Global Navigation Satellite System (GNSS) meta-signals are obtained by combining components from different frequencies. Such combinations can be formed at different receiver processing levels, including the sample, correlator, and measurement levels. This last case involves the combination of single frequency measurements, which are related to the meta-signal observations through a Hadamard transform. This paper discusses the concept of synthetic meta-signal observations with an arbitrary number of components. In this respect, synthetic meta-signal pseudoranges and carrier phases are defined. A matrix transform is first applied to map original carrier phase measurements into the meta-signal domain. This transform is defined by a Hadamard matrix when the number of components is a power of two. In the general case, a truncated Hadamard matrix is used. The carrier phases obtained through this matrix transform can be interpreted in terms of a narrow-lane combination and several wide-lane combinations. The meta-signal pseudorange is obtained by solving the integer ambiguities of the wide-lane combination with the smallest equivalent wavelength. A strategy based on Hatch-Melbourne-Wübbena (HMW) combinations is proposed.
It is recognized that the transform used for the meta-signal measurement reconstruction introduces parity constraints on the integer ambiguities of the different carrier phase combinations. For instance, in the dual-frequency case, narrow- and wide-lane integer ambiguities have the same parity. The knowledge of the parity of the dual-frequency wide-lane combination allows one to determine the parity of the narrow-lane combination, thereby doubling the decision region of its integer ambiguities. These constraints are generalized to a larger number of components. For instance, it is shown that in the quad-frequency case, all the narrow- and wide-lane combinations must have the same parity. Moreover, when the ambiguities of the three quad-frequency wide-lane combinations are known, one can determine the narrow-lane ambiguities modulo four. These constraints are used to improve the ambiguity resolution process for the meta-signal pseudorange reconstruction. The meta-signal carrier phase is also defined by normalizing the narrow-lane combination by an appropriate power of two. Parity constraints are used to solve for possible fractional cycle ambiguities introduced by the normalization.
Parity constraints are general and valid outside the meta-signal context and can be adopted for the processing of narrow- and wide-lane carrier phase combinations. Theoretical results have been supported by experimental analysis involving both static and dynamic data. For the static experiment, two Septentrio PolaRx5S multi-frequency, multi-constellation receivers were set up in a zero-baseline configuration. Triple- and quad-frequency measurements from the Galileo system were collected and combined to obtain the different meta-signal combinations. The validity of the parity constraints was experimentally validated. Their application to the integer ambiguity resolution process through the use of the HMW combinations was also demonstrated. The dynamic experiment was conducted in an urban environment in the proximity of the campus of the Warsaw University of Technology and also involved the collection of multi-frequency, multi-constellation measurements from a Septentrio PolaRx5S. To implement differential processing, reference data from the JOZE00POL reference station were used. Static and dynamic results demonstrate the effectiveness of the proposed framework.

Global Navigation Satellite Systems (GNSS) provide precise positioning, navigation, and timing (PNT) capabilities for intelligent transportation and safety-critical applications. In this context, improving GNSS robustness and code-based accuracy has become a central topic in the literature, leading to advanced signal processing strategies such as the use of wideband signals, high-order BOC modulations, and meta-signal combinations. While these meta-signals and high-order BOCs are proposed to enhance performance, their large bandwidth makes the development of design and performance-evaluation tools particularly challenging. Such tools aim to assess performance against interference and multipath and to evaluate code-tracking precision and theoretical limits. Achieving this requires running multiple tests and extensive Monte Carlo simulations across wide parameter sweeps; hence, efficient and accurate correlation models are essential to keep these large-scale evaluations computationally feasible.

To address this challenge in the context of GNSS signal processing, modeling, and simulation, alternative correlation models have been proposed in the literature. In addition to the classical definition of the correlation between two sampled signals, two additional models can be considered. The first leverages Fourier Transform properties to accelerate the correlation computation using the Fast Fourier Transform (FFT). The second describes the correlation results as an analytical function of the signal's parameters (code delay, Doppler shift, amplitude, attenuation, and phase), no longer relying on the individual signal samples.

This paper presents a comparative analysis of these correlation models for wideband GNSS signal processing, focusing on both accuracy and computational complexity. To this end, it compares the correlation error and computational burden for GPS L1/CA and Galileo E5 signals under different sampling frequencies and RF front-end bandwidths. The results show that all models achieve comparable accuracy. However, the third model offers significantly lower computational complexity and introduces appealing linearity properties that facilitate the separation of the contributions of different signal components. For these reasons, this model stands out as a strong candidate for modeling and simulation tasks.

Finally, these results are illustrated through two applications: (i) the computation of theoretical pseudotrue parameters and the Misspecified Cramér–Rao Bound (MCRB) in the presence of multipath, and (ii) the estimation of the Maximum Likelihood Estimator within Monte Carlo simulations. Both applications require precise and numerous cross-ambiguity function (CAF) evaluations and correlations, which become highly time-consuming when high sampling rates and FFT-based models are employed.
The results demonstrate an order-of-magnitude reduction in simulation time when using the third correlation model.
Specifically, the proposed methodology for identifying multipath scenarios to conduct Monte Carlo simulations may require computing times that extend to several days for wideband signals when utilizing FFT-based modeling. Consequently, the model derived from the evaluation of the summation within the correlation definition presents a noteworthy alternative to conventional FFT-based techniques for the simulation and processing of wideband signals.

Prompted by the necessities of safety-critical applications, significant effort is continually brought forth to improve performance metrics of navigation systems. Similarly, this work is focused on providing mathematical tools for decreasing the computational cost of optimizing Direct Position Estimation (DPE), while minimizing any resultant decreases in its accuracy. Without a constraint on computation time or energy consumption, a brute force approach may be employed as a trivial global optimizer. Therefore, though accuracy is important, its practical significance requires the specification of computational cost. DPE is an optimization problem introduced in [1] to produce navigation solutions with signals from Global Navigation Satellite Systems (GNSS). Its accuracy and reliability are known to be theoretically better than traditional GNSS signal processing architectures [2], but at the cost of significantly greater computation time. Given a state vector and known satellite trajectories, all line-of-sight signals may be directly calculated and correlated with the received signal. The DPE objective function is defined as the sum of correlation power across all channels, as a function of the position, velocity, and time (PVT) states. The use of such a model leads to the performance gains of DPE, since it does not treat channels independently of one another. Multiple different optimization methods have been proposed to address the issue of computational complexity, but none have analyzed the use of gradient-based optimization methods applied to DPE. This work proposes a set of gradient-based methods for optimizing the DPE problem with a relatively small and fixed number of correlations per channel. A recently developed GPS correlation covariance model [3] is used to characterize the optimal placement of these correlations and the resulting solution covariance for GPS signal processing. Results include analytic and simulated performance of proposed methods, including computation time, PVT accuracy, and received signal power tracking threshold.

[1] P. Closas, C. Fernández-Prades, and J. A. Fernández-Rubio, “Maximum likelihood estimation of position in gnss,” IEEE Signal Processing Letters, vol. 14, no. 5, pp. 359–362, 2007.
[2] P. Closas, C. Fernández-Pradesz, and J. A. Fernández-Rubioy, “Direct position estimationapproach outperforms conventional two-steps positioning,” in 2009 17th European Signal Processing Conference, pp. 1958–1962, IEEE, 2009.
[3] B. Baker and S. Martin, “Derivation and validation of a higher-fidelity gps correlator model,” in 2025 IEEE/ION Position, Location and Navigation Symposium (PLANS), pp. 583–594, IEEE, 2025.

Direct position estimation (DPE) is an alternative global navigation satellite system (GNSS) receiver technique that has been originally proposed by [1]. In contrast to a conventional two-step approach, DPE estimates the position, velocity, and time (PVT) solution directly from the sampled received signal without intermediate ranging measurements. However, solving the resulting multivariate optimization problem can be computationally demanding.

Different implementation strategies have been proposed in the past [2, Chapter 21], including the use of maximum likelihood (ML) estimators [1] or Bayesian filters [3], [4]. Attempts have been made to reduce computational complexity, for example through linearization of the signal model [5] or low duty-cycles [6]. All of these approaches rely on the assumption that the PVT is found where the received signal best matches the local replicas, i.e., where the correlation output is greatest. This rather weak assumption results in good multipath mitigation characteristics [7].

This work proposes a low-complexity implementation of a DPE receiver. Therefore, a post-correlation signal model is derived. In this process, the assumption of nominally shaped auto-correlation functions has been made. A navigation filter has then been developed to estimate, based on the correlator outputs of all satellites, the PVT solution. Since the estimator goes straight from correlator outputs, i.e., a compressed representation of the received signal, to the PVT solution, this approach is considered a DPE solution. The performance of the proposed solution is evaluated with synthetic as well as authentic signals.

References
[1] P. Closas, C. Fernandez-Prades, and J. A. Fernandez-Rubio, “Maximum Likelihood Estimation of Position in GNSS,” IEEE Signal Processing Letters, vol. 14, no. 5, pp. 359–362, May 2007. doi: 10.1109/LSP.2006.888360.
[2] Y. J. Morton, F. van Diggelen, J. J. Spilker Jr., B. W. Parkinson, S. Lo, and G. Gao, Position, Navigation, and Timing Technologies in the 21st Century: Integrated Satellite Navigation, Sensor Systems, and Civil Applications. John Wiley & Sons, 2021, isbn: 978-1-119-45841-8. doi: 10.1002/9781119458449.
[3] P. Closas, C. Fernandez-Prades, D. Bernal, and J. A. Fernandez–Rubio, “Bayesian Direct Position Estimation,” Sep. 19, 2008, pp. 183–190.
[4] J. Dampf, “Probability analysis for bayesian directposition estimation,” Graz University of Technology, Graz, Feb. 2021, 244 pp.
[5] J. Liu, X. Cui, M. Lu, and Z. Feng, “Direct position tracking loop based on linearised signal model for global navigation satellite system receivers,” IET Radar, Sonar & Navigation, vol. 7, no. 7, pp. 789–799, 2013. doi: 10.1049/iet-rsn.2012.0307.
[6] Y. Ng and G. X. Gao, “Computationally Efficient Direct Position Estimation via Low Duty-Cycling,” in Proceedings of the 29th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2016), Sep. 16, 2016, pp. 86–91. doi: 10.33012/2016.14580.
[7] S. Tang, H. Li, and P. Closas, “Assessment of Direct Position Estimation Performance in Multipath Channels,” in Proceedings of the 37th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2024), Sep. 20, 2024, pp. 3705–3714. doi: 10.33012/2024.19799.