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### Session F1: Advanced Software and Hardware Technologies for GNSS Receivers

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**A General Multi-Dimensional GNSS Signal Processing Scheme Based on Multicomplex Numbers**

*Daniele Borio, European Commission Joint Research Centre (JRC)*

**Date/Time:** Wednesday, Sep. 18, 11:03 a.m.

Peer Reviewed

Modern Global Navigation Satellite Systems (GNSS) broadcast signals on several frequencies

allowing advanced applications, such as the estimation of the ionospheric delay and fast converging Precise Point Positioning (PPP) algorithms. Four components are currently broadcast by Galileo whereas five signals are transmitted by third generation BeiDou Navigation Satellite System (BDS) satellites. These frequency components are broadcast synchronously and coherently and are affected by similar delays and proportional Doppler frequencies, once received on the ground. This fact has motivated the development of joint processing schemes where signals from several frequencies are jointly tracked in order to improve receiver sensitivity and obtain high-accuracy measurements and precise position solutions. A possible approach for multi-frequency GNSS signal processing is based on the so-called meta-signal paradigm, where components from different frequencies are treated as a single entity. In thy way, a signal with a Gabor bandwidth larger than those of the single-band components is obtained. This is a precondition to have high accuracy GNSS signals and measurements. The meta-signal paradigm has been mainly developed considering two frequencies, which are jointly represented as the product of three terms: a code, a carrier and a subcarrier component. The subcarrier splits the two signal components from a common centre frequency to the sideband frequencies. A dual-frequency meta-signal can thus be processed using a triple-loop architecture where a Delay Lock Loop (DLL) is used to track the common code delay. The meta-signal carrier and subcarrier frequencies are tracked using a Phase Lock Loop (PLL) and a Subcarrier Phase Lock Loop (SPLL), respectively. In this way, the semi-sum and semi-difference of the sideband Doppler frequencies are tracked. Thus, both sideband components contribute to the estimation of the meta-signal carrier and subcarrier Doppler frequencies and phases.

Dual-frequency meta-signals have been recently analysed and processed using bicomplex numbers (Borio 2023), a bi-dimensional extension of complex numbers. Bicomplex numbers allow one to jointly represent signals from two different frequencies as a single entity and effectively generalize single frequency concepts to the dual-frequency case. For instance, it was possible to derive a dual-frequency Cross-Ambiguity Function (CAF) expressed with respect to the meta-signal carrier and subcarrier frequencies. The dual-frequency CAF is at the basis of the design of advanced acquisition algorithms and effective triple-loop tracking architectures.

Despite these results, meta-signal theory has been essentially limited to the dual-frequency case: additional work is required to extend these results to a larger number of frequencies. This is the goal of this paper, which extends the meta-signal theory to the more general case with 2^n components from different frequencies. This generalization is performed by using multicomplex numbers (Segre 1981, Price 1991), which are multi-dimensional extensions of complex numbers with commutative products. Complex and bicomplex numbers are multicomplex numbers of order one and two, respectively. Multicomplex numbers of higher order are obtained through a recursive construction: at each iteration, the number of components generating the multicomplex space is doubled. For instance, tricomplex numbers, i.e. multicomplex numbers of order three, features four complex components. Signals from different frequencies are mapped to the different complex components of a multicomplex number: given the recursive construction of multicomplex numbers, only cases featuring 2^n components are considered.

We first show that a set of 2^n signals from different Radio Frequencies (RFs) can be represented as the real part of the product between a multicomplex signal and 2^(n-1) complex carriers that split the different components into the different RFs. In this way, a general multi-component modulation formula is obtained. Moreover, the different RF signals are jointly represented as a single multicomplex baseband signal. The frequencies used to modulate the multicomplex baseband signal are obtained as the Hadamard transform (Yarlagadda and Hershey 1997) of the vector containing the original RFs. The dual-frequency case, obtained using bicomplex numbers, perfectly fits into this paradigm: the carrier and subcarrier frequencies are obtained as the semi-sum and semi-difference of the original sideband frequencies, which are equivalently obtained through a Hadamard transform of order 2.

In the Galileo case, four components are broadcast into the E5a, E5b, E6 and E1 frequencies. According to the multicomplex modulation formula, these four components are brought to their respective RFs by first bringing them to a common centre frequency, equal to 1309.44 MHz. Then the four components are split into two groups by adding and subtracting a first subcarrier frequency equal to 117.645 MHz. Two additional subcarrier frequencies are added and subtracted from the different components that are brought to the final Galileo RFs. Thus, in the case of four frequencies, three subcarriers are needed to split the different components into their respective RFs. Similarly, when eight components are considered, seven subcarriers are needed.

The modulation formula obtained has direct implications for the processing of multiple GNSS signals: instead of tracking the different carrier independently, one can track the average carrier component and the resulting subcarriers. In this way, all original components contributes to the joint estimation of the carrier and subcarrier parameters.

The multicomplex baseband signal formula is then used to derive a model jointly representing the GNSS signals received from different frequencies. This model is then used to formulate a Maximum Likelihood (ML) estimation problem for the received signal parameters. A multicomplex CAF is obtained. Its maximization allows one to estimate the signal code delay and the carrier/subcarrier Doppler frequencies and phases.

As in traditional GNSS signal processing the multicomplex CAF can be maximized through a two-step approach including the acquisition and tracking stages. In this respect, optimal, in the ML sense, acquisition and tracking algorithms are found for the processing of signals from multiple frequencies.

Note that multicomplex numbers feature an orthogonal decomposition: any multicomplex number can be expressed as the linear combination of orthogonal components. It is shown that in the case considered, the orthogonal components correspond to a scaled version of the original single-band components. Thus, the acquisition and tracking algorithms derived can be expressed in terms of the different single-band components. The acquisition algorithm derived is a form of weighted linear combining where the moduli of the single-band CAFs are aggregated. The weights of the combination depend on the relative received power of the single-band components. While, in principle, the acquisition search space should include all the carrier and subcarrier Doppler frequencies, the fact that these frequencies are related through a proportionality relationship can be exploited to reduce it to a bidimensional space as in the single frequency case.

In tracking, a common carrier term is processed considering the average carrier frequency. Moreover, the different subcarrier components are tracked using dedicated SPLLs. In the Galileo case, with four signals, three SPLLs are adopted. Note that all the single-band components contribute to the tracking of the subcarrier components. Moreover, each subcarrier features significantly lower Doppler frequencies than the original single-band components, facilitating the tracking of the signal dynamics. Thus, the bandwidth of the corresponding SPLL can be lower. Aiding between the different channels can also be implemented.

Full details on the acquisition and tracking algorithms developed using multicomplex numbers will be provided in the full version of the paper.

In order to demonstrate the theoretical findings obtained using the multicomplex paradigm, the reception of signals from the four Galileo frequencies has been considered. An experimental setup involving four HackRF One front-ends (Ossmann 2024) has been implemented. The four HackRF One front-ends were connected to a multi-frequency GNSS antenna through a RF splitter. The antenna was placed under open sky static conditions. The four front-ends were synchronized through the procedure outlined by Bartolucci et al. (2016). The first front-end was equipped with a Temperature Compensated Crystal Oscillator (TXCO), which was shared with the other HackRF One devices through their clock ports. Moreover, the hardware trigger of the first front-end was shared with the other devices. In this way, synchronous data grabbing was implemented. Using this setup, it was possible to collect synchronous In-phase/Quadrature (IQ) data from the different Galileo frequency bands. A common sample frequency of 20 MHz was adopted. Each front-end was tuned on a different Galileo frequency. The data were stored to disk and processed using a custom Software Defined Radio (SDR) receiver developed in Python implementing the acquisition and tracking scheme derived using the multicomplex number paradigm.

Since four signals were considered, a five-loop tracking architecture was implemented comprising a DLL, a PLL and three SPLLs. A single DLL was used to track the delay of the four single-band components. To avoid performance losses, a calibration stage, running during the first tracking epochs was also implemented to estimate and compensate for potential biases between the delays of the different channels. The PLL was used to track the common centre frequency whereas the three SPLLs tracked the three subcarriers.

Experimental results confirm the validity of the approach proposed that allows one to jointly track the different signal components. A complete description of the acquisition and tracking algorithm developed using multicomplex numbers will be provided in the final version of the paper.

The theory developed provides a general framework for processing 2^n components from different frequencies. Moreover, multicomplex numbers allow a compact and elegant multi-signal representation that generalizes results obtained for the single frequency case. Theoretical findings also have direct implications on measurement generation and provide a general framework for constructing, for instance, quad-frequency measurement combinations. These aspects are studied in a companion paper. Theoretical findings are supported by experiments considering four Galileo frequency components.

REFERENCES

Bartolucci M., Del Peral-Rosado J. A., Estatuet-Castillo R., Garcia-Molina J. A., Crisci M. and Corazza G. E. (2016) “Synchronisation of low-cost open source SDRs for navigation applications,” 2016 8th ESA Workshop on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (NAVITEC), Noordwijk, Netherlands, pp. 1-7

Borio D. (2023) “Bicomplex representation and processing of GNSS signals,” NAVIGATION, the Journal of the Institute of Navigation, pp. 1–33

Ossmann M. (2024) “Great Scott Gadgets HackRF One” Available on-line https://greatscottgadgets.com/hackrf/one/ [last accessed 13 February 2024]

Price G. B. (1991) “An introduction to multicomplex spaces and functions”, 1st ed. CRC Press

Segre C. (1891) “Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici,” Math. Annalen, vol. 40, pp. 413–467, 1891. Available on-line: http://www.bdim.eu/item?fmt=pdf&id=GM_Segre_CW_2_338

Yarlagadda R. K. R. and Hershey J. E. (1997), “Hadamard Matrix Analysis and Synthesis”

Springer New York, NY, USA

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