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### Session F3b: GNSS Robustness to Vulnerabilities 1

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**Enhanced Tracking with Improved Code Autocorrelation Function via Fractional Fourier Domain**

*Yiran Luo, University of Calgary; Yi-Fen Tseng, AUROXAT Inc.; Naser El-Sheimy, University of Calgary*

**Date/Time:** Thursday, Sep. 19, 9:43 a.m.

Peer Reviewed

Objectives

In challenging environments, such as dense urban areas, global navigation satellite system (GNSS) signals can become highly distorted. It is understood that the carrier component of GNSS signals is also significantly affected by versatile environments. Nevertheless, this paper will focus exclusively on GNSS code signal processing.

One of the primary reasons for GNSS challenges in urban areas is the severe multipath effects [1]. This interference can cause the behavior of the received baseband signal to deviate significantly from standard models. Consequently, the receiver may provide biased estimations against the actual signal behavior. Another contributing factor is the characteristics of GNSS signals. Specifically, the accuracy of code error estimation depends on the autocorrelation function (ACF) of a designated code sequence (see pages 454 to 456 of [2]). In the convolution of two identical code sequences, the signal will be less susceptible to interference from short non-line-of-sight (NLOS) signal rays when the autocorrelation peak degrades more rapidly concerning time delay/code phase offset.

Based on the discussions above, we aim to enhance the ACF of spreading code sequences used in GNSS signals. A code sequence with such characteristics will possess an improved ability to mitigate multipath effects by resisting incoming NLOS rays with a narrower main lobe in the code autocorrelation domain. For instance, high-rate code sequences with 10230 chips will outperform low-rate ones with 1023 chips in multipath mitigation in GNSS signals.

Existing approaches have extensively worked on the standard code ACF to enhance multipath mitigation ability by modifying the early-minus-late (E-L) code chip space and/or increasing the number of code correlators in the early/late channels. This modification aims to estimate less biased code errors corresponding to the design of these code correlators. Typical examples include narrow correlators [3], multiple correlators [4], MEDLL [5], etc. However, to the best of our knowledge, almost none of these approaches or their variations affect the ACF over the code correlating process. This implies that the performance of a given code sequence is limited by a deterministic code ACF once it is designed. Is this true? Can we make it changeable? This work will address these questions. To achieve this goal, the objectives of this research work will include:

1. Investigate how the code ACF can be influenced by leveraging the advantages of fractional Fourier domain (FRFD).

2. Explain the direct (traditionally used in the tracking stage) and indirect (often used in the acquisition stage) approaches in a tracking loop in cases where the code signals can be wiped off.

3. Based on the indirect approach as mentioned above, suggest a method for code tracking utilizing the potential of the fractional Fourier domain.

4. Assess the performance of the proposed code tracking loop.

5. Discuss and conclude this research endeavor.

Methodology

In this section, we will explore how the code ACF can be altered through a modified parallel-code search (PCS) employing fractional Fourier transform (FRFT). Subsequently, we will introduce a code-tracking architecture leveraging the advantages of FRFD to enhance tracking capabilities.

Definition of the FRFT

For the integrity of the context, we first give the definition of the pth order FRFT via a linear integral operation way (see pages 118 to 119 of [6])

f_p (u)??_(-?)^(+?)??K_p (t,u)f(t)dt?

with

K_p (t,u)={?(C_? e^j?[(t^2+u^2 ) cot??-2ut csc?? ] ,&??n?@?(t-u),&?=2?n@?(t+u),&?=(2n+1)?)?,n?Z

C_???(1-j cot?? ), ??p?/2

where K_p (t,u) represents the transformation kernel of FRFT and j the imaginary unit; p is the FRFT order; ? denotes a rotation angle from the time-frequency plane in terms of the ordinary Fourier transform (FT) to an extended plane; f(t) is the input signal varying with the variable t and f_p (u) is the representation of FRFT in terms of f(t) with respect to the order p, varying with the variable u.

A Novel Code ACF in FRFD

Traditionally, the standard code ACF is computed through convolution operations between two code sequences. During the start of a GNSS receiver, a parallel-code search (PCS) approach is utilized to significantly reduce the computational load in code phase acquisition. This reduces computational load efficiently to ?(Nlog_2 N) instead of ?(N^2 ), where N corresponds to the number of code correlators. Figure 1 compares two computational methods for obtaining the code ACF, where “FFT” and “IFFT” represent fast Fourier transform and inverse fast Fourier transform operations, respectively.

However, this research does not primarily focus on exploring the advantages of the PCS approach in reducing computational complexity. Instead, our investigation centers on the PCS’s capability to bridge the standard code autocorrelation domain (accessible through the methods illustrated in Figure 1) to a novel code autocorrelation domain.

Figure 1 Two standard methods to obtain the code ACF related to the ordinary Fourier domain. a) convolution (top); b) PCS (bottom).

To access this “novel code correlation domain,” we employ the approach outlined in Figure 2. In the following sections, we will investigate how the calculation method presented in Figure 2 impacts the code ACF.

Figure 2 A novel method to obtain the code ACF in FRFD.

As depicted in Figure 3, comparisons between the standard and proposed novel code ACFs are presented. Several noteworthy observations arise from these findings:

1. The number of sampling points and FRFT order affect the code ACF in the FRFD, but they do not apply to the standard ACF.

2. Increasing the number of sampling points used in the FRFT enhances the ability of the new ACF to reject interferences to the main lobe.

3. Employing a higher FRFT order in the digital FRFT operation improves the capability of the new ACF to reject interferences to the main lobe.

Figure 3 Comparison between the standard and proposed novel code ACFs is illustrated. “FFT-PCS” and “FRFT-PCS, p=1.25” or “FRFT-PCS, p=1.5” correspond to the calculation approaches depicted in Figure 1 (bottom) and Figure 2, respectively. “p=1.25” and “p=1.5” represent the digital FRFT calculations with FRFT orders of 1.25 and 1.5, respectively.

Now, let’s rephrase the findings mentioned above in plain language. That is to say, the GPS L1C/A/L1C, Beidou B1I/B1C, and Galileo E1B/C signals may have the opportunity to be empowered by the multipath mitigation capability naturally designated to the GPS L5/ Beidou B2a/b/ Galileo E5a/b signals.

More excitingly, the mathematical expression of the discrete FRFT can be straightforwardly derived from its theory. Considering that the GNSS baseband processor is constructed based on FT fundamentals and that FRFT is a generalized form of FT, adapting the tracking architecture to follow the FRFT rule can be easily achieved.

A Code Tracking Channel Design in the FRFD

Inspired by the discussions above, we propose a novel tracking architecture to enhance the tracking performance of old GNSS signals to match the boosted capabilities found in modernized GNSS signals.

To provide a clearer understanding of the proposed tracking architecture and its distinctions from traditional ones, firstly, we quickly introduce the classic tracking architecture designed with four code tracking branches: very early (VE), early (E), late (L), and very late (VL) branches, as depicted in Figure 4. Components shared between the proposed and traditional tracking architectures, such as carrier signal tracking channel designs, discriminators, and loop filters, are deliberately omitted for simplicity. Only the code signal tracking channel designs for a single in-phase (I) or quadrature (Q) branch are retained.

Figure 4 Traditional code tracking channel design within a single I/Q component with the four VL, L, E, and VE branches.

Likewise, a novel four-branch code tracking channel design, as introduced earlier, is provided in Figure 5.

Figure 5 A novel code tracking channel design within a single I/Q component with the four VL, L, E, and VE branches in the FRFD.

It is worth noting that, as shown in Figure 3, only when the sampling point number within one code chip exceeds two will the new code ACF in the FRFD have a more evident improvement to the standard ACF. It means that a minimum of four branches will be required to guarantee the superiority of the proposed code-tracking architecture for GPS L1 C/A signals. Therefore, we design a four-branch code tracking architecture in Figure 5.

Anticipated results

This section will propose an evaluation strategy for the new code-tracking approach in the FRFD. We plan to collect intermediate-frequency (IF) GPS L1C/A signals from real-world scenarios and/or simulate IF GPS L1C/A signals in various scenarios to test the proposed tracking method using our GNSS software-defined radio (SDR) receiver. Statistical analysis will be conducted on the estimated tracking errors.

Ultimately, we anticipate the following outcomes based on this research:

1. In an open-sky scenario, we expect to observe a reduction in the variance of code tracking errors estimated using the approach outlined in Figure 5 compared to the method depicted in Figure 4.

2. In a typical dense-urban scenario, where incoming signals are interfered with by NLOS signal rays, we anticipate improvements in the mean and variance of code tracking errors through the proposed code tracking method.

Conclusions

This research work proposes a code tracking method in the fractional Fourier domain (FRFD), enabling old GNSS signals (such as GPS L1C/A) to benefit from the exceptional multipath mitigation capabilities found in modernized GNSS signals. These modernized signals are modulated with higher-rate spreading code sequences with longer chip lengths.

Since the fractional Fourier transform (FRFT) is a generalized form of the Fourier transform (FT) and has an analytical expression, the novel tracking approach can be directly applied to current GNSS receiver designs.

Based on anticipated results, the performance of the proposed code tracking in the FRFD has the potential to greatly surpass that of traditional code tracking techniques commonly employed in the ordinary Fourier domain.

Key innovations

The key innovations of this research work include:

1. The discovery that the traditional autocorrelation function of spreading code sequences can be enhanced in the fractional Fourier domain.

2. The proposal of a meaningful application of this discovery to elevate the tracking performance of old GNSS signals to a level comparable to modernized GNSS signals.

3. The design of a code-tracking architecture in the fractional Fourier domain for a practical satellite navigation receiver.

The significance of this work

The significance of this research work includes, but is not limited to:

As the core of a GNSS receiver, upgrading the baseband processor will significantly impact the receiver’s overall performance. Therefore, this research may signal the potential for commercial GNSS receivers to enter a new era of improved positioning, navigation, and timing accuracy, especially in challenging environments.

This research focuses on a highly general methodology that can be easily applied to other navigation signal receiver designs, such as those for LEO satellite signals, 5G/LTE signals, and beyond.

References:

[1] M. S. Braasch, “Multipath Effects,” in Global Positioning System: Theory and Applications, vol. 1, B. W. Parkinson, J. J. Spilker Jr, P. Axelrad, and P. Enge, Eds., Washington D.C.: American Institute of Aeronautics and Astronautics, Inc., 1996.

[2] E. D. Kaplan and C. Hegarty, Understanding GPS/GNSS. Principles and Applications, 3rd ed. Artech house, 2017.

[3] M. S. Braasch, “GPS multipath model validation,” in Proceedings of Position, Location and Navigation Symposium - PLANS ’96, IEEE, 1996, pp. 672–678. doi: 10.1109/PLANS.1996.509144.

[4] G. A. Mcgraw and M. S. Brassch, “GNSS Multipath Mitigation Using Gated and High Resolution Correlator Concepts,” Proceedings of the 12th - International Technical Meeting of the Satellite Division of the Institute of Navigation, ION GPS99, pp. 333–342, 1999.

[5] R. D. J. van Nee, J. Siereveld, P. C. Fenton, and B. R. Townsend, “The multipath estimating delay lock loop: approaching theoretical accuracy limits,” in Proceedings of 1994 IEEE Position, Location and Navigation Symposium - PLANS’94, IEEE, 1994, pp. 246–251. doi: 10.1109/PLANS.1994.303320.

[6] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier transform with applications in optics and signal processing. Wiley, 2001.

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