Research on ELoran Demodulation Algorithm Based on Multiclass Support Vector Machine

Abstract

Demodulation and decoding are pivotal for the eLoran system’s timing and information transmission capabilities. This paper proposes a novel demodulation algorithm leveraging a multiclass support vector machine (MSVM) for pulse position modulation (PPM) of eLoran signals. Firstly, the existing demodulation method based on envelope phase detection (EPD) technology is reviewed, highlighting its limitations. Secondly, a detailed exposition of the MSVM algorithm is presented, demonstrating its theoretical foundations and comparative advantages over the traditional method and several other methods proposed in this study. Subsequently, through comprehensive experiments, the algorithm parameters are optimized, and the parallel comparison of different demodulation methods is carried out in various complex environments. The test results show that the MSVM algorithm is significantly superior to traditional methods and other kinds of machine learning algorithms in demodulation accuracy and stability, particularly in high-noise and -interference scenarios. This innovative algorithm not only broadens the design approach for eLoran receivers but also fully meets the high-precision timing service requirements of the eLoran system.

1. Introduction

Since the 21st century, global navigation satellite systems (GNSSs) have become utilized for high-precision positioning, navigation, and timing (PNT) services, playing a crucial role in modern life [1,2,3]. However, GNSSs are susceptible to various interference and spoofing attacks, which limits their application in related fields to some extent [4,5,6,7]. In order to ensure the reliability of PNT services and information delivery, it is essential to research reliable alternative or complementary technologies. With countries around the world re-examining the vulnerability of GNSSs and proposing the enhanced Loran (eLoran) terrestrial backup system concept, many countries have begun modernizing and upgrading the original Loran-C systems. The eLoran system, as an internationally standardized ground-based radio system [8,9], can provide robust PNT services with strong anti-interference capabilities and wide coverage. Its stability and excellent performance make it a promising and important supplement and backup system for GNSSs. In recent years, many countries around the world have carried out or resumed the development of eLoran system. The establishment and advancement of the eLoran system not only meets the needs of time and frequency system construction of countries, but also becomes an indispensable part of the national space–Earth integration PNT system [10,11,12].

Demodulation is one of the core modules of signal reception and processing in the eLoran system, and it directly affects the accuracy of timing and information transmission. Scholars from various countries have conducted extensive research on demodulation technology for eLoran systems.

Currently, eLoran receivers commonly employ envelope phase detection (EPD) demodulation technology [13,14]. This method extracts the phase difference by sampling the envelopes of two orthogonal signal paths and achieves demodulation through a majority decision. However, this method has limitations in terms of anti-interference and noise resistance. In 2007, Lo et al. proposed a signal matching correlation–pulse position detection (SMC–PPD) method based on time-domain signal sliding correlation [15]. However, the error frequency of peak position detection increases significantly with the enhancement of noise and interference, leading to a higher bit error rate. In 2020, Yuan et al. proposed the envelope correlation–phase detection (EC-PD) algorithm, which combines two envelope correlation schemes: moving average–cross correlation (MA–CC) and matched correlation (MC), respectively, for skywave and no-skywave scenarios. Their study focused on the SNR gain impact of skywave and its effect on the demodulation performance, and adaptively switched two schemes according to the SNR gain, which can improve the modulation performance to a certain extent. However, it still falls within the category of phase demodulation [16]. In 2022, Lyu et al. proposed a new algorithm based on log likelihood ratio (LLR), which calculated the probability that each bit in the transmitted information was “0” or “1”, directly solved the modulation code, and achieved certain improvement in the decoding success rate. However, it should be noted that this method does not consider the influence of various interference environments on probability calculation, which needs further derivation and verification [17].

The above research demonstrates the diversity and challenges of current demodulation techniques in eLoran systems. In order to improve the performance of existing technologies in complex electromagnetic environments such as continuous wave interference and strong noise, this study conducts an in-depth analysis of eLoran signal characteristics and proposes a pioneering multiclass support vector machine (MSVM) algorithm for demodulation. Developed alongside other machine learning (ML) techniques such as the random forest (RF) and K-nearest neighbors (KNN) and a time-domain cancellation residual detection (CRD) algorithm, these methods serve as a comprehensive benchmark for evaluating demodulation performance. Our research aims to demonstrate the superiority of the MSVM algorithm over traditional methods like EPD and offer new insights in eLoran signal processing. This study has significant theoretical and practical contributions, as follows:

Firstly, by proposing a novel multiclass support vector machine (MSVM) algorithm, this study provides theoretical support for the innovation and development of eLoran demodulation technology, and lays a foundation for promoting its wide application in critical infrastructure and military fields.

Secondly, by optimizing the demodulation process, the anti-interference and anti-noise performance can be enhanced, and the stability and reliability of the eLoran system can be improved, so as to optimize the service quality of the system. This is of great significance for the eLoran system as an effective auxiliary and backup for GNSSs.

Finally, the innovative methods and theoretical results of this study have important theoretical reference value for the application of machine learning algorithms in the field of navigation technology. It provides valuable experimental data and theoretical support for researchers and engineers in related fields, and helps to promote the progress and application of navigation technology.

2. Background Principles

According to the regulations of the International Telecommunication Union (ITU), the operating frequency of Loran-C signals is mainly between 90 and 110 kHz, with a carrier center frequency of 100 kHz. The normalized standard unmodulated Loran-C pulse waveform can be represented by Formula (1) [18,19].

$$s\left(t\right)=\left\{\begin{array}{c}A{\left(t-\tau \right)}^2\exp \left(-2{\displaystyle \frac{\left(t-\tau \right)}{65}}\right)\sin \left(2\pi f_{c}t+P_c\right),\hspace{1em}\tau \ll t\ll 65+\tau \\ 0,\hspace{1em}t<\tau \end{array}\right.$$

(1)

where A is a normalization constant related to the peak amplitude; t is the time in μs; τ is the envelope-to-cycle difference in µs; $f_c$ = 100 kHz is the carrier frequency; and $P_c$ = 0 or π is the phase code parameter, in radians.

The Loran-C signals are transmitted in the form of pulse group, and the data transmission is realized through the information modulation of each pulse. In the same group repetition interval (GRI), the master station sends nine pulses in each group, with a 1 ms interval between the first eight pulses and a 2 ms interval between the eighth and ninth pulses. Each substation sends only the first eight pulses. According to Formula (1), the waveforms of Loran-C pulse and master station pulse group are shown in Figure 1. In order to facilitate identification and timing requirements, a unique GRI is designed for each chain, and different phase coding formats are also designed for the master station and substation [20,21].

The eLoran system is an upgraded version of the Loran-C system, which incorporates the Eurofix data link technology to modulate the information on the Loran-C pulse group [8,22]. By controlling the transmission time of the signal, the eLoran system performs a three-state pulse position modulation (PPM) of 0 or ±1 μs on the third to eighth pulses of each pulse group [23,24], as shown in Figure 2. This enables the dissemination of timing information and lays the foundation for future sharing of ASF differential correction values and other additional data information.

According to the PPM mode, the three modulations correspond to the characters “00”, “01”, and “10”, respectively. The six pulses are combined into a 12-character binary value, for a total of 3⁶ = 729 possible modulation combinations, of which 141 conform to the principle of balanced modulation. We select 128 different combinations, which can correspond to the 128-character ASCII code encoding method commonly used in computers. Each combination corresponds to a 7-bit binary ASCII code, also known as a “code word” [13,25].

The Eurofix data link mode defines that in the eLoran system, one frame of time code message data contains 210 bits of binary encoded information, including 56 bits of data information, 14 bits of cyclical redundancy check (CRC) code, and 140 bits of Reed–Solomon (RS) forward error-correcting code. That is, 30 consecutive pulse groups need to be transmitted and demodulated [26]. CRC and RS codes are used to correct different types of errors. CRC codes use the principle of division and remainder to calculate syndromes for error location detection [27,28]. The RS codes are used for error correction and have good resistance to burst interference [29,30]. The combined use of these two encodings can correct up to 10 character encoding errors, providing a reliable guarantee for the demodulation and decoding accuracy of eLoran data information.

The radio system obtains data by demodulating the modulation codes and combining them into a complete frame information. According to the above introduction, the eLoran signal will have a phase shift of 0° or ±36° through PPM modulation. By comparing the phase difference between the sampling point and the reference point, it will be converted into the corresponding modulation code word, and then decoded to the time code information. This basic logic is common in current eLoran receivers. It can be seen that the key to successful decoding lies in the accuracy of demodulating the modulation codes.

However, based on the signal characteristics of the eLoran system, we can observe that there are some problems with this method. Since the eLoran signal is an AM signal on the 100 kHz carrier, and each sampling point has different amplitude multiplication factor, the 36° phase detection method itself has principle deviation. In addition, the eLoran signal is susceptible to noise and continuous wave interference (CWI), especially high-power CWI, which can cause different distortion effects on the eLoran sample signals and the reference signal, resulting in significant error in phase detection, and ultimately affecting the success rate of decoding. In order to optimize the signal processing capability of receiving terminal, this paper takes the demodulation method as the core research object, and proposes a novel MSVM classification demodulation algorithm.

3. Method Description

This chapter will introduce the traditional envelope phase detection (EPD) demodulation techniques used by conventional eLoran receivers, and delve into the MSVM algorithm proposed in this paper. Furthermore, we will evaluate several alternative feasible methods that were conceptualized and tested as part of our research to serve as benchmarks against the MSVM algorithm. These alternatives were originally designed specifically for comparative analysis, allowing us to highlight the strengths and limitations of the MSVM method. We will provide a thorough description and comparison of the processing logic, algorithmic details, and the advantages and disadvantages of each method.

3.1. Envelope Phase Detection

Phase detection demodulation technology is currently widely used in eLoran receivers. It realizes the demodulation and decoding of the modulated signal through filtering processing and phase difference discrimination.

According to the technical block diagram of EPD algorithm shown in Figure 3, the main steps of the PD algorithm are as follows:

• Phase difference detection. After the phase tracking process stabilizes, the two branches I and Q undergo low-pass filtering, retaining only the sine and cosine components of the sum of modulation phase and phase offset, and the quadrature sampled data is ($a_i,b_i$). Similarly, the reference signal only retains the signal phase offset, while the quadrature sampled data are ($a_i^{ref},b_i^{ref}$), and the phase difference can be calculated using the following formulas:

  $$\left\{\begin{array}{c}\mathit{tan}{\alpha }_i=\mathit{sin}{\alpha }_i/\mathit{cos}{\alpha }_i=a_i/b_i\\ \mathit{tan}{\beta }_i=\mathit{sin}{\beta }_i/\mathit{cos}{\beta }_i=a_i^{ref}/b_i^{ref}\end{array}\right.$$

  (2)

  $$\mathit{tan}\left({\alpha }_i-{\beta }_i\right)={\displaystyle \frac{\mathit{tan}{\alpha }_i-\mathit{tan}{\beta }_i}{1+\mathit{tan}{\alpha }_i\mathit{tan}{\beta }_i}}={\displaystyle \frac{a_i{b}_i^{ref}-b_i{a}_i^{ref}}{a_i{a}_i^{ref}+b_i{b}_i^{ref}}}$$

  (3)

  $${\alpha }_i-{\beta }_i=arctan\left({\displaystyle \frac{a_i{b}_i^{ref}-b_i{a}_i^{ref}}{a_i{a}_i^{ref}+b_i{b}_i^{ref}}}\right)$$

  (4)

  where ${\alpha }_i$ and ${\beta }_i$ represent the phase of the modulated signal and the reference signal at the i-th sampling point, respectively. ${\alpha }_i-{\beta }_i$ represents the corresponding phase difference.
• Polarity decision. We set the threshold to ±18° and perform multithreshold processing on the phase difference calculated using Formula (4).
  • When $-18°\le \alpha -\beta \le 18°$, it is classified as “0” modulation;
  • When $\alpha -\beta >18°$, it is classified as “+” modulation;
  • When $\alpha -\beta <-18°$, it is classified as “−” modulation.
• Majority decision and balance error correction. Based on the results of the statistical N sampling points, the modulation is determined as “0”, “+” or “−” by majority decision. In the case of an equal number of occurrences, the error correction can be further decided according to the principle of balanced modulation [31].

The EPD algorithm is an intuitive and effective demodulation scheme widely used in traditional eLoran receivers. However, it also has obvious limitations. Due to uncontrollable factors such as hardware and electromagnetic interference, the eLoran signal may experience distortion and phase distortion in the transmission, propagation, and reception process. Especially when there is significant phase distortion in the received signal, it can severely affect the demodulation results. In addition, the method itself has some logical errors. Even in an ideal noise-free environment, due to the different carrier multiplication terms of the “+” and “−” modulated signals and the reference signal at the sampling points, the phase difference is not exactly 36°. Moreover, eLoran signals are also susceptible to in-band CWI, resulting in fluctuations in the phase difference. Figure 4 provides an example, showing the waveform distortion caused by two CWIs on the first 4 standard modulated pulse signals of an eLoran pulse group in a noise-free environment and the corresponding impact on demodulation.

As depicted in Figure 4a, in the presence of interference, the waveform distortion of each pulse signal is severe and the degrees are different. Figure 4b visually illustrates the varying levels of perturbation and fluctuation in the phase difference calculation by the EPD algorithm for the same modulated signal. Among them, the 3rd pulse is correctly demodulated as “+”, while the phase difference of most of the sampling points of the 4th pulse are below the threshold line, resulting in the incorrect demodulation as “0”. With the increase in CWI and noise power, as the example shows, the fluctuations become more pronounced and uncertain, resulting in decoding failures that occur frequently in practical applications. In order to improve the accuracy and reliability of demodulation, the MSVM classification algorithm based on machine learning (ML) is studied and proposed.

3.2. MSVM Algorithm

Support vector machine (SVM) is a traditional binary classification model widely used in various fields such as optical character recognition, handwritten digit recognition, text classification, land type classification, forest fire risk prediction, grain yield prediction, and more [32,33,34,35,36]. It achieves classification by finding a hyperplane to divide samples, with the principle of maximizing the margin, i.e., maximizing the distance between samples of two classes [37,38]. In a two-dimensional space, the hyperplane can be represented by a straight line, as shown in Figure 5a.

In Figure 5, the horizontal axis represents the feature $x_1$, and the vertical axis represents the feature $x_2$. These axes are designed to visually demonstrate the position and orientation of the hyperplane in relation to the data points. The primary purpose of this figure is to illustrate the classification capability of the hyperplane. Consequently, these axes do not carry traditional physical meanings or dimensions; instead, the focus is on the hyperplane’s effectiveness in classifying the data. The goal of SVM is to find a hyperplane equation:

$${\mathit{\omega }}^T\mathit{x}+b=0$$

(5)

where $\mathit{x}={\left(x_1,x_2,\cdots ,x_d\right)}^T$ is the point on the hyperplane, $\mathit{\omega }={\left({\omega }_1,{\omega }_2,\cdots ,{\omega }_d\right)}^T$ is the normal vector, which determines the direction of the hyperplane, and $b$ is the offset term. The hyperplane divides the space into two parts, and the distance from the sample ${\mathit{x}}_i$ to the hyperplane can be written as

$$r={\displaystyle \frac{\left|{\mathit{\omega }}^T{\mathit{x}}_i+b\right|}{‖\mathit{\omega }‖}}$$

(6)

Typically, positive class samples are labeled as y = +1, and negative class samples are labeled as y = −1 in SVM. In this way, for any sample point, its category label on the hyperplane side multiplied by the distance from the sample point to the hyperplane is always greater than or equal to 1. Assuming that the hyperplane can be correctly classified, there are

$$\left\{\begin{array}{c}{\mathit{\omega }}^T{\mathit{x}}_i+b\ge +1, y_i=+1\\ {\mathit{\omega }}^T{\mathit{x}}_i+b\le -1, y_i=-1\end{array}\right.$$

(7)

where $y_i$ is the category label of the feature sample ${\mathit{x}}_i$, which, together, form the sample points in the space. The several sample points when the equal sign of Formula (7) is true are called support vectors. The margin distance is determined by the two nearest different-class support vectors and can be expressed by Formula (8).

$$\gamma ={\displaystyle \frac{2}{‖\mathit{\omega }‖}}$$

(8)

The goal of SVM is to find the optimal solution of hyperplane equations, that is, to find parameters $\mathit{\omega }$ and $b$ that maximize the margin distance. Figure 5b shows the optimal hyperplane with the best generalization ability and the strongest robustness. Then, the optimization problem model of SVM can be established:

$$\begin{array}{c}\underset{\mathit{\omega },b}{\max }{\displaystyle \frac{2}{‖\mathit{\omega }‖}}\Rightarrow \underset{\mathit{\omega },b}{\min }{\displaystyle \frac{1}{2}}{‖\mathit{\omega }‖}^2\\ \mathrm{s}.\mathrm{t}. y_i\left({\mathit{\omega }}^T{\mathit{x}}_{\mathit{i}}+b\right)\ge 1, i=\mathrm{1,2},\cdots ,m.\end{array}$$

(9)

Formula (9) is a convex function problem. By introducing the Lagrange multiplier $\mathit{\alpha }=\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_m\right)$, a Lagrange function can be obtained, and, thus, converted into a parameter optimization problem.

$$L\left(\mathit{\omega },b,\mathit{\alpha }\right)={\displaystyle \frac{1}{2}}{‖\mathit{\omega }‖}^2+\sum _{i=1}^m{\alpha }_i\left(1-y_i\left({\mathit{\omega }}^T{\mathit{x}}_i+b\right)\right)$$

(10)

By zeroing out the partial derivatives of Formula (10), performing a series of transformations, and applying the Karush–Kuhn–Tucker (KKT) conditions for constraints, the optimization objective of the following duality problem can be obtained.

$$\begin{array}{c}\underset{\mathit{\omega },b}{\max }\left({\displaystyle \sum _{i=1}^m}{\alpha }_i-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^m}{\alpha }_i{\alpha }_j{y}_i{y}_j\kappa \left({\mathit{x}}_i,{\mathit{x}}_j\right)\right)\\ \mathrm{s}.\mathrm{t}.{\displaystyle \sum _{i=1}^m}{\alpha }_i{y}_i=0\\ \begin{array}{c}{\alpha }_i\left(y_i\left({\mathit{\omega }}^T{\mathit{x}}_{\mathit{i}}+b\right)-1\right)=0\\ y_i\left({\mathit{\omega }}^T{\mathit{x}}_{\mathit{i}}+b\right)\ge 1\\ {\alpha }_i\ge 0,\hspace{1em}i=1,2,\cdots ,m.\end{array}\end{array}$$

(11)

Then, the sequential minimal optimization (SMO) algorithm can be used to iteratively optimize the values of the Lagrange multipliers, gradually approaching the optimal solution. After the training is completed, only the multipliers ${\alpha }_i$ corresponding to the support vectors are nonzero, which determines the position and shape of the hyperplane, while other non-support-vector sample points have no influence on the hyperplane. This is also an important feature of SVM, as it reduces the computational complexity and realizes efficient classification [39].

For nonlinear divisible data classification problems, kernel functions can be used to map data to higher dimensional spaces. Kernel function plays a key role in SVM; its core idea is to use inner product to avoid explicit calculation of data mapping in high-dimensional feature space, thus greatly reducing the computational complexity. The most commonly used kernel functions include linear kernel, polynomial kernel, and Gaussian kernel, etc. [40,41,42].

To solve the 3-class modulation problem of eLoran signal, we propose the MSVM classification algorithm. MSVM is an innovative demodulation scheme based on SVM, which converts multiclassification problems into multiple binary classification problems and classifies the data according to the confidence score of the sample in each training model. The confidence score is usually obtained by calculating the distance from the sample to the hyperplane. In FPGA implementation, the computational resources can be saved by parallel integrated learning, and the decoding accuracy and reliability can be improved.

As shown in the basic technical block in Figure 6, the MSVM method is completely different from the EPD method in that it does not need to calculate the phase information of the two branches. The main steps of MSVM demodulation algorithm are as follows:

• Feature vector construction. Based on the pulse groups and modulation features of the eLoran signal, the feature points are collected within an 80 μs range near the maximum peak position P (P − 32.5 μs~P +47.5 μs), including the following 3 types:
  • Approximate slope of peak position: The value difference of lag and lead 1 μs sampling points for each pulse’s 16 positive and negative peaks (16 dimensions).
  • Peak position value: 16 positive and negative peak position values (16 dimensions).
  • Zero-crossing position value: Values of the sampling points 2.5 μs before the 16 peak positions (16 dimensions).

This refined feature collection approach, focusing on 8 consecutive carrier cycles, capitalizes on their heightened power to reduce the influence of noise and interference, thus bolstering the dependability of the extracted features. Experimental results have demonstrated that this configuration achieves a superior balance in performance, ensuring elevated demodulation precision without incurring the high computational costs associated with an excessive number of features. Moreover, selecting 8 carrier cycles also considers the traditional range utilized in the EPD algorithm, guaranteeing equitable grounds for lateral comparisons.

2.

Sample collection: Under the influence of added CWI and white noise, the first pulses of 100 consecutive GRI pulse groups are collected as the “0” modulation feature samples, according to step 1. Then, samples are collected with a 1 μs lag as the “+” modulated samples and with a 1 μs lead as the “−” modulated samples. These samples, together, form the training set, which consists of 300 samples obtained from 100 groups of signals, each group containing 3 pulse groups. In addition, the same position sampling points of the 3rd to 8th pulses of each pulse group, together with the modulation information, form the test set.

3.

Kernel function selection: The selection of kernel function has a significant impact on the classification result. Reference [43] describes the kernel technique in detail. Linear kernel is generally used for linear problems, while nonlinear problems need to be tested according to the actual situation to choose the appropriate kernel function. The experimental results in Section 4.2 show that linear kernel is more suitable for the eLoran demodulation problem in this study.

4.

Training the model: To address the PPM modulation characteristics of the eLoran signal, the multiclassification problem is transformed into three binary classification problems. SVM models are trained separately for each, and then combined to form the MSVM model.

5.

Confidence-based classification: The distance between the sample and the hyperplane of each binary classification model is calculated as the confidence score, or it is transformed into probabilities using functions such as the Sigmoid function, and the data are classified and demodulated based on the principle of high confidence.

The MSVM algorithm, based on the principles of SVM, exhibits strong generalization capability and robustness. It demonstrates fast computation speed and training time savings when dealing with small sample problems such as eLoran signal demodulation. By selecting appropriate kernel functions and feature vectors, the MSVM algorithm showcases high performance advantages in handling both linear and nonlinear problems. However, a potential drawback of this algorithm is the need for extensive experimentation and adjustment when selecting kernel functions and feature vectors.

Figure 7 shows the different features of the eLoran signal within one carrier cycle in a noninterference environment. Among them, the red dots and the black connecting lines, respectively, correspond to the peak position value, the zero-crossing position value, and the approximate slope, which is consistent with the content introduced in step 1 of MSVM algorithm in this section. In Section 4, the effects of using different feature vector schemes and different kernel functions will be fully verified and compared.

3.3. Alternative Algorithms

This section will introduce three different algorithms, all of which can be used to solve the demodulation problem of eLoran signals, and each has different characteristics, advantages, and disadvantages. RF and KNN, two widely used ML algorithms, are often compared with SVM in various scenarios [33,35,44], while the CRD algorithm is a simple and easy-to-implement basic method.

3.3.1. Random Forest Algorithm

RF is a powerful ML algorithm based on ensemble learning principle, which has a wide range of applications in solving classification problems [45]. In this study, we also test the effectiveness of this algorithm on the eLoran signal demodulation problem, and compare it with the MSVM method proposed above.

According to Figure 8, the specific steps of the RF scheme are as follows:

• Data augmentation: The RF algorithm constructs multiple decision trees by introducing randomness. For each tree, k training samples are randomly selected with replacement from the original training dataset. This subset selection method is called Bootstrap sampling [46].
• Feature selection: From all M features, m features are randomly selected, and the optimal one is selected from these m features each time the tree is split. Randomness can be increased by randomly selecting m features. Generally, the heuristic selection of $m\approx \sqrt{M}$ is chosen, which can reduce the complexity of the model and increase the generalization ability of the model [47].
• Decision tree training: Decision tree training is performed by recursively splitting the dataset using the classification and regression tree (CART) algorithm, without pruning, to maximize the purity (Gini index or entropy) of each child node until no further partitioning can be performed [46,47,48]. In this way, an RF model composed of N decision trees can be obtained.
• Ensemble prediction: In the prediction process, the RF algorithm will calculate the prediction result of each decision tree and obtain the final classification result by majority voting. This can reduce the misjudgment of a single decision tree and improve the accuracy of the model.

The RF algorithm has the advantages of high accuracy and robustness, processing high dimensional data, and it is not easy to overfit [49,50]. However, RF also has some drawbacks, for example, poor performance on small datasets, sensitivity to imbalance samples and noise, poor model interpretation, long training, and prediction time. Additionally, the computational complexity increases with the depth of the decision tree.

The number of trees N and m value are the key parameters that affect the performance of the RF algorithm [51]. In the experiment, it can be found that increasing the number of decision trees can improve the classification accuracy to some extent, but it also increases the amount of computation. Through repeated debugging, we found that 100 decision trees can achieve good classification performance while keeping the amount of computation within an acceptable range. Other optimal setting parameters include the following: the minimum sample number of leaf nodes is 1, the feature selection method is “best”, and the depth is set to “maximum depth”.

3.3.2. K-Nearest Neighbors Algorithm

The KNN algorithm is an instance-based nonparametric method, widely used in data mining and ML fields, as well as a basic classification and regression method [52,53,54]. The core idea is that if most of the K most similar (i.e., closest neighbors in feature space) samples of a sample belong to a certain category, then the sample belongs to the same category as well [53,55]. The block diagram of the eLoran system demodulation scheme based on the KNN algorithm is shown in Figure 9, and the specific steps of KNN demodulation of eLoran system are as follows:

• Distance calculation: The key to the KNN algorithm lies in calculating the distance between unknown samples and known samples. The Euclidean distance (straight-line distance), Manhattan distance, and Chebyshev distance are among the most commonly used metrics methods in the KNN algorithm [56,57]. In practical applications, the most suitable distance metric can be selected based on the characteristics of the data and the requirements of the problem, so as to obtain the best classification or regression performance.
• K value selection: During the training process, a suitable K value needs to be selected, that is, the K known samples closest to the unknown samples considered in the calculation. Too small a K value may result in overfitting, while too large may reduce the accuracy of the classification. In experiments, the appropriate K value can be selected through methods such as cross-validation [54,57,58].
• Classification decision: Count the number of occurrences of the K known samples in each category. The category of the unknown sample is determined based on the majority vote of the categories of the K nearest neighbors [53,55].

Unlike other algorithms, the KNN algorithm does not have an explicit model in the training process, and its “training result” actually refers to the storage of training data stored in memory for subsequent prediction use. Therefore, the main advantages of the KNN algorithm are that it is simple and easy to implement, there is no need for complex training, and it can deal with multiclass problems.

However, the KNN algorithm also has some shortcomings, such as high computational complexity for high-dimensional and large-scale datasets, and sensitivity to imbalanced data and noise. In experiments, we verify that the KNN algorithm may not achieve the same classification effect as the RF and MSVM algorithms in some cases.

The main parameters of the KNN algorithm include K value, distance metric method, and feature weight [59]. In solving the classification demodulation problem of eLoran signals, a smaller K value can be used due to the small sample size, and odd values should be chosen to avoid ties [60]. Through extensive parameter tuning experiments, we determined the optimal parameter selection as follows: K = 5, Euclidean distance metric, and employing average weights.

3.3.3. Cancellation Residual Detection Algorithm

The CRD algorithm is a common signal processing technology that is mainly used to detect and eliminate signal interference caused by multipath propagation. It is also an effective scheme for eLoran signal demodulation. The technical block diagram of the scheme is shown in Figure 10.

The basic idea of the CRD algorithm is to demodulate by estimating and comparing the residual power of signals in different channels. Specifically, the steps of the CRD scheme are as follows:

• Signal cancellation: By means of delayed sampling, the prompt, 1 μs lagging, and 1 μs leading signals of the reference signal correspond to “0”, “+”, and “−” modulation, respectively. The three signals are cancelled with the reference signal. The phase code of the sample signal needs to be confirmed by the capture module at the front end, and consistent with the reference signal by flipping or remaining unchanged.
• Residual power calculation: The root mean square (RMS) of the 3 channels after cancellation is used as the residual power estimation. Theoretically, the residual power of the branch is the smallest when the modulation signal is matched.
• Demodulation verdict: Based on the principle of minimum power, determine which of the signals best matches the sample signal, and select it as the optimal match channel for demodulation corresponding to the modulation.

Compared with phase detection method and machine learning algorithms, the cancellation detection algorithm has the advantage of simple implementation without phase calculation and iterative learning. In addition, it can reduce the influence of noise and carrier interference on signal demodulation to a certain extent.

However, in the complex strong interference and noise environment, the performance of the cancellation detection algorithm may be poor, and it cannot meet the requirements of high-stability signal demodulation. In particular, the near-frequency interference may cause the distortion of sample signal and reference signal, resulting in mismatching and demodulation errors.

4. Experimental Results and Performance Analysis

This chapter will compare the proposed MSVM algorithm, the traditional EPD algorithm, and several other feasible schemes. Among them, the sampling data used by several schemes remained consistent. The experimental objectives include the following:

• Effects of MSVM when using different kernel functions.
• Influence of different feature vectors.
• Comparison of demodulation accuracy of various methods in different SNR and CWI environments.

4.1. Experiment Design

The experimental environment design is given below:

• Experiment environment: Matlab R2017, Windows 10 operating system, Intel Core i7 processor, 16 GB memory.
• Input signal: Continuous modulated pulse group signals with a GRI of 60,000 μs. This GRI value was chosen primarily for experimental consistency, and the robustness of the experimental results is independent of the specific GRI value.
• Data modulation: Standard time code frame data.
• Sampling rate: 2 MHz.
• Bandpass filter: 95~105 kHz.
• SNR: −20~10 dB white noise (before bandpass filtering); step size: 1 dB.
• CWI: 0~4 single-frequency interferences, labeled as C₁, C₂, C₃, and C₄. The specific frequencies and signal-to-interference ratio (SIR) for each interference are as follows: C₁ at 97.2222 kHz with an SIR of −7 dB, C₂ at 101.3333 kHz with an SIR of −10 dB, C₃ at 103.6666 kHz with an SIR of −9.21 dB, and C₄ at 99.1111 kHz with an SIR of −8.24 dB. Since the bandpass filter and de-jamming module of the front-end signal processing can eliminate most of the interference, we set a moderate interference power. Each interference is introduced with a random initial phase to simulate a realistic and variable interference environment. In experiments involving CWIs, they are labeled and combined sequentially; “1 CWI” refers to C₁, and “3 CWIs” refers to the combination of C₁, C₂, and C₃. This labeling scheme is applied uniformly throughout our experiments and results descriptions in this paper.
• MSVM kernel functions: Linear kernel, Gaussian kernel, and polynomial kernel.
• Repetitions: 90,000 demodulation samples (500 frames) were used for validation under each distinct interference environment setting, involving 10 independent experiments. Each experiment utilized 50 consecutive frames to ensure randomness and reduce the contingency of experimental results.
• Evaluation index: This study employs the demodulation accuracy ratio (DAR) as the primary evaluation metric to comprehensively assess the demodulation performance of various demodulators across diverse scenarios.

4.2. Kernel Function Performance Evaluation

This section primarily conducts experiments to compare and analyze the performance of the MSVM demodulation algorithm under different kernel functions. Building on the sample preprocessing introduced in Section 4.1, we focus on studying the linear, Gaussian, and polynomial kernels in different noise and interference environments.

The experimental results, as illustrated in Figure 11 and detailed in

Table 1. Average DAR (%) difference of each kernel to linear kernel in the MSVM algorithm.

Kernel Function	No CWI	1 CWI	2 CWIs	3 CWIs	4 CWIs
Linear	0	0	0	0	0
Gaussian	−0.71	−1.00	−1.57	−1.88	−1.63
2-order polynomial	−0.74	−0.87	−1.70	−2.52	−2.52
3-order polynomial	−2.73	−3.05	−4.57	−4.85	−5.73

, indicate that the linear kernel demonstrates superior demodulation accuracy, consistently outperforming other kernel functions; the DAR increases by an average of 0.71%, 0.74%, and 2.73% at least when using the linear kernel compared to other options. In relatively low SNR conditions (−20~0 dB), these improvements are particularly pronounced, reaching 1.05%, 1.09%, and 4.03% respectively. Furthermore, as interference increases, the advantage of the linear kernel becomes even more pronounced. The Gaussian kernel, as a suboptimal choice, slightly outperforms the two-order polynomial kernel and can handle a certain degree of nonlinear relationships. However, its performance in eLoran signal demodulation is mediocre and may fail to effectively capture the characteristics of eLoran signals.

Although Table 1 does not include detailed statistical data for SNR segments due to the complexity of the data, it offers a comprehensive analysis across a broad range of SNR levels. It is evident from the table that the linear kernel function employed in the MSVM algorithm demonstrates a clear advantage in addressing the eLoran demodulation problem. This advantage is particularly pronounced in signal environments that pose varying degrees of challenge. The tables in the following sections will also utilize a similar approach, drawing directly from experimental result data.

The linear kernel is based on the linear divisibility of the samples in a high-dimensional feature space, assuming that samples of different classes can be efficiently separated by a hyperplane. In eLoran signal demodulation, the noise effect is predominantly additive, meaning that the signal can be represented by a linear function. Thus, the linear kernel can better capture this linear relationship and improve demodulation accuracy.

Furthermore, in low-SNR environments, noise has a more pronounced impact on the signal. The linear kernel function possesses a certain level of noise suppression. By employing linear combinations, it can suppress features or noises unrelated to classification tasks, thereby reducing the influence of noise and improving the model’s robustness and generalization capability. In contrast, higher-order polynomial kernel functions map feature vectors into higher-dimensional spaces, where they exhibit more sensitivity to noise and unimportant features. This can lead to overly complex models that overfit the training data and fail to generalize well to unknown data, potentially resulting in a model that is too complex and overfits the training data, hindering its ability to generalize.

In summary, the linear kernel better captures the characteristics of eLoran signals and exhibits superior adaptability in eLoran signal demodulation. It can effectively manage linear relationships, suppress noise, and capture signal features. Therefore, in the proposed MSVM algorithm, the linear kernel is the optimal choice. This conclusion serves as a reference for addressing similar problems in the future.

4.3. Feature Vector Selection

In ML algorithms, selecting the appropriate sample features is essential for optimal performance. Section 3.2 presents the feature acquisition methods, which consist of 16 slope value features, 16 zero-crossing position value features, and 16 peak position value features extracted from each pulse. Utilizing these features, we evaluated three distinct feature integration schemes:

• Using only approximate slope values (16 dimensions).
• Using approximate slope values and zero-crossing positions values (32 dimensions).
• Using approximate slope values, zero-crossing positions values, and peak positions values (48 dimensions).

As illustrated in Figure 12a–c, the results consistently indicate that across various ML methods, scheme 3 achieves the highest demodulation accuracy. This finding underscores the substantial benefit of combining diverse feature types within ML algorithms. By augmenting the number and variety of features, richer information is furnished to the learning process, thereby enhancing its performance.

Table 2. Average DAR (%) difference of each feature selection scheme to scheme 1.

Method	Feature Selection Scheme	No CWI	1 CWI	2 CWIs	3 CWIs	4 CWIs
MSVM	Scheme 1	0	0	0	0	0
	Scheme 2	2.34	2.48	2.56	2.58	2.71
	Scheme 3	4.03	5.01	4.99	4.60	4.64
RF	Scheme 1	0	0	0	0	0
	Scheme 2	2.92	3.18	3.20	3.23	3.45
	Scheme 3	4.11	5.76	6.26	4.88	4.69
KNN	Scheme 1	0	0	0	0	0
	Scheme 2	2.32	2.49	2.58	2.61	2.89
	Scheme 3	3.66	5.23	5.98	4.72	4.93

provides a detailed statistical analysis of the DAR differences among these approaches under different interference conditions. Scheme 3 demonstrates a significant performance advantage in various interference scenarios, with an average DAR improvement of 3.36% to 4.93% over scheme 1. At lower SNR conditions (−20~0 dB), this enhancement is even more pronounced, ranging from 4.91% to 7.92%. In addition, scheme 3 also has obvious improvement compared with scheme 2.

4.4. Cross-Method Performance Analysis

Section 4.2 and Section 4.3 discuss the optimal kernel function and the feature vector selection scheme of the MSVM algorithm, and the optimal parameters of other comparison methods are introduced in Section 3. This section will compare the demodulation and decoding performance of several methods mentioned in this paper under different noise and interference environments.

• No CWI environment

Firstly, the influence of noise on various methods is discussed in an environment without carrier interference. Figure 13 shows the DAR curves of various methods.

The statistical data in

Table 3. Average DAR (%) difference of each algorithm to EPD algorithm.

Method	No CWI	1 CWI	2 CWIs	3 CWIs	4 CWIs
EPD	0	0	0	0	0
MSVM	7.84	15.16	12.87	11.70	12.18
RF	6.79	13.14	10.66	9.87	9.84
KNN	6.48	12.35	11.08	10	10.04
CRD	2.63	7.39	4.45	3.65	2.94

indicate that the performance of each method improves with the increase in SNR. When the SNR is above −10 dB, the accuracy rate of all methods exceeds 80%. Among them, the ML algorithms, especially the MSVM algorithm, demonstrate the best demodulation performance, with the DAR significantly outperforming the traditional EPD method. For instance, at relatively low SNR levels, such as −20 dB, the DAR difference even exceeds 20%, while the DAR of the MSVM algorithm can reach 100% when above −2 dB. Notably, at an SNR of −10 dB, the decoding success rate achieves a perfect score of 100%, underscoring the superior performance of the MSVM algorithm compared to its counterparts.

b.

Different number of CWI environment

Since the receiving front end will be processed by anti-interference such as BPF and notch filter, only the in-band CWIs are considered in the experiment, and the SIR will not excessively high. According to the design in Section 4.1, the number of interference is set from 1 to 4. Figure 14 shows the comparison of DAR of each demodulation method under different conditions.

The above figures clearly demonstrate that even in the presence of interference, the MSVM algorithm maintains optimal demodulation performance. As detailed in Table 3, the DAR of the MSVM algorithm is, on average, at least 11.7% higher than that of the EPD algorithm across a range of interference conditions, significantly outperforming other algorithms. This performance advantage is particularly evident at low SNR levels. Furthermore, Figure 14a–d highlight the substantial negative impact of CWI on demodulation performance. High interference power can prevent the achievement of ideal demodulation accuracy, even in high-SNR environments.

4.5. Overall Evaluation and Discussion

In this chapter, we conducted a comprehensive experimental design, result analysis, and performance comparative evaluation of the MSVM-based eLoran demodulation algorithm proposed in this paper, and reached the following conclusions:

• Experimental results clearly show that linear kernel functions perform excellently in our study. This means that our demodulation problem may be more suitable for the linearly separable case. Linear kernel has the advantages of high computational efficiency and model simplicity, while also exhibiting good noise resistance. However, other kernels (such as polynomial kernel, Gaussian kernel, etc.) may be more efficient in other cases. Therefore, it is crucial to select the appropriate kernel function according to the characteristics of specific problems.
• The research results on feature vector selection indicate that the demodulation accuracy of the eLoran signal can be significantly improved by using a combination of multitypes of features. However, it is also necessary to consider the computational complexity that can be encountered when dealing with high-dimensional data.
• Through comparing the demodulation and decoding performance of several methods in different environments, the proposed MSVM algorithm stands out in various aspects. It demonstrates significant advantages in classification accuracy and computation efficiency (except CRD). In contrast, the RF algorithm has slightly lower accuracy and unsatisfactory computational efficiency. The traditional EPD algorithm performs poorly in strong interference environments, while the KNN algorithm, despite its simple principle, lacks advantages in all aspects. In addition, although the CRD algorithm is simple to implement, it has poor stability and is rarely seen in practical applications.

In summary, the MSVM algorithm is a robust and efficient ML method that performs well in eLoran demodulation and decoding problems. Future studies need to further validate and test real scenarios to explore more effective parameter selection and optimization methods to further improve model performance.

5. Conclusions

With the rapid development of computer technology, ML algorithms have been widely used in the field of communication. In this study, we explored the application of ML to the eLoran demodulation problem, and proposed an MSVM demodulation algorithm.

This study introduced the principles, advantages, and disadvantages of traditional EPD algorithms, CRD algorithms, and various other machine learning algorithms for demodulation, followed by a comprehensive experimental comparison analysis. Firstly, the performances of different kernel functions in the proposed MSVM algorithm were analyzed. The experimental results reveal that the linear kernel outperformed others in handling the problem of eLoran signal demodulation and can effectively capture the characteristic information of eLoran signals. Secondly, the issue of feature vector selection was studied, and it was found that the combination of multifeature categories has a positive effect on the demodulation performance, which can significantly improve the accuracy and stability of the demodulation model. Finally, through the horizontal comparison of various methods in different environments, the superior comprehensive performance of the proposed MSVM algorithm in eLoran signal demodulation was verified, especially in terms of the classification accuracy and calculation speed.

However, despite the promising outcomes, we acknowledge the challenges for practical deployment. The complexity of the MSVM algorithm may present difficulties for efficient and stable implementation on the FPGA and DSP platforms typically utilized in eLoran receivers. To leverage hardware capabilities effectively for real-time processing, further optimization and customization of the algorithm are necessary. Moreover, while we have conducted extensive experiments in simulated settings, the absence of broad testing and parameter tuning in complex real-world environments might limit our comprehensive understanding of the algorithm’s versatility and effectiveness.

In conclusion, the implications of this study are twofold: enhancing the current eLoran system’s robustness and effectiveness while paving the way for the integration and innovation of artificial intelligence with the field of communication. Our findings lay a solid groundwork for future signal processing studies and indicate the transformative impact of artificial intelligence on communication technology. The MSVM algorithm exemplifies the potential of machine learning to innovate and optimize signal processing tasks. Looking ahead, this study’s extensive applicability invites further exploration. Future research should concentrate on the following:

• Implementation and debugging of the algorithm in receiver.
• Further optimization of the kernel function and feature selection method to improve model performance and stability.
• Conducting more extensive tests on real environments to verify the universality and effectiveness of the MSVM algorithm.

Advancing along these research avenues will deepen our understanding of ML’s role in signal processing and drive the integration of eLoran systems with emerging technologies, ensuring their sustained relevance and progress in the communication sector.