An NLOS Ranging Error Mitigation Method for 5G Positioning in Indoor Environments

Abstract

Positioning based on wireless signals such as mobile communication networks has become an important means to provide high-precision location services in environments where satellite signals are blocked. In complex environments such as indoors and underground, wireless signal propagation is obstructed and non-line-of-sight (NLOS) phenomena appear due to serious occlusion and reflection. The time delay caused by NLOS effects has little impact on communication system but can significantly increase positioning errors in positioning systems. Therefore, the effective suppression of NLOS errors is crucial to improving 5G positioning accuracy. To address the insufficient feature extraction of existing NLOS error suppression methods, the neglect of residual NLOS measurement errors, and poor stability of position estimation results, this paper innovatively proposes an NLOS mitigation and location estimation method for 5G positioning terminals. Simulation and experimental test results demonstrate that the proposed method outperforms the comparative methods both theoretically and practically, achieving an average positioning accuracy of 1.85 m in complex indoor NLOS environments. The method proposed in this paper provides a new strategy for NLOS error suppression in indoor 5G positioning, which can significantly contribute to high-precision location services based on commercial 5G networks.

1. Introduction

Temporal and spatial information is essential to human existence and development. Location-based services (LBS) utilize positioning and navigation technologies to provide users with a series of services based on temporal and spatial locations, playing an increasingly important role in various emerging fields such as smart cities, intelligent transportation, and industrial internet. Ubiquitous and trustworthy location services require the support of advanced positioning technologies. In outdoor environments, GNSSs can provide high-precision location information. However, due to the low penetrability of satellite signals, it is not possible to offer users high-precision location services in non-exposed spaces such as indoor and underground environments [1,2]. Currently, positioning technologies in satellite-denied environments worldwide include Bluetooth, UWB, WLAN, and mobile communication networks [3]. Among these, mobile communication networks are the most widely used wireless networks for ground coverage. In particular, 5G networks, which can provide high-precision location services based on mobile communication networks, have become a research hotspot [4,5,6,7,8]. The primary technical bottleneck in improving 5G indoor positioning accuracy is the NLOS effect [9]. In complex environments such as indoors and underground, wireless signal propagation is severely obstructed and reflected, resulting in NLOS phenomena [10]. Since communication services in mobile communication networks are insensitive to delays, NLOS phenomena have a minor impact on communication quality. However, for positioning, NLOS phenomena can lead to increased positioning observation errors. The errors caused by NLOS phenomena may be much larger than the errors introduced by the ranging capability itself. Therefore, researching NLOS error suppression techniques in mobile communication networks is of great significance for achieving high-precision positioning.

The process of NLOS effect suppression mainly consists of two parts. Firstly, the sources in NLOS states are identified, and then the influence of NLOS effects on the positioning results is suppressed through filtering or compensation methods. A common method for NLOS identification is to use residual magnitudes to judge the accuracy of signal measurements after positioning. Observations with large residuals are identified as being in NLOS states. Steve Bartelmaos et al. found that observations with larger residuals mostly come from signals in NLOS channel states, and they judged whether NLOS observations are used in the current positioning based on the mean residual [11]. Abbas Al-baidhani et al. studied Time-of-Flight (TOF) positioning through first-group observations from different base stations and then performed initial positioning separately. They calculated the residual between the positioning result and the TOF ranging result to identify TOF measurements with large residuals as being in NLOS states [12]. Shixun Wu et al. used a similar method to judge the LOS and NLOS states of Time-Difference-of-Arrival (TDOA) observations [13]. These methods can effectively identify observations in NLOS states, but they require a high redundancy of observation information and cannot be used when there are few observation data points. Wireless signal statistical characteristics can be utilized to identify channel states and thus recognize NLOS effects. For instance, Jie Zhang et al. utilized the kurtosis of signal amplitudes to differentiate between Line-of-Sight (LOS) and NLOS channel states. Results from experiments involving various systems indicate that signals in NLOS states exhibit significantly smaller kurtosis values compared to those in LOS states [14]. Kegen Yu and Y. Jay Guo analyzed the differences in the standard deviation of the angle of arrival (AOA) between LOS and NLOS states. They proposed NLOS identification methods based on the AOA and a joint method based on the Time of Arrival (TOA) and signal strength, which are applicable under different equipment conditions [15]. Chen Huang et al. employed machine learning techniques to combine various signal features such as maximum received power, power kurtosis, power skewness, root mean square delay spread, Ricean K-factor, and angle-of-arrival spread. This approach enhances the dimensionality of information and improves NLOS identification effectiveness [16].

The current mainstream NLOS suppression methods can be categorized into error compensation-based suppression methods and filtering-based suppression methods. Error compensation-based methods involve compensating for measurements from NLOS states and using corrected measurements for positioning. Following a similar approach, Hongji Cao et al. utilized statistical differences in signal delays between LOS and NLOS channel states to identify channel states. They corrected temporary NLOS observations using measurements from LOS states, thereby effectively improving positioning accuracy [17]. Based on filtering-based suppression methods, once the NLOS states of signal sources are identified, they are treated differently from LOS sources. This involves either discarding observations from NLOS states directly or suppressing their influence on positioning results through weighting. Minghao Si et al. utilized the statistical differences in received signal strength characteristics between LOS and NLOS wireless channel states to filter observations. They discarded measurements from NLOS states and selected TOF measurements from LOS states for positioning [18]. Abdo Gaber and Abbas Omar, after identifying NLOS states, assigned smaller weights to NLOS TOA and DOA measurements using weighted least squares to mitigate the impact of NLOS on the positioning results [19]. Kegen Yu et al. considered the impact of filtering results on the geometric configuration while conducting NLOS source filtering. They selected NLOS sources that contributed significantly to the geometric accuracy and assigned smaller weights to them. This approach suppresses NLOS effects while preventing a significant degradation in geometric accuracy [20]. The existing NLOS suppression technology is mainly divided into three categories, as shown in

Table 1. Methods dealing with the NLOS error.

Category	Advantege	Shortcoming	Related Methods
Detection	The method is simple and easy to implement and can detect observations with obvious NLOS errors.	Redundant observations are required. And may cause an unnecessary loss of accuracy.	Consistency check [11,12,21], NLOS identification techniques [13,14,17,22,22]
Mitigation	Relatively low requirement of hardware equipment, and it can achieve high-precision positioning under NLOS error in a real environment.	Poor adaptability to noise and interference; the model is not general enough, and there is currently a lack of comprehensive theoretical analyses.	Machine learning techniques [16,23,24], support vector machines [15,18,25], vector tracking loop [26,27]
Correction	Theoretically, the impact of NLOS errors can be completely eliminated.	It requires a known high-precision map, which has high computational complexity and is difficult to deploy on commercial terminals.	Virtual station method based on ray tracing [28,29], NLOS modeling [19,30,31]

.

The first category is detection, which involves initially identifying the observations of positioning signals. Observations that may be affected by NLOS are either directly removed or minimized in terms of their weight. The advantage of this approach is its simplicity, while its disadvantage is the lack of classification. Ref. [21] proposed the consistency checking method to examine the residual range measurement or navigation solution, which is similar to receiver autonomous integrity monitoring.

The second category is mitigation, which uses multiple features to classify the received positioning signals, and involves performing error suppression based on the observation feature classification results. Technologies such as machine learning and support vector machines have been widely used to classify and perform the feature extraction of GNSSs and wireless communication network observations [23,24,25,26]. Ref. [27] proposed the reception of NLOS signals based on vector tracking loop and initially proved the effectiveness of the proposed method through field tests. However, there is currently a lack of comprehensive theoretical derivation and performance limit analyses for this type of method.

The third type is correction. In this type of technique, NLOS paths are considered useful signals. In utilizing known three-dimensional maps for assisted positioning, potential reflection paths are inferred, and error correction is performed accordingly [28,29,30,31]. This is equivalent to obtaining a new observation quantity that is not affected by non-line-of-sight errors. This type of method can ideally remove the impact of NLOS errors on positioning and can even bring additional improvements in performance. However, the computational complexity is very high, and it is difficult to meet real-time requirements. In addition, its deployment on hardware platform is also a significant challenge.

The existing NLOS mitigation methods suffer from an insufficient feature extraction of wireless signals, lack consideration of residual NLOS measurement errors during the position estimation process, and poor stability in the mixed positioning estimation results [32,33,34]. To address these issues, this study innovatively proposed an NLOS error identification, suppression, and position estimation method for 5G positioning terminals. The primary contributions of this study are outlined as follows:

(1) A method for discriminating and classifying NLOS observations based on decoupling the features of wireless signals is proposed. This method initially divides the indoor propagation channels into multiple categories and utilizes the results of channel classification to assess the severity of the NLOS effects. Subsequently, it establishes selection criteria for filtering the ranging results of 5G positioning signals.

(2) A position estimation method based on a square root unscented Kalman filter (SRUKF) is proposed. The square root of the state covariance is introduced, thereby avoiding the need for covariance reconstruction at each iteration. This approach offers advantages such as numerical stability and positive semi-definiteness of the state covariance. During the fusion process of mixed observations, it ensures the numerical stability of the filter’s position estimation output.

(3) We performed a comprehensive comparison of the existing advanced methods and built a 5G localization system to support the validation of the proposed method in real complex indoor environments. Both the simulation and experimental results demonstrate the superiority of the proposed NLOS mitigation and location estimation method over existing methods.

The remainder of this paper is organized as follows. In Section 2, the 5G positioning reference signal measurement model is presented, and related problems are described. In Section 3, an NLOS mitigation and location estimation method for 5G positioning is proposed, which is composed of the derivation of positioning performance limits, measurement identification, and a location estimation method for NLOS mitigation. In Section 4, the performance of the proposed NLOS error mitigation algorithm is evaluated based on simulations as well as experimental tests. Finally, the discussion of this article is presented in Section 5.

2. Problem Description and System Model

This section introduces the 5G NR time–frequency resource mapping scheme. The 5G positioning reference signal (PRS) has an abundance of time–frequency resources, and it is expected to exhibit a better performance. The pseudorandom code of the PRS signal is generated using gold sequences, and OFDM modulation is employed to achieve reliable data transmission in complex scenarios with a scalable OFDM numerology. The physical signal’s transmissions are organized into signal frames, subframes, and slots in the time domain. Each signal frame has a duration of 10 ms and consists of 10 subframes with a subframe duration of 1 ms. A subframe is formed by one or multiple slots. The resource element consists of one subcarrier in one OFDM symbol, which is the smallest physical time–frequency resource unit. The physical resource block is defined as 12 continuous subcarriers in one time slot.

In this study, factors such as the positioning accuracy and resource occupancy rate were comprehensively considered. The parameter of 30 KHz subcarrier spacing was used in our experiments, and thus, one subframe had two slots, and each slot had 14 OFDM symbols. Comb-4 was adopted for resource mapping. This study took into account a wide range of parameters, including the resource occupancy rate and placement accuracy. In our experiments, we selected a 30 kHz subcarrier spacing setting, which resulted in two slots per subframe, with 14 OFDM symbols in each slot. Comb-4 was utilized for the mapping of resources.

In this section, we analyze the 5G PRS as an example. After appropriate adjustments, the proposed method can also be applied to the suppression of NLOS errors in other wireless signals. The baseband received signal at the terminal for 5G positioning can be represented as follows:

$$\begin{array}{cc}\hfill r\left(t\right)=& \sum _{m=1,m\ne n}^M\left(A^m{C}^m\left(t-{\tau }^m\right)cos\left(2\pi \left(f_{IF}^m+f_d^m\right)t+{\phi }^m\right)\right)\hfill \\ \hfill & +{\alpha }^{\mathrm{NLOS}}{A}^n{C}^n\left(i-{\tau }^{\mathrm{LOS},n}-{\tau }^{\mathrm{NLOS}}\right)\hfill \\ \hfill & \times cos\left(2\pi \left(f_{IF}^n+f_d^n\right)t+{\phi }^n\right)+\omega \left(t\right)\hfill \end{array}$$

(1)

where $r\left(t\right)$ represents the 5G signal sampled at time t, m denotes the identifier of the positioning base station in the network, M represents the total number of positioning base stations. A indicates the amplitude value of the positioning signal, C represents the ranging code sequence, ${\tau }^m$ denotes the delay of the ranging code for a base station, $f_{IF}$ and $\phi$ denote the frequency and initial phase of the local carrier, respectively, $f_d$ represents the Doppler frequency shift, n indicates the identifier of the NLOS base station, ${\alpha }^{\mathrm{NLOS}}$ represents the reflection coefficient of the NLOS signal, ${\tau }^{\mathrm{NLOS}}$ denotes the delay caused by the NLOS effect relative to the direct path signal, and $\omega \left(t\right)$ represents additive Gaussian white noise.

Achieving THE high-precision positioning of 5G PRS signals requires continuous signal tracking at the terminal side. The impact of NLOS effects on the code tracking loop is far greater than that on the carrier tracking loop. In assuming that the terminal can completely strip the carrier from the received signal, the outputs of the early, prompt, and L correlators can be represented as follows:

$$E=A^m·R\left({\tau }_e^m+\frac{d}{2}\right)+{\alpha }^{\mathrm{NLOS}}{A}^n·R\left({\tau }_e^{\mathrm{NLOS}}+\frac{d}{2}\right)$$

(2)

$$P=A^m·R\left({\tau }_e^m\right)+{\alpha }^{\mathrm{NLOS}}{A}^n·R\left({\tau }_e^{\mathrm{NLOS}}\right)$$

(3)

$$L=A^m·R\left({\tau }_e^m-\frac{d}{2}\right)+{\alpha }^{\mathrm{NLOS}}{A}^n·R\left({\tau }_e^{\mathrm{NLOS}}-\frac{d}{2}\right)$$

(4)

where R represents the autocorrelation function, d represents the spacing between the early and late correlators, ${\tau }_e^m={\widehat{\tau }}^m-{\tau }^m$ represents the code phase difference of the direct path signal, ${\tau }_e^{NLOS}={\widehat{\tau }}^{NLOS}-\left({\tau }^{LOS,n}+{\tau }^{NLOS}\right)$ is the code phase difference of the NLOS signal, and ${\widehat{\tau }}^m$ represents the estimated values of the ranging results. Here, we mainly consider the scenario of 2D positioning, but it can be extended to 3D positioning as needed. The estimated values of the ranging results at the terminal can be modeled as follows:

$${\widehat{d}}_m=\left\{\begin{array}{cc}{d}_m+{\omega }_m,\hfill & \mathrm{for}\mathrm{LOS}\mathrm{case}\hfill \\ d_m+{\omega }_m+{\eta }_m,\hfill & \mathrm{for}\mathrm{NLOS}\mathrm{case}\hfill \end{array},m=1,2,\cdots ,M\right.$$

(5)

$$d_m=\sqrt{{\left(x_m-x\right)}^2+{\left(y_m-y\right)}^2}$$

(6)

where $\left(x_m,y_m\right)$ and $(x,y)$ represent the coordinates of the 5G base station and the terminal, ${\widehat{d}}_m$ represents the estimated distance from the mth base station to the terminal, ${\omega }_m$ denotes zero-mean Gaussian noise, and ${\eta }_m$ is an NLOS error term. The ranging results from the M base stations are divided into two categories, ${\Phi }_{LOS}$ and ${\Phi }_{NLOS}$, representing line-of-sight and non-line-of-sight phenomena, respectively.

$${\Phi }_{LOS}=\left\{\left(x_m,y_m,{\widehat{d}}_m\right):{\widehat{l}}_m=1\right\},{\Phi }_{NLOS}=\left\{\left(x_m,y_m,{\widehat{d}}_m\right):{\widehat{l}}_m=-1\right\}$$

(7)

where ${\widehat{l}}_m$ is the indicator of the channel state, 1 represents the LOS channel, and −1 represents the NLOS channel. In an indoor environment, the terminal often needs to provide position estimation results in the presence of a mixture of LOS and NLOS ranging results. The optimal set of ranging results available for the user terminal to choose from can be denoted as

$$\Phi ={\Phi }_{LOS}\cup {\Phi }_{NLOS}^{\prime }\mathrm{and}{\Phi }_{NLOS}^{\prime }\subseteq {\Phi }_{NLOS}$$

(8)

Afterward, various methods can be used to provide position estimation values based on the ranging results. However, existing methods do not consider the incomplete identification of NLOS observations or incomplete suppression of NLOS errors. These residual NLOS errors can lead to a significant increase in positioning errors for 5G positioning.

3. The Proposed NLOS Mitigation and SRUKF-Based Location Estimation Method for 5G Positioning

In this section, an NLOS mitigation and SRUKF-based location estimation method (NM-SLE) for 5G positioning terminals is proposed. Firstly, the performance limits of the TOA estimation in indoor environments with mixed line-of-sight phenomena and NLOS channels is derived. Based on this, a strategy for filtering NLOS observations is proposed, and a position estimation method for NLOS ranging errors based on a SRUKF is introduced. The diagram of the proposed method is illustrated in Figure 1.

3.1. Ziv-Zakai Bound on Time-of-Arrival Estimation with Statistical NLOS Channel Knowledge at the Receiver

In this section, the performance limits of TOA estimation in NLOS channels are analyzed. Analytical expressions for the Cramér–Rao Bound (CRB) and Ziv-Zakai Bound (ZZB) of the root mean square error under different NLOS conditions were derived. These performance limits can serve as effective benchmarks for designing practical TOA estimators. The 5G positioning terminal obtains an estimation of the TOA, denoted as ${\widehat{\tau }}^m$, from the received signal r(t) using Equation (1). We assume that $\tau$ is unknown and randomly distributed within the sampling time interval and that all NLOS components are observed within this sampling interval. The proposed performance boundary based on the ZZB can be derived from the following general equation for a root mean square error estimation:

$$\mathrm{E}\left\{{\xi }^2\right\}=\frac{1}{2}{\int }_0^{\infty }\phi ·\mathrm{P}\left\{\left|\xi \right|\ge \frac{\phi }{2}\right\}d\phi$$

(9)

where $\xi ={\widehat{\tau }}^m-{\tau }^m$ represents the estimation error of the ranging results in the positioning terminal. The core idea of the ZZB is to transform the performance evaluation of an estimation problem into a binary detection problem with equal probability hypotheses:

$$\begin{array}{cc}\hfill & H_1:r\left(t\right)\sim \mathbf{p}\{r\left(t\right)\mid \tau \}\hfill \\ \hfill & H_2:r\left(t\right)\sim \mathbf{p}\{r\left(t\right)\mid \tau +\phi \}\hfill \end{array}$$

(10)

where $\mathrm{p}\{·\}$ represents probability distribution below, and it can be used to obtain the error probability corresponding to the optimal decision rule based on a likelihood ratio test, further derived from Formula (9), to obtain its performance lower bound. First, the value of ${\tau }^m$ is estimated, and then a decision is made between the two hypotheses based on the minimum distance criterion to obtain the performance lower bound by calculating the error probability of the optimal decision rule that minimizes the error probability.

$${\rm Y}\left(r\left(i\right)\right)=\frac{\mathbf{p}\left\{r\right(t)\mid \tau \}}{\mathbf{p}\left\{r\right(t)\mid \tau +\phi \}}\left\{\begin{array}{c}{H}_1:>1\hfill \\ H_2:<1\hfill \end{array}\right.$$

(11)

When ${\tau }^m$ is uniformly distributed within the sampling interval T, then ZZB can be expressed as

$$ZZB=\frac{1}{T}{\int }_0^T\phi (t-\phi )P_{min}\left(\phi \right)d\phi$$

(12)

From the analysis above, it can be seen that the derivation of the ZZB for NLOS channels has been transformed into the performance evaluation of a binary detection problem. Deriving the ZZB requires evaluating the probability error function $P_{min}\left(\phi \right)$ for deciding between the two hypotheses in Equation (11). Under the condition where the channel state information is known, the ideal received signal r(t) can be obtained along with the corresponding channel parameters $\mathit{c}={\left[c_1,c_2,\dots ,c_L\right]}^T$, and likelihood ratio test results can be computed. This process can be equivalently interpreted as evaluating the corresponding error probability $P_{min}(\phi ,\mathit{c})$ for a binary discrimination system.

$$\begin{array}{cc}\hfill & H_1:r(t-\tau ,\mathit{c})\hfill \\ \hfill & H_2:r(t-\tau -\phi ,\mathit{c})\hfill \end{array}$$

(13)

where $\mathit{c}$ is known. Then, the error probability can be expressed as

$$P_{min}(\phi ,\mathit{c})=Q\left(\sqrt{{\kappa }_{\mathrm{snr}}\left({\rho }_r(0,\mathit{c})-{\rho }_r(\phi ,\mathit{c})\right)}\right)$$

(14)

where ${\kappa }_{\mathrm{snr}}=E_{\mathrm{p}}/N_0$ denotes the signal-to-noise ratio, $E_{\mathrm{p}}$ is the average received energy, and Q denotes the Gaussian Q-function. The normalized autocorrelation function is defined as follows:

$$\begin{array}{ccc}\hfill {\rho }_r(\phi ,\mathit{c})& =\frac{1}{E_{\mathrm{p}}}{\int }_{-\infty }^{\infty }r(t-\tau ,\mathit{c})r(t-\tau -\phi ,\mathit{c})dt\hfill & \hfill ={\mathit{c}}^T\mathit{R}\left(\phi \right)\mathit{c}\end{array}$$

(15)

where

$$\mathit{R}\left(\phi \right)\triangleq \left(\left[\begin{array}{cc}cos\left(2\pi f_c\phi \right)\hfill & sin\left(2\pi f_c\phi \right)\hfill \\ -sin\left(2\pi f_c\phi \right)\hfill & cos\left(2\pi f_c\phi \right)\hfill \end{array}\right]\otimes \mathit{R}\left(\phi \right)\right)$$

(16)

By replacing $P_{min}\left(\phi \right)$ with $P_{min}(\phi ,\mathit{c})$ in Equation (12), we can obtain the ZZB under the condition of channel state $\mathit{c}$:

$${\left.ZZB\right|}_c=\frac{1}{T}{\int }_0^T\phi (T-\phi )P_{min}(\phi ,\mathit{c})d\phi$$

(17)

On the basis of Equation (17), in further deriving the idea of averaging on the vector c, the unconditionally restricted ZZB can be obtained:

$$ZZB={\left.\int f_{\mathit{c}}\left(\mathit{c}\right)ZZB\right|}_{\mathit{c}}d\mathit{c}=\frac{1}{T}{\int }_0^T\phi (T-\phi ){\overline{P}}_{min}\left(\phi \right)d\phi$$

(18)

The average error probability function can be expressed as

$${\overline{P}}_{\min }\left(\phi \right)=\int f_{\mathit{c}}\left(\mathit{c}\right)P_{min}(z,\mathit{c})d\mathit{c}$$

(19)

It should be noted that the premise of obtaining Equation (18) is that the positioning terminal has accurate channel state information. Therefore, when the positioning terminal receives signals completely synchronized, the ZZB can serve as a lower bound for the root mean square error of the TOA estimation. Furthermore, Equation (19) represents the average bit error probability using receiver coherent detection with perfect channel state information in the presence of NLOS and multiple paths. Many scholars have studied the coupling relationship between BEP and ZZB under different conditions [35]. Here, we adopted a method similar to that in [36] to solve for ${\overline{P}}_{min}\left(\phi \right)$:

$$Q\left(x\right)=\frac{1}{\pi }{\int }_0^{\pi /2}exp\left\{-\frac{x^2}{2{sin}^2\left(\theta \right)}\right\}d\theta$$

(20)

Substituting Equation (14) and Equation (20) into Equation (19) can yield the expression for the error probability function as follows:

$$\begin{array}{cc}\hfill {\overline{P}}_{min}\left(\phi \right)& ={\mathrm{E}}_{\mathbf{c}}\left\{P_{min}(\phi ,\mathit{c})\right\}\hfill \\ \hfill & =\frac{1}{\pi }{\int }_0^{\pi /2}{E}_{\mathbf{c}}\left\{exp\left\{-\frac{{\kappa }_{\mathrm{snr}}{\mathbf{c}}^T\left[{\mathit{R}}_2\left(0\right)-{\mathit{R}}_2\left(\phi \right)\right]\mathbf{c}}{2{sin}^2\left(\theta \right)}\right\}\right\}d\theta \hfill \\ \hfill & =\frac{1}{\pi }{\int }_0^{\pi /2}{\Phi }_{\rho }\left(-\frac{{\kappa }_{\mathrm{snr}}}{2{sin}^2\left(\theta \right)}\right)d\theta \hfill \end{array}$$

(21)

where ${\Phi }_{\rho }$ represents the moment generating function (MGF). The specific derivation process can be found in [37]. Here, we only need to evaluate an integral with a finite limit. An effective method for evaluating the integral in the form of Equation (21) can be found in [38]. Therefore, under the condition of known accurate channel state information in an additive white gaussian noise (AWGN) channel, the performance ZZB for wireless localization can be expressed as follows:

$$ZZB=\frac{1}{T}{\int }_0^T\phi (T-\phi )Q\left(\sqrt{{\kappa }_{\mathrm{snr}}\left(1-{\rho }_{\mathrm{p}}\left(\phi \right)\right)}\right)d\phi$$

(22)

3.2. 5G NLOS Measurement Identification and Classification Method

When the available LOS signals are not sufficient to support localization, it becomes necessary to design a criterion for filtering ranging results to ensure that only beneficial NLOS signals are included in the positioning computation. The NLOS error at the 5G positioning terminal can be represented as

$${\mathrm{Error}}_{NLOS}=\frac{10\left(\sqrt{{\vartheta }_{nlos}}-1\right)}{{ϵ}_{nlos}}{log}_{10}\left(\frac{{max}_{1<i\le L}\left(a_i^2{\tau }_i^n\right)}{a_{DP}^2{\tau }_{DP}^n}\right)$$

(23)

where ${\vartheta }_{\mathrm{nlos}}$ represents the dielectric constant associated with obstacles in the propagation channel, and ${ϵ}_{\mathrm{nlos}}$ represents the attenuation constant associated with obstructing objects in the channel. L represents the number of paths for a multipath implementation; $a_i$ and ${\tau }_i$, respectively, represent the amplitude and delay; and the subscript DP stands for the direct path signal. It can be seen that the errors introduced by NLOS effects are closely related to the channel conditions. The key to selecting NLOS observations lies in extracting typical classification features from the waveform of the received signal. The extracted features must be strongly correlated with changes in the NLOS channel state.

To ensure the ease of implementation of the proposed method, we selected six common features as well as estimations from the 5G signal: Total Energy (TE), Maximum Peak Value (MPV), signal-to-noise ratio (SNR), Rise Time (RT), root mean square delay spread (RDS), and kurtosis (KUR). Their definitions are shown in Equations (24)–(29), respectively:

$$\zeta =\sum _{i=1}^N{\left|r\left(t_i\right)\right|}^2$$

(24)

$$A_{max}=max\left\{\left|r\left(t_i\right)\right|\right\}$$

(25)

$${\kappa }_{snr}=10{log}_{10}\left(\frac{A_{max}^2}{2{\sigma }_n^2}\right)$$

(26)

$$\begin{array}{c}{\tau }_{\mathrm{rise}}={\tau }_{\mathrm{stop}}-{\tau }_{\mathrm{start}}\\ \left\{\begin{array}{c}{\tau }_{\mathrm{start}}=min\left\{t_i:\left|r\left(t_i\right)\right|\ge 0.1A_{\max }\right\}\hfill \\ {\tau }_{\mathrm{stop}}=min\left\{t_i:\left|r\left(t_i\right)\right|\ge 0.9A_{\max }\right\}\hfill \end{array}\right.\end{array}$$

(27)

$$\lambda =\frac{1}{N{\sigma }_r^4}\sum _{i=1}^N{\left(\left|r\left(t_i\right)\right|-{\mu }_r\right)}^4$$

(28)

$${\tau }_{rms}=\frac{1}{\zeta }\sum _{i=1}^N\left[{\left(t_i-\frac{1}{\zeta }\sum _{i=1}^N\left(t_i{\left|r\left(t_i\right)\right|}^2\right)\right)}^2{\left|r\left(t_i\right)\right|}^2\right]$$

(29)

where N is the total number of sampling points of the signal waveform received by the terminal, and i represents the sampling point index. ${\mu }_r=\frac{1}{N}{\sum }_{i=1}^N\left|r\left(t_i\right)\right|,$ and ${\sigma }_r^2=$$\frac{1}{N}{\sum }_{i=1}^N{\left(\left|r\left(t_i\right)\right|-{\mu }_r\right)}^2$. When a wireless signal encounters obstacles, its total signal energy will inevitably experience significant attenuation. Additionally, LOS channels obstructed by obstacles can lead to significant fluctuations in RT, RDS, and KUR values. Moreover, areas with severe multipath effects often accompany severe NLOS errors, causing RDS to increase with the increase in multipath classification. Finally, KUR can also serve as an important indicator for assisting in judging the quality of NLOS observations. KUR values decrease significantly with the increase in the severity of multipath interference.

The Fuzzy Comprehensive Evaluation was adopted to classify channel states. Let $\widehat{F}={\left\{f_x\right\}}_{x=1}^{\alpha }$ represent the feature set extracted from the signal waveform, and the corresponding distance estimation result be $\widehat{d}$, where $\alpha$ denotes the total number of classification features. The specific method for classifying NLOS channels is as follows.

Find the data closest to the real-time measurements in the local sampled dataset. Let the local dataset under channel p be denoted as $D.\widehat{F}$ and $\mathrm{D}$ can be considered, respectively, the factor domain and evaluation domain. When assuming the weight values $w_k$ for the above features $f_k$ regarding the channel classification, the combination importance of $\alpha$ classification features can be viewed as a fuzzy set $\underset{\sim }{W}$ in the feature domain $\widehat{F}$. The evaluation result is defined as a fuzzy set $\underset{\sim }{S}={\left\{s_x^i\right\}}_{x=1}\left(s_x^i\in [0,1]\right)$ in domain $\mathrm{D}$, which can be obtained through the following fuzzy transformation:

$$\underset{\sim }{S}=\underset{\sim }{W}·\underset{\sim }{R}$$

(30)

$$W={\left\{w_k\right\}}_{k=1}^{\alpha },\mathrm{subject}\mathrm{to}\left\{\begin{array}{c}0\le w_k\le 1\hfill \\ \sum _{x=1}^{\alpha }{w}_k=1.\hfill \end{array}\right.$$

(31)

$\underset{\sim }{R}$ can be expressed in matrix form as $\left[r_{kx}\right]$. In this article, the fuzzy operation operation can be regarded as the inner product of two vectors.

$$s_x^i=\sum _{k=1}^p\left(w_k{r}_{kx}\right)$$

(32)

The sampling data that achieve the maximum value in the evaluation result $\mathrm{S}$ are considered the data closest to the real-time measurements. The corresponding $\widehat{F}$ can be represented as $F_{\mathrm{best}}$. A new local dataset $D_{\mathrm{best}}={\left\{F_{\mathrm{best}}^i,\Delta d_{\mathrm{best}}^i,l_{\mathrm{best}}^i,s_{\mathrm{best}}^i\right\}}_{i=1}^p$ can be generated, where $\Delta d^i$ represents the ranging error, and $l_{\mathrm{best}}^i$ represents the current true channel condition. We identified the propagation channel of online measurements as one of the sample channels in $D_{\mathrm{best}}$ or a mixture of multiple sample channels. When NLOS phenomena are determined, we can further predict the NLOS ranging error using the same method. There is an upper limit on the representative number of channels, and the threshold can be expressed as

$$s_{\mathrm{best}}^i<\beta ,\forall i,i\in \{1,2,\dots ,p\}$$

(33)

According to Equation (33), if the volume of the real-time test data exceeds the discrimination capability of this method, it is necessary to discard all data to avoid outliers that deviate significantly from the local data. Subsequently, the implementation of the NLOS identification and suppression method is as follows:

$$\left\{\begin{array}{cc}\mathrm{nlos},\tilde{d}=\widehat{d}-\Delta d_{\mathrm{best}}^i,\hfill & \mathrm{when}\widehat{l}=l_{\mathrm{best}}^i<0\hfill \\ los,\tilde{d}=\widehat{d},\hfill & \mathrm{when}\widehat{l}=l_{\mathrm{best}}^i>0.\hfill \end{array}\right.$$

(34)

If the above conditions cannot be met, then the positioning channel at this time can be divided into a combination of multiple channels, and these channels all meet the condition

$$s_{\mathrm{best}}^i\ge \beta ,i\in \{1,2,\dots ,p\}$$

(35)

The NLOS suppression method can be denoted as

$$\left\{\begin{array}{c}NLOS,\tilde{d}=\widehat{d}-\sum \left(\frac{s_{\mathrm{best}}^i}{\Psi }\Delta d_{\mathrm{best}}^i\right),\mathrm{when}\\ \widehat{l}=\sum \left(\frac{s_{\mathrm{best}}^i}{\Psi }{l}_{\mathrm{best}}^i\right)<0\\ \mathrm{LOS},\tilde{d}=\widehat{d},\mathrm{when}\widehat{l}=\sum \left(\frac{s_{\mathrm{best}}^i}{\Psi }{l}_{\mathrm{best}}^i\right)\ge 0,\end{array}\right.$$

(36)

where

$$\Psi =\sum s_{\mathrm{best}}^i,\mathrm{subject}\mathrm{to}{s}_{\mathrm{best}}^i\ge \beta$$

(37)

3.3. SRUKF-Based Location Estimation Method for NLOS Mitigation

In the previous section, we presented a method for NLOS mitigation for each base station’s positioning signal. In this section, we select some NLOS observations based on the previous section for fused position estimation. To improve the numerical stability of the output results of the position estimator, we introduced the square root of the state covariance on the basis of the UKF. In the implementation of the SRUKF, the square root of the state covariance is directly propagated, avoiding the need to reconstruct the covariance at each iteration. Additionally, the square root form has the additional advantages of numerical stability and positive semi-definiteness of the state covariance. It is essential to ensure the stability of the filter during the fusion process of LOS and NLOS observations. The implementation of the proposed method is shown in Figure 2.

The initial system state follows a Gaussian distribution with mean m and variance P. From the system state matrix, we can obtain 2L+1 sigma points. Ref. [39] presented the way to calculate sigma points. The nonlinear function is used to propagate these sigma vectors:

$${\mathcal{Y}}_i=h\left(x_i\right)i=0,\dots ,$$

(38)

The covariance of the 5G LOS and NLOS mixed positioning state prediction error can be expressed as

$$P^{-}\left[t_k\right]=FP\left[t_k\right]F^T+Q\left[t_k\right]$$

(39)

where $Q\left[t_k\right]$ is a Gaussian process noise matrix at $t_k$, and $P\left[t_k\right]$ denotes the estimate error covariance and its square root form $R\left[t_k\right]$:

$$P\left[t_k\right]=R\left[t_k\right]{\left(R\left[t_k\right]\right)}^T$$

(40)

The square root of the state prediction covariance can be defined as follows by introducing QR decomposition:

$$R^{-}\left[t_k\right]={\left(qr\left\{{\left[FR\left[t_k\right]\sqrt{Q\left[t_k\right]}\right]}^T\right\}\right)}^T$$

(41)

In the SRUKF algorithm for 5G NLOS mitigation observations, process noise mainly consists of two parts: deterministic noise and stochastic noise. Deterministic noise can be addressed through prior knowledge, while stochastic noise caused by modeling errors should be accurately compensated. Due to the inaccuracies in modeling, the expected mean and covariance of the innovations may differ from the actual values. We propose a stability coefficient to address this issue.

$$R\left[t_k\right]=\sqrt{\phi \left[t_k\right]}{R}^{-}\left[t_k\right]$$

(42)

The estimated innovation covariance $P_{\gamma ,\gamma }\left[t_k\right]$ should be larger than or equal to the actual one $\mathrm{E}\left\{\gamma \left[t_k\right]{\gamma }^T\left[t_k\right]\right\}$ in order to maintain consistent estimations.

$$P_{\gamma ,\gamma }\left[t_k\right]\ge \mathrm{E}\left\{\gamma \left[t_k\right]{\gamma }^T\left[t_k\right]\right\}$$

(43)

$$\gamma \left[t_k\right]=y\left[t_k\right]-\widehat{y}\left[t_k\right]$$

(44)

The innovation covariance has the following expression:

$$P_{\gamma ,\gamma }\left[t_k\right]=H\left[t_k\right]P\left[t_k\right]H^T\left[t_k\right]+Q\left[t_k\right]$$

(45)

where $P\left[t_k\right]$ denotes the covariance of the predicted state, and the cross-covariance between the measurements and the predicted state is $P_{x,y}\left[t_k\right]$:

$$P_{x,z}\left[t_k\right]=P\left[t_k\right]H^T\left[t_k\right]$$

(46)

Combining (45) and (46) leads to (47):

$$P_{\gamma ,\gamma }\left[t_k\right]=P_{x,y}^T\left[t_k\right]P^{-1}\left[t_k\right]P_{x,y}\left[t_k\right]+Q\left[t_k\right]$$

(47)

Incorporating (43) into (47) yields (48), and the fading factor introduced into the predicted state covariance should meet the following requirement:

$$\phi \left[t_k\right]\ge \frac{E\left\{y\left[t_k\right]y^T\left[t_k\right]-Q\left[t_k\right]\right.}{{\left(P_{x,y}^{-}\left[t_k\right]\right)}^T{\left(P^{-}\left[t_k\right]\right)}^{-1}{P}_{x,y}^{-}\left[t_k\right]}$$

(48)

where $P^{-}\left[t_k\right]$ represents the covariance of the predicted state $P\left[t_k\right]=$$\phi \left[t_k\right]P^{-}\left[t_k\right],$ and $P_{x,y}^{-}\left[t_k\right]$ denotes the cross-covariance of the state and measurement without the fading factor, given by $P_{x,y}\left[t_k\right]=\phi \left[t_k\right]P_{x,y}^{-}\left[t_k\right]$.

As a result, it can be confirmed that the lower bound of the stabilized coefficient $\phi \left[t_k\right]$, as obtained in Equation (48), can be utilized to mitigate modeling errors while maintaining the estimation consistency of the innovation. The square root of the predicted measurement covariance can be derived according to Equation (41).

$$R_{y,y}\left[t_k\right]={\left(qr\left\{{\left[\begin{array}{cc}{y}^{\ast }\left[t_k\right]\hfill & \sqrt{U\left[t_k\right]}\hfill \end{array}\right]}^T\right\}\right)}^T$$

(49)

$$y^{\ast }\left[t_k\right]=\left[{\left(y\left[t_k\right]\right)}_0-\widehat{y}\left[t_k\right]\cdots {\left(y\left[t_k\right]\right)}_i-\widehat{y}\left[t_k\right]\cdots {\left(y\left[t_k\right]\right)}_{2L}-\widehat{y}\left[t_k\right]\right]\times diag\left(\sqrt{W^{\left(c\right)}}\right)$$

(50)

Here, $\mathrm{L}$ represents the dimension of the receiver. ${\left(y\left[t_k\right]\right)}_i$, where i ranges from 0 to $2\mathrm{L}$, signifies the propagated sigma point, and $y\left[t_k\right]$ denotes the predicted measurement results. $U\left[t_k\right]$ represents the Gaussian measurement noise matrix at time $t_k$. The term $\sqrt{W^{\left(c\right)}}$ represents the weight of the sigma points. The state covariance derived from QR decomposition can be calculated as follows:

$$R\left[t_k\right]={\left(qr\left\{{\left[{\chi }^{\ast }\left[t_k\right]-y^{\ast }\left[t_k\right]K\left[t_k\right]\sqrt{U\left[t_k\right]}\right]}^T\right\}\right)}^T$$

(51)

$${\chi }^{\ast }\left[t_k\right]=\left[{\left(\chi \left[t_k\right]\right)}_0-\widehat{s}\left[t_k\right]\cdots {\left(\chi \left[t_k\right]\right)}_i-\widehat{s}\left[t_k\right]\cdots {\left(\chi \left[t_k\right]\right)}_{2n_x}-\widehat{s}\left[t_k\right]\right]\times diag\left(\sqrt{W^{\left(c\right)}}\right)$$

(52)

where ${\left(\chi \left[t_k\right]\right)}_i$, where i ranges from 0 to $2\mathrm{L}$, denotes the propagated sigma points, and $K\left[t_k\right]$ represents the filtering gain.

4. Analysis of Simulation and Experiment Results

In this section, we evaluate the performance of the proposed NM-SLE algorithm through simulation and field experiment results. The effectiveness of the proposed method for NLOS observation identification and suppression was validated in real environments. The proposed algorithm can adequately exploit the features of NLOS signals and effectively suppress the residual NLOS measurement errors in the position estimation process. In both simulation and field experiments, we compared the performance of the proposed localization algorithm with those of four existing, advanced NLOS error elimination algorithms: SVM-REM [25], Vector Tracking Loop (VTL) [26], Virtual Base Station (VBS) [28], and Improved Robust Unscented Kalman Filter (IRUKF) [35]. The dataset used in the simulation was composed of 12,000 LOS samples and 5200 NLOS samples.

4.1. Simulation Result Analysis

In Figure 3, the correlation of the candidate features selected for NLOS observations in this study is presented using a correlation coefficient matrix plot. The correlation matrix in the figure is symmetric about the diagonal, and each color within the square corresponds to the same meaning and value. From the analysis of the figure, it can be observed that there are significant positive correlations between the MPV and TE, MPV and SNR, and MPV and KUR, with the MPV showing a particularly strong correlation with the TE. On the other hand, there are noticeable negative correlations between the RT and SNR, RT and MPV, and RDS and SNR. The correlation among other features is relatively weak.

In Figure 4, the overall performances of the different methods are compared in terms of the MSE. From the graph, it can be observed that SVM-REM exhibits a significant performance drop as the number of features increases, while the VBS method performs the best. The performance of the NM-SLE algorithm proposed in this paper is close to that of the VBS algorithm, with both methods showing lower errors compared to the VTL and IRUKF algorithms. The average positioning errors for the proposed method and SVM-REM, VTL, VBS, and IRUKF algorithms are 1.56 m, 8.16 m, 1.91 m, 1.52 m, and 1.85 m respectively.

Figure 5 presents the cumulative distribution function (CDF) of the positioning errors for the NM-SLE method proposed in this paper, as well as those for the SVM-REM, VTL, VBS, and IRUKF methods. It can be observed that the performances of the NM-SLE method and VBS are significantly better than those of the other three methods. At a 60% probability level, both NM-SLE and VBS achieve positioning accuracy better than 2 m. The 90% positioning accuracy for NM-SLE, SVM-REM, VTL, VBS, and IRUKF methods are 3.31 m, 4.59 m, 4.57 m, 3.42 m, and 4.87 m respectively.

The average execution time of different NLOS suppression methods is presented in Figure 6. The data shown in the graph were obtained by running the methods on a laptop. Although the absolute values may not be directly applicable to resource-constrained embedded systems, the relative performance differences between different NLOS suppression methods still reflect their varying computational complexities.

Although this section has demonstrated the effectiveness of the proposed NM-SLE method through simulations, there are still some limitations that need further investigation. Firstly, due to hardware limitations, the simulations only considered data obtained in static scenarios. Further validation is required in dynamic scenarios. Additionally, different shapes and materials may have significant effects.

4.2. Experiment Result Analysis

We built an indoor NLOS 5G positioning test platform based on 5G positioning signal transmitters, clock synchronization modules, atomic clocks, and 5G positioning terminals. The transmitters shared the same atomic clock and achieved time synchronization through fiber optic synchronization modules. The positioning terminals measured TDOA values and transmitted them back to a laptop for position calculation. In Figure 7, the left side shows the physical structure of the 5G positioning signal transmitter used in the experiment, which consists of a power supply module, power amplification module, and positioning signal generation module. The right side shows the structure of the 5G positioning terminal. The terminal performs baseband signal processing and data demodulation in the FPGA, algorithm control and position calculation in the ARM processor, and information transmission through a Bluetooth module.

Figure 8 shows the experimental test scene. A total of six 5G positioning signal transmitters were deployed. There were four pillars in the center of the test area used as wireless signal obstructions. The blue circle in the figure indicates the location of the positioning signal transmitter, and the yellow square indicates the location of the test point where the positioning terminal is placed. The distance between Tx-1 and Tx-3 was 5 m, and the distance between Tx-1 and Tx-2 was also 5 m. The distance between test points was 2.5 m.

In Figure 9, the CDF of the position error is displayed. The performance of each algorithm in the actual environment is slightly lower compared to that in the simulation results. The NM-SLE algorithm proposed in this paper outperforms the other four algorithms significantly. The positioning error was below 4 m for over 90% of the cases. Additionally, the theoretical average values of the positioning error calculated from the derived CDF are presented in

Table 2. Comparision of the positioning error.

Method	Mean Esimation Error
NM-SLE	1.85 m
VBS	1.92 m
VTL	2.18 m
SVM-REM	2.67 m
IRUKF	2.62 m

. Compared to the other four algorithms, the proposed NM-SLE algorithm exhibits the best positioning accuracy in field experiment.

5. Conclusions

In this paper, we propose a method aimed at addressing the challenge of mitigating NLOS errors in 5G indoor localization. Our approach involves several key steps for 5G terminals: firstly, we performed the classification and identification of NLOS observations based on decoupling wireless signal features, where indoor propagation channels are categorized into multiple classes to assess the severity of NLOS effects based on channel identification results. Secondly, we established 5G NLOS measurement identification and classification criteria for filtering ranging results. Finally, we introduced a position estimation method based on the SRUKF to ensure the numerical stability of the filter position estimation output results during the fusion of observations. The theoretical simulations and field experiment demonstrated the superior positioning accuracy compared to four advanced methods: SVM-REM, VTL, VBS and IRUKF. The method proposed in this article provides a new idea for NLOS error suppression in indoor 5G positioning and can also be extended to NLOS error suppression for other wireless signals through appropriate adjustments. The research results of this article show good NLOS suppression effects, and the proposed method can be directly applied in real environments. This can make important contributions to high-precision positioning based on commercial 5G base stations and provide crucial support for location-based services in commercial 5G networks.