver single differences of the HMW combinations. More specifically, Figure 9 shows the single difference quad-frequency HMW combinations for the case with wavelength equal to $0.4$ m.

In this case, most errors observed in the undifferenced case disappear. All the HMW single differences are well aligned along integer values, and their dispersion is significantly reduced with respect to the previous case. When measurements from a second receiver are available, it is possible to correct for most error sources and reliably solve for the integer ambiguities instantaneously. The residual fractional biases and standard deviations obtained in the single-difference case are provided in

Table 5. Residual fractional biases and standard deviations of the quad-frequency single difference between receivers HMW combinations obtained for configuration 1 for the case with wavelength equal to $0.4$ m.

PRN	Fractional Mean (Cycles)	Standard Dev. (Cycles)
	$\lambda =0.40$ m
19	$-4.461\times {10}^{-3}$	$0.043$
20	$6.529\times {10}^{-3}$	$0.028$
22	$-2.060\times {10}^{-1}$	$0.466$
23	$-5.206\times {10}^{-3}$	$0.072$
29	$-2.428\times {10}^{-2}$	$0.164$
32	$1.677\times {10}^{-3}$	$0.075$
35	$-5.163\times {10}^{-3}$	$0.091$
37	$4.441\times {10}^{-16}$	$0.045$

. These results confirm the findings suggested by Figure 9.

An analysis similar to that performed for configuration 1 was repeated for the second case described in Figure 1, which utilizes the B3I component instead of the B1I signal. The wide-lane combinations obtained for this configuration are characterized by wavelengths ranging from 65 cm to $1.09$ m. For such cases, results similar to those discussed for the last combination of configuration 1 are obtained. The wavelengths are too small to allow for reliable single-epoch ambiguity resolution. This fact is analyzed for the combination with wavelength equal to 89 cm in Figure 10, which provides the corresponding undifferenced HMW combinations.

It is possible to reliably solve for the integer ambiguity only for high elevation satellites such as the one with PRN 20. Moreover, it is not possible to completely compensate for the fractional biases, and some of the combinations in Figure 10 do not align with integer values. Similar results were obtained for the other wide-lane combinations of configuration 2. The residual fractional biases and standard deviations of the quad-frequency HMW combinations obtained for configuration 2 are provided in

Table 6. Residual fractional biases and standard deviations of the quad-frequency HMW combinations obtained for configuration 2.

PRN	Fractional Mean (Cycles)	Standard Dev. (Cycles)
	$\lambda =1.09$ m
19	$-0.147$	$0.170560$
20	$0.0$	$0.080459$
22	$0.226$	$1.669126$
23	$-0.140$	$0.213662$
29	$0.212$	$0.579391$
32	$-0.282$	$0.206885$
35	$0.373$	$0.309368$
37	$0.467$	$0.139981$
	$\lambda =0.89$ m
19	$-0.154$	$0.170852$
20	$0.0$	$0.092987$
22	$0.332$	$1.691244$
23	$-0.058$	$0.212089$
29	$0.399$	$0.576749$
32	$-0.161$	$0.212230$
35	$0.429$	$0.312745$
37	$-0.447$	$0.147996$
	$\lambda =0.65$ m
19	$-0.15$	$0.206672$
20	$-0.052$	$0.100400$
22	$-0.403$	$1.900081$
23	$-0.428$	$0.263385$
29	$-0.228$	$0.634435$
32	$0.0$	$0.246402$
35	$-0.056$	$0.380935$
37	$0.259$	$0.172312$

, which confirms that, for these cases, it is not possible to perform instantaneous ambiguity resolution reliably.

Similarly to the case with wavelength equal to 40 cm, single-difference HMW combinations were analyzed, as shown in Figure 11. The differentiation process removes most errors observed in the undifferenced case, and the HMW combinations now align along integer values.

This fact is confirmed by the results provided in

Table 7. Residual fractional biases and standard deviations of the quad-frequency single difference between receivers HMW combinations obtained for configuration 2 for the case with wavelength equal to $0.89$ m.

PRN	Fractional Mean (Cycles)	Standard Dev. (Cycles)
	$\lambda =0.65$ m
19	$-1.164\times {10}^{-3}$	$0.0258$
20	$3.723\times {10}^{-3}$	$0.0163$
22	$-1.484\times {10}^{-2}$	$0.2268$
23	$-1.223\times {10}^{-2}$	$0.0402$
29	$1.041\times {10}^{-3}$	$0.0900$
32	$-2.567\times {10}^{-3}$	$0.0443$
35	$1.540\times {10}^{-3}$	$0.0501$
37	$4.440\times {10}^{-16}$	$0.0253$

, which provides the residual fractional biases and standard deviations of the quad-frequency single difference between receivers HMW combinations obtained for configuration 2 for the case with wavelength equal to $0.89$ m.

The residual fractional biases are negligible, and the standard deviations are in the order of a few hundredths of cycles. Similar results were obtained for the remaining combinations with wavelengths equal to $1.085$ and $0.65$ m, respectively.

7.2. Wide-Lane Carrier Phase Combinations

HMW combinations were analyzed in the previous section, considering undifferenced and single-difference measurements. In this section, quad-frequency wide-lane phase combinations are analyzed considering double differences, which allows the removal of biases and drifts originating, for instance, from uncompensated ionospheric effects. Figure 12 shows the fractional part of the double-difference carrier phase wide-lane combinations obtained considering configuration 1 and indexes $(1,-1,-1,1)$. Satellite with PRN 19 was used as a pivot. Only the fractional part of the double differences is provided to improve the clarity of the figure. Since data were collected in a zero-baseline configuration, double-difference carrier phase measurements are expected to align along integer values [32,42]. This is indeed the case in Figure 12, where all the fractional parts of the double differences align along zero.

Most of the double differences in Figure 12 are within $\pm 0.02$ cycles, which corresponds to about $36.6$ cm in this case, where the wavelength is $18.32$ m. Similar results were obtained for the other cases. For instance, the combinations with a wavelength equal to 40 cm are considered in Figure 13.

Also, in this case, the fractional part of the double differences is well aligned along zero, and most measurements are within the $\pm 0.02$ cycle region. Note that all the wide-lane combinations considered in this paper are characterized by combination signatures with elements all equal to $\pm 1$. This implies that the noise amplification factor in cycles [9,16] is the same for all the combinations. For this reason, carrier phase combinations originating from the same measurements are expected to have the same standard deviation. This fact is clearly visible when comparing Figure 12 and Figure 13: similar measurement dispersions are observed.

The residual fractional biases and standard deviations of the quad-frequency double-difference carrier phase combinations obtained for configuration 1 are summarized in

Table 8. Residual fractional biases and standard deviations of the quad-frequency double-difference carrier phase combinations obtained for configuration 1. A satellite with PRN 19 was used as a pivot.

PRN	Fractional Mean (Cycles)	Standard Dev. (Cycles)
	$\lambda =18.32$ m
20	$3.7\times {10}^{-5}$	$0.0038$
22	$5.72\times {10}^{-4}$	$0.0235$
23	$-1.24\times {10}^{-4}$	$0.0056$
29	$-2.0\times {10}^{-5}$	$0.0112$
32	$-2.38\times {10}^{-4}$	$0.006$
35	$-8.5\times {10}^{-5}$	$0.0067$
37	$-2.8\times {10}^{-5}$	$0.0044$
	$\lambda =6.66$ m
20	$-1.72\times {10}^{-4}$	$0.0038$
22	$4.567\times {10}^{-3}$	$0.0224$
23	$4.1\times {10}^{-5}$	$0.0055$
29	$-8.4\times {10}^{-5}$	$0.0111$
32	$-5.1\times {10}^{-5}$	$0.0059$
35	$-6.8\times {10}^{-5}$	$0.0068$
37	$-8.0\times {10}^{-5}$	$0.0045$
	$\lambda =0.40$ m
20	$-1.6\times {10}^{-4}$	$0.0039$
22	$4.318\times {10}^{-3}$	$0.0202$
23	$-2.21\times {10}^{-4}$	$0.0055$
29	$-2.06\times {10}^{-4}$	$0.0114$
32	$1.04\times {10}^{-4}$	$0.0061$
35	$-1.4\times {10}^{-5}$	$0.0066$
37	$-6.6\times {10}^{-5}$	$0.0044$

, which confirms the findings obtained from Figure 12 and Figure 13. The fractional part of the double differences has a mean close to zero, and significantly reduced standard deviations are observed. As expected, the three combinations analyzed in Table 8 are characterized by practically the same standard deviations.

Similar results were obtained for configuration 2, which is not discussed in more detail here.

8. Discussion

In this paper, quad-frequency wide-lane combinations derived from the GNSS meta-signal paradigm were analyzed. More specifically, the carrier phase combinations obtained are scaled versions of the subcarrier components that split the different meta-signal components on different RFs. These combinations are a first step toward the reconstruction of high-accuracy pseudoranges for meta-signals with four components. In particular, high-accuracy pseudoranges can be obtained by correcting wide-lane carrier phase combinations for their integer ambiguities, which are estimated by rounding the corresponding HMW measurements. Since, in the quad-frequency case, there exist three wide-lane combinations for a given set of four components, the accuracy of the reconstructed measurements depends on the selected combination. While additional work is needed to unequivocally define the concept of synthetic quad-frequency meta-signal pseudorange, a possibility could be based on the wide-lane combination with the largest frequency separation, either dual- or quad-frequency, obtainable with the individual signal components. Its ambiguity could be solved using a cascaded approach based on the other wide-lane combinations with larger wavelengths and through a noise minimization process. This approach would be similar to that adopted in the TCAR process [16,35], where the accuracy of the terms used for ambiguity resolution through rounding is improved at each step.

When BDS open signals are considered, extra wide-lane combinations with wavelengths of several meters are obtained. These allow reliable single-epoch ambiguity resolution, which can be used to reconstruct high-accuracy meta-signal pseudoranges. They can be used together with dual-frequency wide-lane combinations. In this respect, it is important to note that the transformation between quad- and dual-frequency combinations is invertible. Thus, a pair made of a quad- and a dual-frequency combination allows the direct recovery of the complementary quad- and dual-frequency combinations.

Figure 14 shows the undifferenced dual-frequency HMW combinations obtained using measurements from the B1C and B1I signals. This is the GNSS signal combination with the largest wavelength and can be used to reliably solve for the ambiguities associated to the B1C/B1I signal pair. The quad-frequency combination with signature $(1,-1,-1,1)$ and wavelength $18.32$ m has the second largest wavelength obtainable using coefficients all equal to $\pm 1$. Also, the associated integer ambiguities can be reliably solved. From these combinations, it is then possible to directly obtain the ambiguity associated with the ACE-BOC and the complementary quad-frequency combination with signature $(1,-1,-1,1)$. Thus, the optimal sets of combinations can be selected to solve for the wide-lane measurements with the smallest wavelength.

As better discussed in the Section 9, the main limiting factor preventing single-epoch ambiguity resolution in the undifferenced case remains the presence of biases, which cannot be completely corrected using precise PPP products. These biases are effectively removed when considering single differences between receivers.

9. Conclusions

This paper analyzed meta-signal-inspired quad-frequency measurement combinations obtained using BDS open signals. Specific focus was devoted to narrow-lane and wide-lane combinations that are obtained through a Hadamard transform of order four. Quad-frequency HMW combinations were also obtained and characterized. Since five open signals are currently available from BDS-3 satellites, five quad-frequency configurations exist. The analysis performed focused on two such configurations. The first was selected because it leads to two quad-frequency combinations with large wavelengths in the order of several meters. The second configuration is characterized by signals with RF frequencies close to those of the Galileo open signals. From the analysis, it emerged that single-epoch ambiguity resolution can be performed reliably on uncombined measurements from quad-frequency wide-lane combinations with wavelengths equal to $18.32$ and $6.66$ m, respectively. These combinations are obtained by combining signal pairs with individual components closely spaced in frequency. Moreover, their noise can be minimized with a proper selection of the coefficients used to combine the measurements. In these cases, the impact of satellite biases is negligible, and the ambiguities can be solved without correcting them. For combinations with shorter wavelengths, the single-epoch ambiguity resolution process becomes challenging when considering the undifferenced case. The main error source remains the presence of biases, which cannot be completely corrected using PPP products. This fact becomes evident when moving from undifferenced measurements to single differences between receivers. In this second case, most errors common to both receivers canceled out. In the zero-baseline experiment conducted, single-difference HMW combinations aligned along integer values with reduced variations. Future work will include the analysis of cascaded approaches for the single-epoch ambiguity resolution process in the context of GNSS meta-signals.