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### Session E1: Advanced Technologies in High Precision GNSS Positioning

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**Meta-signal Inspired Quad-Frequency GNSS Measurement Combinations**

*Daniele Borio, European Commission Joint Research Centre (JRC); Melania Susi, Topcon Positioning Systems Inc.; Kinga Wezka, Warsaw University of Technology*

**Date/Time:** Wednesday, Sep. 18, 11:48 a.m.

Peer Reviewed

Modern Global Navigation Satellite Systems (GNSS) provide several signals on different frequencies. The European GNSS, Galileo, currently offers Open Service (OS) signals in four frequency bands whereas third generation BeiDou Navigation Satellite System (BDS) satellites broadcast signals with five different centre frequencies. At the same time, modern GNSS receivers are able to simultaneously process most, if not all, the OS GNSS signals broadcast in the different frequencies, delivering multi-frequency measurements. This opens several possibilities where GNSS measurements are effectively combined to remove the impact of the ionosphere, reduce observation noise, and increase the measurement wavelength. Such combinations can be exploited, for example, to enhance the solution accuracy and the convergence performance in Precise Point Positioning (PPP) algorithms and Real Time Kinematic (RTK) using long baselines.

In this respect, substantial work has been performed to obtain optimal dual, triple and quad-frequency (e.g. Duong et al. 2019, Liu et al 2021, Deo et al. 2020) linear measurement combinations. Depending on the application and the subsequent positioning approach, optimality is defined with respect to different criteria and constraints. These criteria include the elimination of the ionospheric delay, the geometry, and the minimization of the measurement noise. In particular, different combinations impose a trade-off between noise reduction and wavelength increase that should be carefully considered for high accuracy applications.

In parallel to the developments in the measurement domain, significant research work has been performed at the receiver signal processing level where advanced acquisition and tracking algorithms have been proposed to jointly process signals from different frequencies. These algorithms exploit the fact that signals broadcast by a single satellite share similar delays and proportional Doppler frequencies. Thus, joint processing schemes can be devised to improve receiver sensitivity and provide more accurate measurements. A possible approach to multi-frequency GNSS signal processing is the so called meta-signal paradigm (Ortega et al. 2020), which has been mainly developed for the dual-frequency case. A GNSS meta-signal is obtained by considering as a single entity two side-band components from two different frequencies to generate a signal with a wider bandwidth and leading to improved performance. Using this approach, the two side-band components can be acquired and tracked together estimating common signal parameters. An example of GNSS meta-signal is the Galileo Alternative Binary Offset Carrier (AltBOC), which is obtained by combining the E5a and E5b components. The AltBOC can be processed as a single signal with single estimates for its code delay and carrier phase. These estimates are then converted into AltBOC pseudorange and carrier phase measurements.

Jointly tracking side-band components from different frequencies is not free from challenges, such as the presence of possible ambiguities in the joint code delay estimates. These ambiguities are generated by the interaction between the side-band components that need to be aligned in phase to constructively combine the side-band received power. A possible solution is to introduce an additional tracking loop, which is used to track the phase difference between the two side-band components that can be finally aligned in phase. This loop, often denoted as Subcarrier Phase Lock Loop (SPLL), provides differential phase estimates that can be used to generate additional measurements denoted as subcarrier phases.

In a GNSS receiver, the measurement and signal processing domains are strictly related and a concept arising in one domain often has its equivalent in the other. This is the case, for instance, of the measurements generated by meta-signal processing algorithms. In particular Borio and Gioia (2023) showed that it is possible to reconstruct meta-signal measurements from side-band observations. The meta-signal carrier phase can be computed as the narrow-lane linear combination of the side-band carrier phases. Similarly, the meta-signal subcarrier phase correspond to the wide-lane linear combination of the side-band carrier phases. Meta-signal pseudoranges are mixed code-carrier observations and can be reconstructed using an approach similar to the Hatch-Melburne-Wubbena (HMW) code-carrier combination (Hatch 1982). Side-band pseudoranges are at first combined to reduce noise. Then a code-carrier combination is formed using the subcarrier phase. This combination converges to the HMW for small side-band frequency separations and it is used to estimate the subcarrier phase ambiguities and obtain high-accuracy pseudoranges that fully exploit the meta-signal benefits.

While significant work has been performed considering dual-frequency meta-signals, limited research has been conducted to extend the theory to a higher number of signals. Recent work (Borio 2024) has shown that the meta-signal paradigm can be actually generalized to a number of frequencies equal to a power of two. This includes the quad-frequency case, where four signals are jointly processed. In such case, a quad-frequency meta-signal is obtained with a common carrier and three subcarrier components. The meta-signal extension to four frequencies has a direct impact in the GNSS measurement domain since quad-frequency generalizations of the narrow-lane and wide-lane carrier combinations are obtained.

This is the main goal of this paper that aims at defining and characterizing quad-frequency GNSS measurement combinations, which generalize to four dimensions the narrow- and wide-lane combinations. Moreover, a generalization of the HMW code-carrier combination with four signals is also provided.

The meta-signal generalization considering a number of signals equal to a power of two is obtained through the Hadamard transform (Georgiou et al., 2003), which maps the signal parameters to be estimated in a new set of code delays, Doppler frequencies and carrier phases. All the signal components from the different frequencies are used to estimate jointly the new set of signal parameters. The Hadamard transform, which is built recursively for the different powers of two, can be used to obtain quad-frequency generalizations of the narrow and wide-lane combinations.

When the dual-frequency case is considered, the narrow and wide-lane carrier phase combinations are obtained as the sum and difference of the side-band carrier phases in cycles, respectively. This is equivalent to applying a matrix transformation to the vector with the side-band carrier phases. When observations are expressed in cycles, the matrix defining such transformation has two rows: the first with two ones and the second with a one and a minus one. The first row computes the narrow-lane combination, the second the wide-lane one. This matrix corresponds to a Hadamard matrix of order two. Thus, the generalization to four frequencies follows by considering a Hadamard matrix of order four, which is obtained as the Kronecker product of two Hadamard matrices of order two. The Hadamard matrix of order four is made of four rows, the first with all ones and the second with different combinations of two ones and two minus ones. In this way, one quad-frequency narrow-lane carrier phase combination is obtained along with three quad-frequency wide-lane combinations.

The three quad-frequency wide-lane carrier phase combinations are further combined with the pseudoranges to obtain generalized HMW composite observations. The coefficients of the code measurements have been determined in closed-form by requiring the preservation of the properties of the HMW combination (iono-free, geometry-free). The approach adopted first expresses the quad-frequency code-carrier combinations as a linear combination of dual-frequency HMW composite observations. In this way, two independent solutions for the code coefficients are found for each wide-lane combination. These solutions can be further combined to find a full family of coefficients leading to iono-free, geometry-free combinations that can be used to estimate the wide-lane ambiguities. Among these possible solutions, the one minimizing the noise variance of the resulting combination is finally selected.

It is noted that in the meta-signal approach, normalized Hadamard transforms are also considered. In this case, the original Hadamard matrix is normalized by the number of rows/columns. If this normalization were directly applied to carrier phase measurements, the integer nature of the carrier ambiguities would be broken. However, this transformation can still be applied if computed in a recursive way and by correcting half-ambiguities at each step when dual-frequency combinations are computed. Half-ambiguity resolution can be implemented using the HMW combinations.

The different options originating from the meta-signal paradigm are fully analysed in the final version of the paper, in terms of equivalent wavelengths, ionospheric error compensation and noise reduction.

The analysis is conducted using data collected from two Septentrio PolaRx5S multi-frequency, multi-constellation receivers, which use an ultra-low phase noise oscillator, OCXO (Oven Controlled Oscillator). Two receivers were connected through a Radio Frequency (RF) splitter to a choke-ring antenna mounted on the rooftop of the main building of the Warsaw University of Technology (52°13’15’’N, 21°00’37’’E). The receivers were set up to track the signal across the full GNSS spectrum with a 1 Hz rate under static clear-sky conditions in a zero-baseline configuration. In this way, it was possible to characterize the obtained combinations as uncombined measurements, single-differences and dual-differences.

The analysis shows the advantages and limitations of using these types of quad-frequency measurements. The approach developed generalizes the synthetic meta-signal reconstruction process and allows one to obtain quad-frequency meta-signal observations.

REFERENCES

Borio D. (2023) “Bicomplex Representation and Processing of GNSS Signals”, NAVIGATION, the Journal of the Institute of Navigation, Volume 70, Issue 4, Winter 2023

Borio D. and Gioia C. (2023) “Reconstructing GNSS Meta-Signal Observations Using Sideband Measurements”, NAVIGATION, the Journal of the Institute of Navigation, Volume 70, Issue 1, Spring 2023

Borio D. (2024) “A General Multi-Dimensional GNSS Signal Processing Scheme Based on Multicomplex Numbers” Submitted to the 2024 ION GNSS+ Conference

Deo M. and El-Mowafy A. (2020) “Precise Point Positioning with Decimetre Accuracy Using Wide-Lane Ambiguities And Triple-Frequency GNSS Data” Journal of Applied Geodesy, Vol. 14, No. 3

Duong V., Harima K., Choy S. et al. (2019) “An Optimal Linear Combination Model to Accelerate PPP Convergence Using Multi-Frequency Multi-GNSS Measurements”. GPS Solutions Vol. 23, No. 49

Georgiou, S., Koukouvinos C. and Seberry, J. (2003). “Hadamard Matrices, Orthogonal Designs and Construction Algorithms.” Boston: Kluwer. pp. 133–205

Hatch R. (1982) “The Synergism of GPS Code and Carrier Measurements.” In Proceedings of the third international symposium on satellite Doppler positioning, Las Cruces, NM, USA

Liu L., Pan S., Gao W., Ma C., Tao J. and Zhao, Q. (2021) “Assessment of Quad-Frequency Long-Baseline Positioning with BeiDou-3 and Galileo Observations.” Remote Sensing.

Ortega L., Medina D., Vilà-Valls J., Vincent F., Chaumette E. (2020) “Positioning Performance Limits of GNSS Meta-Signals and HO-BOC Signals.” Sensors

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