Analysing the ML-Based Algorithms for GNSS Data Bit Synchronization and Decoding

Abstract

The position, velocity, and time global navigation satellite systems are vulnerable to signal interference, distortion, jamming, and multipath, which could potentially render the entire system inoperable due to the generally weak signal strength in these conditions. Due to these problems, the Global Navigation Satellite Systems receiver is rendered inoperable by an exceptionally strong navigation frequency band signal along the satellite path. Since global navigation satellite systems are currently widely used, there is a significant increase in the risks of interference, distortion, and jamming. Multipath concerns have been the subject of extensive research with a variety of approaches. But, first, the level of the Global Navigation Satellite Systems multipath must be estimated using a sample of the total signal that the navigation space satellite emits. The satellite constellation and environmental errors have a significant impact on the navigation system. The maximum likelihood estimation technique is presented in this paper along with an evaluation of its consistency and reliability when the Global Navigation Satellite Systems signal multipath is present. As this paper discusses, multipath signals can be numerically discriminated using maximum likelihood estimation techniques based on receiver measurements without the need for additional devices. The measurements and output data derived from the Global Navigation Satellite Systems receiver configuration parameters are used in the maximum likelihood estimation. It was found that the overall performance of the space data synchronization is determined by the number of data bit transitions rather than the total number of bits. An observed state-space representation, lower signal $C/N_0$, and greater Doppler frequency inaccuracy require more data bits for estimation and computation. It was also observed that, in the majority of cases, if not all of them, the bit evolution occurs with a probability equal to 60% upper. With $C/N_0$ in marginal power estimated to 20 dB-Hz without Doppler error, it is probably going to reach the 95%–100% range. If the Doppler error is less than 6 dB-Hz, the signal attenuation caused by the Doppler inaccuracy is insignificant, and the maximum tolerance of the Doppler inaccuracy is 30 dB-Hz.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Since satellite navigation signals are susceptible to signal distortion, interference, jamming, and multipath, the main issue in recent years has been their reliability. Thus, it is essential and a topic of continuous research to be able to detect the signal multipath [1, 2]. Global navigation satellite systems ($\text{GNSS}$) interferences can include signal jamming, which uses higher signal intensities from navigation frequency bands to disable the receiver, and spoofing, which uses signals that are similar to the real navigation signals to give the $\text{GNSS}$ receiver an incorrect output in terms of position and time [3]. Even if the user disrupts the spoofing signal, the $\text{GNSS}$ receiver is built to continuously generate position and time information, so it might not detect abnormalities in the signal right away. Inaccurate timing and placement could have detrimental effects on both military and economic applications. The majority of operations in the military and commercial domains rely on precise positioning and time synchronization [4, 5]. Better methods must therefore be created to identify the spoofing signal and suggest potential fixes to allay such worries.

Glonass, Galileo, Beidou, and the Global Positioning System (GPS) are the four well-known $\text{GNSS}$ constellations. The user can access precise navigation, velocity, and timing data from these systems anywhere in the world [6, 7]. Yet, because of the sharp rise in demand for $\text{GNSS}$ services for indoor applications, urban canyons, and dense foliage, processing weak $\text{GNSS}$ signals has been accelerating. New receivers for the High-Sensitivity Global Navigation Satellite System ($\text{HSGNSS}$) are made to measure the pseudo-range of all satellite signals within a 30 dB attenuation range. In order to achieve significantly higher sensitivity, cross correlation, avoid squaring losses, and mitigate signal multipath, the $\text{HSGNSS}$ receiver can prolong the integration time between processing stages [8, 9]. However, it is expected that the data bits that are part of the navigation message will limit the amount of time needed for coherent integration. In order to prevent energy losses during the process that are brought on by the transition of the $\text{GNSS}$ data bits, the navigation-related message and the $\text{GNSS}$ data bits will also be erased. Understanding the $\text{GNSS}$ data bit limitations and data bit values is necessary for the thorough wiping off of the $\text{GNSS}$ data bit in order to ensure smooth processing. Accordingly, bit synchronization is the process of extracting the bit values and identifying the positions of the data bit boundaries [10, 11].

In certain scenarios, it may take a long time to process the $\text{GNSS}$ navigational data, and outside processing resources may be needed. The signal acquisition with the carrier to noise ($C/N_0$) density of 12, 17, 22, and 32 dB-Hz is thus detected in 800 microseconds, according to the external source of signal processing. However, in certain situations, the $\text{GNSS}$ receiver’s capabilities may be jeopardized, and autonomy may be lost, as the signal acquisition process occasionally necessitates the assistance source to expedite data processing. It is anticipated that using an outside source for data processing will raise operational expenses and complexity. Kalman filter (KF) is a widely used numerical method for handling $\text{GNSS}$ multipath in terms of data synchronization and decoding. In many situations, the $\text{KF}$ and $\text{MLE}$ are included to make it easier to estimate the replica signals and quickly locate standalone $\text{GNSS}$ integrated devices using data bits and values. If $\text{GNSS}$ multipath estimates are needed, the maximum likelihood estimation ($\text{MLE}$) is expected to be the answer for weak $\text{GNSS}$ signals. In environments with weak signals and multipath, the $\text{MLE}$ bit synchronization has been shown through numerical processes to perform better than predictable histogram techniques [12, 13].

The $\text{MLE}$ bit decoding and synchronization will outperform other algorithms in simplicity and technical performances. However, the issue of multibits to be processed and decoded has not been verified, and possible Doppler error has not been assessed either that prompt the need to evaluate the consistency of the $\text{MLE}$ algorithms in such scenarios [14]. Thus, the evaluation of the $\text{MLE}$-based bit decoding and synchronization algorithms in challenging scenarios is crucial to understanding their true potential and limitations.

This paper seeks verify the multi bits processing and decoding as well as the possible Doppler error. It will assess the consistency and efficacy of the MLE algorithms in such challenging scenarios. Hence, it investigates the algorithms’ performance in handling multiple bits, as well as their resilience to Doppler errors, to determine their consistency and suitability for $\text{GNSS}$ applications. The results will provide valuable insights into the trade-offs between algorithm complexity, technical performance, and robustness to various impairments, enabling informed decisions on the deployment of these algorithms in the processing of the $\text{GNSS}$ multipath navigational data.

2. The Algorithms of Decoding and Synchronization of the GNSS Data Bit

This section presents an analysis of the $\text{GNSS}$ data bit decoding and synchronization using MLE techniques. The concept being developed from this section will be maintained throughout the paper.

2.1. The MLE Parameters Estimation Model for GNSS

When the $\text{GNSS}$ encounters an unknown prior distribution of the estimated parameters from the signals emitted by the satellite, the $\text{MLE}$ is employed to estimate the unknown parameter, anticipating the smallest variance in errors if the parameters are estimated in a symmetric and unimodal distribution. According to [15, 16], the linear $\text{KF}$ is the best filter for linear discrete time systems. The $\text{MLE}$ can be used in conjunction with the KF to accurately estimate the loop’s parameters [17]. Combining the two methods can also enhance the tracking loop’s functionality. In contrast, the carrier loop’s parameters are estimated nonlinearly using the MLE, and the resulting parameters are then smoothed using the KF.

2.2. The Basic Principle of the GNSS MLE

The MLE, which is defined as the estimator of the phase angle value (${\theta }_{MLE}$) that maximizes the likelihood function of the $\text{GNSS}$ signal, is frequently used in this situation to estimate the unknown parameters of the $\text{GNSS}$ signal. The probability density function (PDF), also known as the likelihood function, is represented by the ${PDF}_{MLE}t$ of the observed vector, while the estimated phase angle of the $\text{GNSS}$ signal is represented by the symbol ${\theta }_{MLE}t$. The MLE is explained as follows: the $\text{GNSS}$ signal has the chosen phase angle ${\theta }_{MLE}$, the probability function ${PDF}_{MLE}tdx$ is taken into consideration, and that $x$ function falls within the small area.${\theta }_{MLE}t$ estimates that the corresponding ${\theta }_{MLE}$ will be the largest. We considering the $\text{GNSS}$ signal with the ratio of ${\mathcal{R}}_{GNSS}$ between the period of the ranging code $T_{R-\text{code}}$ and the navigation data bit period $T_{D-\text{bit}}$. As a result, (1) represents the ratio.$$\begin{array}{}\text{(1)}& {\mathcal{R}}_{GNSS}=\frac{T_{D-\text{bit}}}{T_{R-\text{code}}}.\end{array}$$

As a result, the correlator output that is computed over the single code period is $k^{th}$ and is sampled with $1/T_{R-\text{code}}$ Hz for the length $N_{\text{data}}$ of the periodical data sequence and given by$$\begin{array}{}\text{(2)}& {\mathcal{H}}_k\mathrm{\Delta }\tau =A_{GNSS}{\mathfrak{R}}_k\mathrm{\Delta }\tau {\chi }_{k,ld}+n_k.\end{array}$$

In (2), the ${\mathfrak{R}}_k\mathrm{\Delta }\tau$ represent the normalized ranging code correlation function of the $\text{GNSS}$ signal, the $\mathrm{\Delta }\tau$ represent the error that is locally generated for a ranging code, and $A_{GNSS}=A_{GNSS}{e}^{j\mathrm{\Delta }\phi }$ represents the complex form for the signal amplitude.

The domain of the satellite communication inclusive of the $\text{GNSS}$ is also governed by the phase different $\mathrm{\Delta }\sigma$ between the space generated signal and the replica or carrier signal generated locally. In such cases, the amplitude of the space signal is likely to approach the real value from complex form. The ${\chi }_{k,ld}$ is the navigation data bit for a specific transition of the data samples relative to the MLE synchronisation bit data. If we anticipate the inaccuracy in the phase code, the numerical estimation tends to be $\mathrm{\Delta }\tau$ = 0, and (2) is rewritten as follows:$$\begin{array}{}\text{(3)}& {\mathcal{H}}_k\mathrm{\Delta }\tau =A_{GNSS}{\chi }_{k,ld}+n_k,\text{where}\text{k}=1,2,3\dots ,N_{\text{data}}{\mathcal{R}}_{GNSS}.\end{array}$$

As we proceed, we also expect that, as demonstrated in (4), it is acceptable to have the fewest errors possible in the $\text{GNSS}$ domain for all replica signals in both the upper and lower frequency bands. The algorithm in (3) is rewritten to take into account the effect of the Doppler shift associated with the frequency error.$$\begin{array}{}\text{(4)}& {\mathcal{H}}_k\mathrm{\Delta }\tau =A_{GNSS}{\chi }_{k,ld}{e}^{-j2\pi \mathrm{\Delta }{f}_{d}k{T}_{R-code}-\mathrm{\Delta }\phi }\sin c\pi \mathrm{\Delta }{f}_d{T}_{R-\text{code}}+n_{l,Q},\end{array}$$where the

• $\mathrm{\Delta }{f}_d$ denote the Doppler or frequency error between the original and replica $\text{GNSS}$ signal

• $n_{l,Q}$ represent the Gaussian noise that is known as additive white noise always found in $\text{GNSS}$ signals.

• $\sin ct$ is the function that represent the carrier spectrum

3. The Bit Numerical Decoding for GNSS Signal and Data

By bit decoding the navigation data to verify correct synchronization, the bit values of the $\text{GNSS}$ data are essential for data synchronization between the space satellite and receiver. The inner product of $T_{D-\text{bit}}$, the correlator output, starting from the bit boundary, and the locally generated bit combinations is the probability of the functions used in machine learning algorithms [18]. Decoding N bits yields a total of $2_N-1$ possible bit combinations, with the correct bit combination expected to have the highest energy [19]. The energy required for ML bit decoding detects the bit transition rather than the actual bit values. However, this is sufficient for the required data wipe off in order to extend the coherent integrating time. The bit value combination matrix B_bit is $2^{N-1}x{N}_{\text{data}}$.$$\begin{array}{}\text{(5)}& B_{\text{bit}}=\begin{array}{cccc}1& 1& \dots & 1\\ 1& 1& \dots & -1\\ \dots & \dots & \dots & \dots \\ 1& -1& \dots & -1\end{array}.\end{array}$$

To decode the number of bits in sequential order, the product between the transitions is$$\begin{array}{}\text{(6)}& {\xi }_m\mathrm{\Delta }{f}_d={\mathfrak{R}}_N\mathrm{\Delta }{f}_d.b_m,{m=1,2,\dots ,2}^{N-1}.\end{array}$$

The vector $b_m$ represents the $m^{th}$ row from (5). As a result, the machine learning algorithm estimates the bit values by maximizing the inner product and the energy required for a smooth transition. The values for ML bit synchronization are obtained using the following expression:$$\begin{array}{}\text{(7)}& \widehat{b}=\arg \max {\xi }_m\mathrm{\Delta }{f}_d;b_m.\end{array}$$

The following is a summary of the $\text{MLE}$ technique used to adhere to the $\text{GNSS}$ data bit synchronization process, as well as the advantages associated with the $\text{MLE}$ techniques.

• We track $\text{GNSS}$ signals for both phase-locked loop ($\text{PLL}$) and the frequency lock loop ($\text{FLL}$).

• The correlator produces coherent and simple output samples across bit intervals.

• We use coherent integration to improve correlation and store event sequences for correlator outputs.

4. The GNSS Data Bit Synchronization

We present and analyse the detection of location bit boundaries using the likelihood estimating function [20]. The ratio of the data bit to the ranging code period is used to ensure proper $\text{GNSS}$ data synchronization between the space satellite and the receiver. Through the analysis and evaluation of MLE techniques, we discovered that the basic concept of $\text{GNSS}$ data synchronization is that the data bit and value transition during correlation are likely to ease the detection of the data bit boundaries and further improve the state-space representation (SSR) [21]. This is numerically expressed as follows:$$\begin{array}{}\text{(8)}& {\widehat{w}}_k=1,k=1,2,\dots ,{\mathcal{R}}_{GNSS}.\end{array}$$

This filter serves as a cross-correlation between the functions of ${\mathfrak{R}}_N\mathrm{\Delta }{f}_d$ in (6). Hence, the ${\widehat{w}}_k$ is expressed as follows:$$\begin{array}{}\text{(9)}& C_{n,\overline{lb}}\mathrm{\Delta }{f}_d={\displaystyle \sum }_{k=1}^{N_{\text{data}}{\mathcal{R}}_{GNSS}}{\mathfrak{R}}_k\mathrm{\Delta }{f}_d{\widehat{w}}_{k-NR+\overline{lb}}N=0,1,N-1.\end{array}$$

The correlation between $\text{GNSS}$ data and signals from (6)–(9) can provide the total values of the cross correlation, which is expressed as follows:$$\begin{array}{}\text{(10)}& {\Psi }_{\overline{lb}}\mathrm{\Delta }{f}_d=\frac{1}{N_{\text{data}}}{\displaystyle \sum }_{n=0}^{N_{\text{data}}-1}{C}_{n,\overline{lb}}\mathrm{\Delta }{f}_d.\end{array}$$

Therefore, the ML estimate of the bit boundaries is derived and estimated as follows:$$\begin{array}{}\text{(11)}& {\widehat{l}}_b=\tan {\delta }_{\text{Bit}}{Ƈ}_{\max }{\Psi }_{\overline{lb}}\mathrm{\Delta }{f}_d.\end{array}$$

5. The Model for Theoretical Performance and Processing

The next step is to examine the theoretical performance of data bit harmonization and decoding, which will be used later to assess the consistency of MLE techniques in terms of signal multipath mitigation [22]. The previous section provided brief mathematical details to demonstrate that the algorithms can work with either FLL or PLL. However, from (4), we discovered that carrier tracking has a significant influence on frequency error and is represented by the sinc function, which is used to attenuate the power and energy that passes through the tracking loop. Bit decoding and data synchronization require the pseudo-range noise (PRN) code to be rendered inaccessible via the delay lock loop (DLL), resulting in insignificant signal power losses. As a result, the error for better 0.5 chips is expected to lose a maximum of 6 dB. The assumption is $\Delta \tau$ = 0, which means the code tracking error is zero and the SSR is expected to be autonomous from the tracking methods and other related signal parameters [23]. Therefore, the resultant SSR will be demonstrated as ${C/N}_0$. In Figure 1, it is shown that the ${\Psi }_{\overline{lb}}$ with the $\text{GNSS}$ signal parameter is $T_{D-\text{bit}}=20\text{ms}$, $T_{R-\text{code}}=1\text{ms}$, and ${\mathcal{R}}_{GNSS}=20$, assuming no Doppler error. According to this, the data bit transition location is always set to the middle of each bit that passes through.

[figure(s) omitted; refer to PDF]

The likelihood of successful data synchronization is represented in the following equation:$$\begin{array}{}\text{(12)}& S_{\text{probability}}=S{C}_{n=1,\overline{lb}}{P}_{lb}{>P}_{\overline{lb}}.\end{array}$$

The $P_{lb}$ is the output of the bit boundaries$$\begin{array}{}\text{(13)}& {\overline{l}}_b={\mathrm{\Delta }}_{lb}+l_b.\end{array}$$

This refers to the difference between the estimate ${\overline{l}}_b$ and actual $l_b$. Successful ML bit synchronization requires an output in the bit boundaries that is expected to be higher than all other outputs. The numerical expressions below represent the summary of processing. The vector is created, and it contains the differences between the nonbit boundaries and the absolute values of the cross-correlation for ML bit synchronization at each transition.$$\begin{array}{}\text{(14)}& X_{ML}=\begin{array}{ccc}{P}_{lb}-& P_{lb}+& 1\\ P_{lb}-& P_{lb}+& 2\\ P_{lb}-& P_{lb}+& \frac{{\mathcal{R}}_{GNSS}}{2}\end{array}.\end{array}$$

From this matrix, the probability of the effective bit synchronization is rewritten:$$\begin{array}{}\text{(15)}& S_{\text{probability}}=S^2{X}_{ML}>0.\end{array}$$

In this context, the cross-correlation output presented in (4) is expected to be Gaussian distributed when considering the original signal generated by the space satellite$$\begin{array}{}\text{(16)}& P_{\text{coherent}}=\delta ·\sqrt{\frac{{2E}_t}{N_d}},\text{(17)}& P_{\text{coherent}}=\delta ·\sqrt{{T_{\text{coi}}·2\mathrm{·}10}^{0.1.{C/N}_0}}.\end{array}$$

The numerical expressions in (16) and (17) are representing the bit error rate (BER) of the coherent decoding for binary phase shift keying (BPSK) and binary offset carrier (BOC) signals where

• $E_t$ is the energy of the transmitted signal per bit.

• $N_d$ is the spectral density of the specified noise that is in the space.

• $T_{coi}$ is denoting the coherent integration time and is considered to be the length of the data bit.

6. Test Simulation Outcomes and Data Analysis

To assess the performance, consistency, and dependability of the differential approach when estimating the $\text{GNSS}$ multipath using the Monte Carlo simulating package, a thorough examination of the simulation results is conducted. The first step in mitigation is to estimate the current replica signal. Additionally, the $\text{GNSS}$ receiver platform has been used to evaluate the numerical algorithms.

The signal response for BPSK and BOC modulations is displayed in Figures 2 and 3. Actually, a set of support points is produced by the numerical algorithms in a manner that assesses reliability and the ML cost function. The findings in Figures 2 and 3 demonstrate that the overall performance of bit decoding and data value synchronization significantly improves with the use of more data bits. However, since more bits are expected to be added, this is true if there are more transitions.

[figure(s) omitted; refer to PDF]

The dual approach’s features for estimating the first path delay are shown in Figure 4. Every tracked space satellite undergoes two-dimensional ML bit synchronization and decoding for positioning, velocity, and timing. Therefore, the positioning depends on the ML technique’s cost function, and accurate estimation requires at least three satellites to be visible. The true channel and the ML cochannel that Figure 5 estimates do not match exactly.

[figure(s) omitted; refer to PDF]

The a priori information about the navigation message predates the bit value and transition in the actual $\text{GNSS}$ data. The nearly equal outcomes of the fourth and fifth data bit orders are shown in Figures 4 and 5. The MLE technique’s actual performance and consistency results for bit synchronization and decoding are focused on the phase lock mode as shown in Figures 6 and 7. Thus, the evaluation of the phase lock mode and frequency lock mode is presented here. Figures 2, 3, 4, and 5 demonstrate that the BPSK and BOC modulated signals are related. The BOC is used for the Galileo constellation in many applications, while the BPSK is used for the GPS. These two systems are therefore compatible with one another. Quality control and remote data retrieval can be carried out based on the results. The Galileo frequencies are accessible to GPS users.

[figure(s) omitted; refer to PDF]

Using the parameters determined using MLE and KF as illustrated in Figure 6, the modelling aims to reduce the effects of the large frequency error and carrier loop adjustments.

As illustrated in Figure 7, the $\text{GNSS}$ receiver of the same mode can track the signal power up to 30 dB-Hz when the coherent integration is set between 1 ms and 20 ms. Additionally, it is expected that a coherent integration of 20 ms will yield superior Doppler estimation results. These findings highlight the necessity of extending the coherent integration time in order to properly estimate the unknown parameters. By improving tracking models like vector tracking and Doppler tracking, the threshold’s tracking is expanded. As we wrap up, Figures 5, 6, and 7 also offer three suggestions for $\text{GNSS}$ schemes that can be applied when $\text{GNSS}$ receivers are implementing MLE bit synchronization.

• The scheme that makes use of various bit counts for synchronization based on signal strength and potential Doppler errors.

• The scheme that will use a lot of bits for synchronization, regardless of the signal power and other relevant factors.

• The scheme that will employ the number of bits according to the estimated ${C/N}_0$ and the relative $\text{GNSS}$ parameters.

7. Conclusion

In order to assess the effectiveness, dependability, and consistency of $\text{MLE}$ bit decoding and synchronization in $\text{GNSS}$ infrastructure, a systematic analysis of $\text{MLE}$ techniques is presented in this paper. The performance and Doppler error of standalone $\text{GNSS}$ receivers with various tracking modes are estimated using the random $\text{GNSS}$ data analysis concept. Additionally, the statistical modelling that is experimentally evaluated is used to provide a theoretical analysis of the $\text{GNSS}$$\text{MLE}$ bit decoding and synchronization performances. In conclusion, we found that the overall performance of the space data synchronization is determined by the number of data bit transitions rather than the total number of bits. An observed SSR, lower signal ${C/N}_0$, and greater Doppler frequency inaccuracy require more data bits for estimation and computation. It was also observed that, in the majority of cases, if not all of them, the bit evolution occurs with a probability equal to 60% or higher. With ${C/N}_0$ in marginal power estimated to 20 dB-Hz without Doppler error, it is probably going to reach the 95%–100% range. If the Doppler error is less than 6 dB-Hz, the signal attenuation caused by the Doppler inaccuracy is insignificant, and the maximum tolerance of the Doppler inaccuracy is 30 dB-Hz.

Funding

The research leading to these results has received funding from the Department of Science and Innovation (DSI), South Africa. This was facilitated through the Space Science and CNS Research Centre at Durban University of technology.

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