Wavelet and Neural Network-Based Multipath Detection for Precise Positioning Systems

Abstract

Multipath errors are significantly challenging in radio navigation systems. In particular, multipath errors in indoor environments cause significant errors in the position domain because not only the building materials that surround the environment but also all objects inside the building can reflect the navigation signals. Multipath errors in outdoor environments, such as in global navigation satellite system (GNSS) signal applications, have been widely studied for precise positioning. However, multipath studies for indoor applications have rarely been conducted because of the complicated environment and the many objects made of various materials in small areas. In this study, multipath mitigation methods using a shallow neural network and a transfer learning-based deep neural network were respectively considered to overcome the complexity caused by the reflected signals in indoor environments. These methods classify each measurement according to whether the measurement exhibits a severe multipath error. Carrier-phase measurements broadcasted from the transmitter were used for the wavelet transform, and the magnitude values after the transform were used for neural network-based learning. Shallow and deep networks attain approximately 87.1% and 85.6% detection accuracies, respectively, and the positioning error can be reduced by 10.4% and 9.4%, respectively, after multipath mitigation.

1. Introduction

To achieve precise positioning irrespective of the environment, the multipath signals, namely, radio signals reflected from unexpected obstacles, are a major error source that degrades the positioning performance of radio navigation systems. The reflected signals confound the range measurement, generating an unexpected bias by distorting the correlation peak [1]. For the global navigation satellite system (GNSS), both pseudo-range and carrier-phase measurements can be contaminated by multipaths [2].

Various studies have been conducted to mitigate this multipath effect. In particular, multipath studies for GNSS have been widely performed to improve the navigation performance in multipath vulnerable areas, such as deep urban canyons [3,4,5,6,7,8]. Recently, artificial intelligence-based approaches such as machine learning and deep learning have been applied to mitigate multipath effects for outdoor GNSS measurements. A support vector machine (SVM) technique was used to separate the line-of-sight (LOS) multipath from non-LOS (NLOS) signals trained by measurements and C/N0 [9]. Classification with machine learning techniques using the received signals from both right- and left-hand circularly polarized antennas was also presented and analyzed [10]. This technique utilizes the fact that the reflected signals usually exhibit reverse-polarization characteristics. A convolutional neural network (CNN) was used to learn the features of the carrier-phase multipath to identify contaminated data [11]. The correlation function can also be utilized for training. SVM and neural networks (NNs) were applied to extract the features from the correlation function to detect NLOS measurements and assess their mitigation [12,13]. An SVM-based nonlinear non-line-of-sight and multipath prediction model using only the relative position information of the user and GNSS satellite to remove the multipath error directly was also presented [14].

Although there have been many studies on outdoor multipath errors, studies on indoor multipaths have not been commonly conducted. Compared with outdoor applications, the path between the transmitter and receiver is very short in indoor environments with denser obstacles between them, which are surrounded by various building materials. Therefore, more complicated paths of reflected signals exist in indoor environments [15]. These characteristics of indoor multipaths make it more difficult to mitigate errors. For instance, the ray-tracing approach applied for outdoor GNSS applications [16] is difficult to apply directly to indoor environments because there are too many irresistible paths for the reflected signals. Multipath mitigation techniques, such as 3D mapping and shadowing, have already been studied for outdoor GNSS signals, which cannot be used for indoor applications because the objects in the environment can vary frequently and easily, resulting in a time-varying multipath effect.

Few studies have been conducted on indoor multipath positioning. Indoor localization using wireless LAN also suffers from multipath errors regarding both signal strength and time-of-flight-based positioning; therefore, a fingerprinting approach to mitigate the multipath error was presented and analyzed [17]. In ultrawideband signal-based indoor positioning, the maximum likelihood estimator was adapted to recognize and resolve the multipath components in the measurements [18]. A multipath study, also considering using optical signals, was conducted to characterize and measure the effects of the multipath on the angle of arrival and phase shift in the signal arrival for indoor positioning [19]. However, there is a lack of multipath studies on the carrier phase in radio navigation signals in indoor environments. In addition, there have been no attempts to apply machine learning techniques for indoor multipath detection.

To overcome the characteristics of indoor multipath errors, this study presents novel approaches that use neural networks as an algorithm-based multipath mitigation solution. Because it is almost impossible to model all the reflected signals in indoor circumstances, machine learning techniques were applied, in which the algorithm learns the reflected signal features to detect the signals, including severe errors. To extract the multipath features in the signals, a wavelet transform from the carrier-phase measurement was applied to generate the input of the network. A continuous wavelet transform (CWT) was utilized for the measured experimental data to form the scalogram, which denotes the absolute value of the signal CWT coefficients [20]. The magnitudes of the coefficients after the transformation were used in the supervised learning process. The output of the network was obtained from the Softmax classifier to detect contaminated measurements.

Two neural network types, namely, shallow neural networks (SNNs) and transfer learning-based deep neural networks (DNNs), were, respectively, considered in this study. The SNN, including one hidden layer, and arrays of the transform coefficient magnitudes were used for the learning. The DNN for transferred learning using AlexNet [21] uses the red, green, and blue (RGB) values of the scalogram image to learn the multipath existence. High multipath detection accuracies were achieved with both SNNs and DNNs, and positioning errors were reduced after eliminating contaminated signals. The results obtained using each network are compared and discussed.

The remainder of this paper is organized as follows. The background knowledge regarding wavelet transform for multipath study, transferred learning, and indoor positioning system utilized in this study are described in Section 2. In Section 3, a mathematical model of multipath errors and methodologies of supervised learning frameworks applied to multipath detection are presented. In Section 4, the experimental environment and settings for the learning process are described, and validation results are presented. The discussion and conclusions are in Section 5.

There are no similar techniques for indoor applications for comparison purposes. In the literature, most multipath mitigation studies have been performed for pseudorange measurements in outdoor GNSS applications. There is an article that utilizes a convolutional neural network for the multipath detection of carrier-phase measurements [11]. However, that work was also designed for outdoor applications, and valid pseudorange measurements are required for the presented process. In general, multipath errors in carrier-phase measurements are less significant in outdoor applications because their impact on the positioning error is quite small, at about a centimeter level. However, their effect cannot be ignored in indoor environments, particularly for single-transmitter-based positioning systems. There have been studies to mitigate multipath errors in indoor environments [17,18,19], but these works also cannot be utilized for comparison groups because they did not use carrier-phase measurements. The ray-tracing approach is also difficult to apply to indoor environments because many reflected signals need to be modeled.

2. Background

Before describing the methodologies applied to multipath detection, the following subjects must be described in advance.

2.1. Wavelet Analysis and Multipath

Wavelet transform represents a signal with a family of wavelet functions, or wavelet basis, generated from a prototype function by translation and dilation operations [22,23]. Wavelet-based techniques have been widely applied in signal processing, data compression, and data analyses in engineering, sciences, finance, etc. [24].

Wavelet transform is also an effective tool for mitigating multipath errors; therefore, various studies have been conducted utilizing wavelet characteristics. Wavelet-based denoising and signal extraction have been applied for GNSS multipath mitigation because wavelet filtering can accurately represent both time and frequency characteristics. Wavelet decomposition was applied with the code minus carrier technique to provide high-fidelity multipath estimation and removal to enhance real-time performance [25]. The wavelet transform was also used to decompose the double-differenced GNSS measurements to separate the high frequencies occurring owing to multipath effects [26].

Techniques using wavelet characteristics have been used to mitigate the multipath errors in static GNSS receivers considering repeated time and orbit. Wavelet decomposition was also used to extract multipaths from global positioning system (GPS) observations, verifying the presence of multipaths observed from a static receiver on different days [27]. To mitigate GPS multipath effects, a cross-validation method for automatically identifying wavelet signal layers was developed to separate noise from signals in a data series [23]. Wavelet decomposition was also applied for multipath filtering of carrier-phase measurements considering repeatable satellite orbits for static high-precision positioning [24].

In other studies, extraction techniques for multipath mitigation in the carrier-phase measurement domain using wavelet multi-resolution analysis were presented [28], and wavelet analyses were applied to multi-constellation and multi-frequency GNSS signals to act as a tide gauge utilizing multipath frequency [29].

2.2. Transfer Learning

Transfer learning or transferred learning is a technique to improve the learning of a new task using knowledge obtained from a different and related task that has already been learned [30]. This technique also attempts to imitate the human learning processes where the knowledge accumulated from previous experience can help learn new work. An application area for transfer learning is DNN-based image processing. In general, a DNN requires a vast labeled dataset to effectively train with good performance; however, it is time-consuming and costly. Therefore, the transfer of knowledge obtained from a different task, which is already fully learned using different but similar data, can be an effective approach to overcome the data shortage when learning a new task [31,32].

The transfer learning technique was applied to detect changes in remote sensing images, which make it difficult to obtain sufficient labeled data [33]. In this study, a change detection framework consisting of pre-training and fine-tuning stages was developed to achieve effective detection performance using transfer learning. The medical image analysis field also suffers from a lack of sufficient datasets [34]. Therefore, transfer learning has been applied in various medical applications. Transfer learning using AlexNet as its basis model was used to classify multiple sclerosis brain images [35]. The performance results were compared with those of state-of-the-art classification approaches, yielding a better performance. Pneumonia detection in chest X-ray images [36,37], diabetic retinopathy image classification [38], stenosis detection [39], and skin and breast cancer classification tasks [34] exhibited good detection and classification performance with transfer learning. In addition, transfer learning also has been used in various fields. It was applied to sarcasm detection [40] in language processing and applied to damage detection and fault diagnosis of rotating machines [41,42].

AlexNet is a representative convolutional neural network (CNN) architecture that uses a DNN for image classification. The network has 650,000 neurons in five convolutional layers, consisting of max-pooling layers, fully connected layers, and a Softmax function. AlexNet competed and won the ImageNet contest in 2012, proving its performance [21]. It is one of the most influential architectures and has been applied in many fields, including transfer learning [35].

2.3. Single Transmitter-Based Positioning System

A single-transmitter-based positioning system, termed the “Mosaic system” in this study, was suggested for non-GNSS environments, such as alternative position, navigation, and timing [43,44,45,46] and indoor navigation systems [47,48,49], owing to its special features. The features are listed as follows:

• A single-transmitter system can provide a positioning service that minimizes the number of transmitters.
• Standalone positioning is available without requiring external information such as a database or map.
• Continuous positioning is available to keep track of user trajectory.
• As the user calculates the position, not the server or other external components, privacy and security are maintained.

In an indoor Mosaic system, the transmitter generates multichannel signals and broadcasts them through a single antenna array. In this study, the signal generator is a pseudolite that broadcasts the GPS L1 signals on the ground [50]. Users calculate the positions utilizing the time difference of arrival (TDOA) of carrier-phase measurements. Another feature of this system is carrier-phase measurements, whose noise level is significantly low, which are used to overcome the poor dilution-of-precision (DOP) environment, which is unavoidable because only a single transmitter covers the target field. Although the multipath error of carrier-phase measurements is mathematically bounded to less than a quarter of the carrier wavelength [16], even small-scale errors can cause considerable positioning errors in a poor DOP environment. In previous research, the positioning error due to multipath was observed in the repeated trajectory and its characteristics were analyzed [48].

A similar concept for using a single transmitter in indoor positioning has also been proposed by other research groups. A multichannel pseudolite with a hyperbolic positioning method was proposed and tested to overcome the problems associated with attempting indoor navigation using a conventional pseudolite system [51]. A pseudolite-based eight-channel single transmitter was configured and tested for indoor applications, achieving a high accuracy [52,53].

3. Multipath Detection Methods

In this section, neural network-based multipath detection methods are derived from the mathematical equations of multipath error and wavelet transform and the methodology of supervised learning.

3.1. Mathematical Model of Multipath Error

The general carrier-phase measurement obtained from the GNSS satellite measurements at a ground receiver in an outdoor environment is modeled as (1) [1]:

$${\varphi }^j=d^j+N^j\lambda +B-b^j+t^j-i^j+m^j+{\epsilon }^j$$

(1)

where $\varphi$ is the measured carrier phase in metric form, d is the distance between the satellite and the user, N is an unknown integer term of cycle ambiguity, λ is the wavelength of the carrier, B is the receiver clock offset, b is the satellite clock offset, t is the tropospheric delay, i is the ionospheric delay, m is the multipath error, ε is other noise, and superscript j represents the measurement made by the j-th satellite.

In this study, a multichannel pseudolite-based single transmitter was utilized for the multipath study. The pseudolite broadcasts the navigation signal at L1 (=1575.42 MHz) frequency in an indoor environment. The wavelength of the carrier was approximately 19.03 cm in this case. Equation (1) can be reduced to (2) considering the pseudolite system in an indoor environment as follows:

$${\varphi }^j=d^j+N^j\lambda +B-b^j+m^j+{\epsilon }^j$$

(2)

where b is the pseudolite clock offset and t and i are zero because tropospheric and ionospheric delays can be ignored. Superscript j represents the measurement obtained by the j-th pseudolite. The multipath error m of the pseudolite carrier-phase measurement in the phase-lock loop can be mathematically modeled as stated in Equation (3). When there are multiple reflected signals, they are denoted with the index l [16,43]:

$$m=\lambda \cdot {\tan }^{-1}\left[\frac{{\displaystyle \sum _l{\alpha }_l{A}_l({\tau }_l)\sin {\theta }_l}}{1+{\displaystyle \sum _l{\alpha }_l{A}_l({\tau }_l)\cos {\theta }_l}}\right]$$

(3)

where $\alpha$ is the damping factor of a reflected signal relative to a direct signal, A is the code correlation function with a time delay of $\tau$, and $\theta$ denotes the phase shift owing to the extra distance from the reflected signal compared to the direct signal. As described in Equation (3), the basis of the mathematical multipath model is composed of trigonometric functions.

For multipath analysis, the time-differenced carrier-phase measurement is applied, which is expressed in Equation (4):

$${\Delta }_{k,k-1}{\varphi }^j\equiv {\varphi }^j(k)-{\varphi }^j(k-1)={\Delta }_{k,k-1}{d}^j+{\Delta }_{k,k-1}B-{\Delta }_{k,k-1}{b}^j+{\Delta }_{k,k-1}{m}^j+{\Delta }_{k,k-1}{\epsilon }^j$$

(4)

where ${\varphi }^j\left(k\right)$ represents the carrier-phase measurement obtained by j-th pseudolite at the k-th epoch. The integer cycle ambiguity term N is eliminated because the value does not change with time if continuous signal tracking is performed. In (4), the time-differenced measurement has a distance rate and clock drift terms for the receiver and pseudolite. The reason for the time difference is to remove the consecutive increase or decrease in the measurements, mainly caused by clock drifts, to prepare for the frequency domain analysis. The time-differenced noise term stated in Equation (4) does not affect the frequency analysis if white noise is considered. Time-differenced multipath errors $∆m$ still exhibit the characteristics of multipath error because each reflected signal has trigonometric functions as their bases, as described in Equation (3). The basis functions for the denominator and numerator inside the arctangent are trigonometric functions, whose derivatives or differences are also trigonometric functions. Therefore, time-differenced measurements are still suitable to be used as inputs for the learning process.

3.2. Wavelet Transform

One-dimensional CWT was performed for the time-differenced carrier-phase measurements of all epochs from beginning to end. In mathematics, CWT provides a complete representation of a signal with the translation and scale parameters of wavelets, which vary continuously [32]. CWT of a function $f\left(t\right)$ is expressed in Equation (5):

$$CWT(p,q)=\frac{1}{\sqrt{p}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(t){\psi }^{*}\left(\frac{t-q}{p}\right)dt}$$

(5)

where $p$ and $q$ are the scale parameter and the translation parameters of wavelets, respectively, and $\psi \left(t\right)$ is a wavelet function. In this study, the function $f\left(t\right)$ in Equation (5) is the time-differenced carrier-phase measurements, and $\psi \left(t\right)$ is a Morse wavelet.

For the CWT, the “cwt” function of MATLAB (version 2022a) software by MathWorks is used. In the “cwt” function, a generalized Morse wavelet is utilized with a symmetry parameter of three meanings that it is symmetric in the frequency domain. The time bandwidth was 60. Voices per octave of 48 were considered; in other words, 48 intermediate scales for each octave were considered.

The transformation was performed individually for each pseudolite. Therefore, five scalograms were generated for each experimental scenario through the wavelet transform because five channel pseudolites were considered during the experiments. Because an entire epoch of measurements of each pseudolite is utilized for the transform, the multipath mitigation method described in this study is based on post-processing analysis.

The preprocessing step used to prepare the dataset for machine learning is shown in Figure 1. A magnitude scalogram was generated from the time-differenced carrier-phase measurement. The X-axis of the scalogram denotes the time epochs for the experimental scenario, and the Y-axis is the scale for frequency conversions in hertz, whose range is from 0.1016 to 4.3412 Hz, divided into 261 steps (or pixels in the image). The magnitude of the transform coefficients for each epoch was applied for the SNN and DNN for transfer learning in a different manner.

At each epoch, 261 × 1 vector of magnitudes were generated. For the SNN, the magnitude values in these vectors for all epochs and pseudolites were used in the learning process.

For DNN, transfer learning with AlexNet is utilized; therefore, an RGB image set of 224 × 224 pixels is required for the learning task. Magnitude values of 261 × 1 at a certain epoch were converted to an image using a jet colormap. Because RGB data are considered, the image has a size of 261 × 1 × 3. This vertical one-column image was horizontally extended to form a square image of 261 × 261 × 3 pixels. After the extension, the image was resized to 224 × 224 × 3. The images obtained from all epochs and pseudolites were utilized as the input of the DNN.

3.3. Supervised Learning

For both the SNN and DNN, a supervised learning technique was applied to learn the networks. All learning and validation processes were carried out on a computer with an Intel i7-12700K processor with 32 GB memory. MATLAB (version 2022a) software with statistics and machine learning toolbox and deep learning toolbox are used for SNN and transfer learning, respectively.

All experiments were conducted by the research group. Three experimental scenarios are considered in this study. In Figure 2, the black square of route A has dimensions of 2.7 m × 2.7 m, and the yellow rectangle of route C has dimensions of 3.9 m × 3.3 m. The scenarios for the test are described as follows:

• Scenario A: The rover follows route A.
• Scenario B: The rover follows route B.
• Scenario C: The rover follows route C.

The numbers of epochs in experimental scenarios A, B, and C were 1219, 1649, and 1258, respectively. The number of datasets for each scenario and the total amount of data used for training and validation are listed in

Table 1. Number of epochs and datasets for training and validation processes.

	Scenario	Number of Epochs	Number of Total Epochs	Number of Data Set
Training data	A	610	2079	10395
	B	840
	C	629
Validation data	A	609	2077	10385
	B	839
	C	629

. In each scenario, 50% of the dataset was used for training, and the rest was used for validation. The training data (50% of the dataset) were selected randomly. Because five-channel pseudolites were considered in this study, five datasets for network learning were generated at each epoch.

Training datasets from all scenarios were utilized to learn the networks. A single model for each SNN and DNN using transfer learning was trained using the whole training data from all scenarios. All scenarios contain reference user positions derived from recorded videos [49]. For supervised learning, reference trajectories are used to calculate the positioning error to determine whether a certain measurement is contaminated by the multipath error. Positioning error using the measurements of all pseudolites and all measurements except one pseudolite can be calculated as stated in Equations (6) and (7), respectively:

$${\overline{R}}_{err}(k)\equiv \Vert {\overline{R}}_u(k)-{\overline{R}}_u^{ref}(k)\Vert$$

(6)

$${\overline{R}}_{err}^{-i}(k)\equiv \Vert {\overline{R}}_u^{-i}(k)-{\overline{R}}_u^{ref}(k)\Vert ,$$

(7)

where ${\stackrel{-}{R}}_u\left(k\right)$ is a least square (LS)-based positioning result at the k-th epoch using all pseudolites, ${\stackrel{-}{R}}_u^{ref}\left(k\right)$ is a reference position at the k-th epoch, and ${\stackrel{-}{R}}_u^{-i}\left(k\right)$ is a LS position except the i-th pseudolite measurement at the k-th epoch. i ranges from one to five because five-channel pseudolites are used in this study. Equations (6) and (7) are compared for all i to label the status of the measurements as $P_{ref}^{-i}\left(k\right)$ in Equation (8):

$$P_{ref}^{-i}(k)=\{\begin{array}{c}\text{<}\mathrm{clean}\text{>}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}:\hspace{0.17em}}{\overline{R}}_{err}(k)\le {\overline{R}}_{err}^{-i}(k)\\ \text{<}\mathrm{multipath}\text{>}:{\overline{R}}_{err}(k)>{\overline{R}}_{err}^{-i}(k)\end{array}$$

(8)

where <clean> indicates that the measurement from the i-th pseudolite is clean, and <multipath> indicates that the measurement from the i-th pseudolite is contaminated, usually by multipath signals. All training datasets calculate the label presented in (8) for all pseudolites at all epochs, the values of which are utilized for supervised learning. During the supervised learning, labels of $P_{SNN}^{-i}\left(k\right)$ and $P_{DNN}^{-i}\left(k\right)$ from SNN and DNN models, respectively, are compared to $P_{ref}^{-i}\left(k\right)$. The learning process is summarized in Figure 3.

During the validation process, two learned networks were tested using the validation dataset. The validation dataset also calculates the label presented in Equation (8); the result is compared to the classification result obtained from the neural networks to evaluate the detection accuracy. A flowchart for the validation process to investigate the detection accuracy is described in Figure 4.

3.4. Shallow Neural Network

The dataset in the form of a 261 × 1 vector, as described in Figure 1, and labeling using Equation (8) are used for the SNN-based supervised learning. In this study, a fully connected SNN including one hidden layer was considered, and hyperparameter tuning was performed empirically with hyperparameter optimizations based on Bayesian statistics. The optimization and learning process were carried out using the “fitcnet” function in the MATLAB software. After the hyperparameter optimization process, 686 layers were derived based on the cross-entropy loss for the cost function and rectified linear units (ReLU) for the activation function. The Softmax function is applied as a binary classifier, and the standardization of the input data is considered to decrease the sensitivity in the scales of the data. The limited-memory Broyden Fletcher Goldfarb Shanno algorithm (LBFGS) is considered a parameter estimation solver. The main hyperparameters applied for the proposed approach are summarized in

Table 2. Hyperparameters for SNN.

Hyperparameters	Value
Layer size	686
Cost function	cross-entropy
Activation function	ReLU
Output classifier	Softmax
Standardization	Applied
Parameter estimation solver	LBFGS
Regularization parameter (lambda)	0
Mini-batch	Not applied

, and the diagram for the SNN is shown in Figure 5. The computing time of the SNN-based learning with the hyperparameters as in Table 2 was 53 s.

3.5. Deep Neural Network

In this study, AlexNet, a deep CNN, was used for transfer learning to detect multipath contaminated measurements. The original AlexNet consists of eight main layers: five convolutional layers and three fully connected layers with a 1000-way Softmax function. To apply AlexNet to multipath detection, the number of nodes in the last fully connected layer is reduced to two as a binary classification, and these two values are the inputs for the Softmax function. A diagram of the DNN using transfer learning is presented in Figure 6.

Because AlexNet is used as the basis architecture in this study, most of the hyperparameters are derived from the original AlexNet: ReLU for the activation function, dropout, overlapping pooling, and local response normalization. For the training process, some parameters were empirically set after the iterations. The mini-batch size used for each training iteration was 128 and the maximum number of epochs used for training was 50. The initial learning rate was set to 0.0001. As an optimizer, a stochastic gradient descent with a momentum algorithm was applied. The computing time using the “trainNetwork” function of MATLAB for the transfer learning process was 4247 s.

4. Results Analysis

The results are described in terms of detection accuracy for multipath contaminated signals and improved positioning performance. In this section, the experimental settings, detection accuracy, and positioning results are presented for both SNN and transfer learning-based DNN.

4.1. Experiment Configuration

Experiments with a multi-channel single-transmitter and rover were conducted in an office environment, as depicted in Figure 7. The test site is located on the 5th floor of the building of the Institute of Advanced Machined and Design at Seoul National University. An antenna array was installed on the ceiling of the room. The user was a mobile rover on the floor, whose speed was approximately 15 cm/s. A Tallysman VP6000 antenna and GPS receiver developed by Telace Inc. were used for the rover. The same receiver was used as a reference. From the GPS receiver, 10 Hz measurements were generated during the test, and carrier-phase measurements from each receiver were collected and used for the indoor multipath study.

The test environment was not cleaned for radio propagation because the structure of the building was a steel frame and the main material of the wall was concrete. In addition, there are various objects in the room, such as bookshelves, tables, drawers, metal window blinds, and other metal structures, as shown in Figure 7a.

As shown in Figure 7b, the pseudolite antenna array had a pentagonal shape. The distance between the antennas was 28.5 cm = 1.5 λ of the L1 carrier signal (=1575.42 MHz). Five pseudo random noises (PRNs) from one to five GPS signals were set for each pseudolite.

The concept of a single transmitter-based positioning system considers a time-synchronized multi-channel transmitter. However, the pseudolites used in this study were not fully synchronized. Instead, a reference receiver was used to eliminate the clock and channel biases between transmitters [49].

SNN- and DNN-based algorithms for detecting multipath contaminated signals are used for carrier-phase measurements from the user receiver. The algorithms were not applied to the reference receiver because the geometry of the signal paths from the transmitters to the static reference receiver did not vary sufficiently during the experiments.

Three experimental scenarios, described in Section 3.3, were considered in this study. Each scenario adopts different routes, as shown in Figure 2. Scenario A was set as the ordinary square trajectory. Scenario B displays the positioning results below the transmitter. Scenario C was additionally considered to investigate the positioning performance near the surrounding objects. It is expected that scenario C has a poor DOP and large multipath errors compared with scenarios A or B. In the figure, the origin of the local coordinates and the position of the reference receiver antenna are also displayed.

4.2. Multiapth Detection Accuracy

All carrier-phase measurements at each epoch for all scenarios taken from the user receiver are labeled as <multipath> or <clean> based on Equation (8). Half of them were utilized for the learning process and the rest were used for the validation process to assess detection accuracy. The confusion matrices of the multipath detection results obtained using the SNN- and DNN-based approaches are described in Figure 8 and Figure 9, respectively. It is noted that a single model of SNN, which was trained using the whole training dataset from all scenarios, was utilized to evaluate detection accuracy for each scenario and for the whole scenario, and its results are depicted in Figure 8. The results of a single DNN model using transfer learning are also described in Figure 9.

The row of the inner confusion matrix indicates the predicted results through the neural networks, and the column of the matrix represents the actual labels. The numbers in the main matrix represent the number of samples and their percentages in each row and column.

The total number of validation datasets used was 10,385. Among them, 65% are labeled as <clean>, and the remaining 35% are categorized as <multipath> using Equation (8). This indicates that the indoor environment considered in this study exhibited severe multipath effects during the experiments. The overall detection accuracies of SNN and transfer learning-based DNN were 87.1% and 85.5%, respectively. The detection accuracies of these scenarios are summarized in

Table 3. Multipath detection accuracy for SNN and transfer learning-based DNN.

Scenarios	Detection Accuracy
	SNN	Transfer Learning-Based DNN
Scenario A	87.9%	86.9%
Scenario B	88.4%	87.2%
Scenario C	84.5%	82.0%
All scenarios	87.1%	85.5%

. According to the results, the SNN-based detection algorithm attains a slightly better performance for each scenario than the DNN-based algorithm.

Scenarios A and B, which expect low multipath errors, have better detection accuracy than scenario C, which expects large multipath errors. However, both algorithms achieved more than 80% detection accuracy, even for scenario C.

4.3. Positioning Performance

Carrier-phase measurements, which are labeled <clean> through SNN- or DNN-based algorithms, are utilized for positioning. For two-dimensional positioning, at least four carrier-phase measurements were used with one redundancy measurement to reduce the effect of the undetected contaminated signal. If more than two measurements are classified as <multipath>, the one with the lowest CN0 is eliminated for positioning. The LS algorithm is used for positioning; a detailed algorithm with mathematical formation is described in this study [49]. During the positioning process, it was assumed that the cycle ambiguity problem of carrier-phase measurements was resolved. A flowchart for the positioning process is summarized in Figure 10.

Figure 11 and Figure 12 describe the positioning results for the three scenarios using SNN- and DNN-based detection algorithms, together with the positioning results using all measurements without the detection process. The trajectories depicted in these two figures include positioning results from only the validation dataset; positioning results from the training set are eliminated.

In Figure 11 and Figure 12, most of the neural network-based positioning methods have similar trajectories as the results without detection algorithms, because the SNN and transfer learning-based DNN detection algorithms predict that 64.6% and 66.9% of the signals are <clean> and there is no need for modifications, as described in Figure 8 and Figure 9. In particular, scenario B attains the highest rate of <clean> signals; therefore, Figure 11b and Figure 12b depict almost the same trajectories compared to the results obtained using all measurements.

There are multiple positioning results that decrease the positioning error; the blue circle is closer to the reference trajectory compared to the red dot, which is the expected result when the neural network-based detection algorithms function appropriately. However, there are points that increase the positioning error using the detection algorithm, because neural network-based algorithms sometimes mis-detect <multipath> signals or false alarm <clean> signals; 12.9% and 14.5% are incorrectly detected portions for SNN and DNN, respectively.

Although there are misdetection and false-alarm cases, the overall positioning performance is improved when SNN- or DNN-based multipath detection algorithms are applied, as reported in

Table 4. Positioning error of validation epochs for each scenario.

Scenarios	2D Positioning Error (RMS)
	w/o Detection Algorithms	w/SNN	w/DNN
Scenario A	0.197 m	0.177 m	0.180 m
Scenario B	0.143 m	0.125 m	0.126 m
Scenario C	0.225 m	0.206 m	0.208 m

. Approximately 10% of the positioning errors in the root mean square (RMS) are decreased.

When SNN and DNN are compared, their positioning performance is quite similar at the millimeter level, even if the detection accuracy of SNN is a few percent better than that of DNN, as reported in Table 3. A small percentage variation in detection accuracy does not make a significant difference in the position domain.

5. Discussion

When SNN- and DNN-based detection algorithms for multipath contaminated signals were applied, both algorithms achieved a detection accuracy of more than 85%. However, in the position domain, there was no significant improvement in the trajectories. Statistically, the positioning errors were reduced by 10%. The reasons for this are not only misdetection and false alarms of the detections but also multiple detections in one epoch. According to the positioning algorithm applied in this study, when multiple measurements are predicted as <multipath>, only one of them is eliminated from the positioning process. The rest of them are still used for positioning, even though they are considered <multipath> signals.

Figure 13 shows a histogram of the number of <multipath> signals at each epoch. The histogram is constructed using Equation (8), not the predicted results. The total number of epochs for the validation process of scenarios A, B, and C was 2077. In the histogram, 66.7% of the epochs had more than two <multipath> labeled signals. Therefore, the high detection accuracy of multipath contaminated signals has limitations in improving the position performance.

According to Figure 13, only 12.7% of the epochs had all <clean> measurements, meaning that 87.3% of the epochs had at least one contaminated signal. In addition, 3.6% had all signals categorized as <multipath>. These results also indicate that the indoor environment causes severe multipath errors.

With respect to the detection accuracy, SNN exhibits better performance compared to transfer learning-based DNN, even though the DNN utilizes a multi-layer deep network. One of the reasons for this is that AlexNet is optimized to categorize images; hence, the magnitude scalogram at each epoch is enlarged and resized to a 224 × 224 RGB image to fit AlexNet. During these processes, some information in the scalogram may be lost and decrease the detection accuracy because the resolution of the image is reduced and the value of the magnitude scalogram is transferred to the RGB color using a certain color map.

In terms of efficiency, SNN exhibits better performance by utilizing a single-layer network with an optimization process. Transfer learning-based DNN achieves a similar performance improvement in the position domain; however, it requires heavier computational loads owing to having multiple convolutional layers and fully connected layers. However, this study considers only three operational scenarios for the learning and validation processes. There is a possibility that DNN has more potential if various scenarios with more datasets are considered for practical application. This topic remains to be addressed in future studies when large datasets are acquired.

In a previous study [49], low-cost inertial measurement units (IMUs) of accelerometers and gyroscopes were utilized and combined with Kalman filtering to improve positioning accuracy and mitigate the multipath effect. Positioning trajectories obtained with IMUs depicted a more aligned line with less cursivity, and their two-dimensional accuracy in RMS was also improved. However, additional sensors are required to increase the system’s complexity. However, the results of this study can be achieved without additional sensors or equipment, applying only its own measurements for the neural network process.

One of the limitations of this study is that the wavelet transform used for neural network-based multipath detection cannot be utilized for real-time processing. To apply this study to real-time operation, instead of the wavelet transform, a real-time representation method to characterize the multipath errors in signals is required. One of the candidates is the short-time Fourier transform.

Another interesting application of this study is GNSS-based outdoor positioning. Both pseudorange and carrier-phase measurements in an urban canyon can be processed via a wavelet transform and neural network-based approach to mitigate the multipath effect.