# An NLOS Ranging Error Mitigation Method for 5G Positioning in Indoor Environments
## 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.  
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 Cramer-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 \(\hat{\tau}^{m}\), 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:  
\[\mathrm{E}\Big{\{}\xi^{2}\Big{\}}=\frac{1}{2}\int_{0}^{\infty}\varphi\cdot \mathrm{P}\Big{\{}|\xi|\geq\frac{\varphi}{2}\Big{\}}d\varphi \tag{9}\]  
where \(\xi=\hat{\tau}^{m}-\tau^{m}\) 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:  
\[ \begin{split} H_{1}:r(t)&\sim\mathbf{p}\{r(t)\mid \tau\}\\ H_{2}:r(t)&\sim\mathbf{p}\{r(t)\mid\tau+\varphi \}\end{split}  \tag{10}\]  
where \(\mathrm{p}\{\cdot\}\) 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. \[\mathrm{Y}(r(i))=\frac{\mathbf{p}\{r(t)\mid\tau\}}{\mathbf{p}\{r(t)\mid\tau+ \varphi\}}\left\{ \begin{array}{l}H_{1}:>1\\ H_{2}:<1\end{array} \right. \tag{11}\]  
When \(\tau^{m}\) is uniformly distributed within the sampling interval \(T\), then \(ZZB\) can be expressed as  
\[ZZB=\frac{1}{T}\int_{0}^{T}\varphi(t-\varphi)P_{\min}(\varphi)d\varphi \tag{12}\]  
Figure 1: Flowchart of the proposed NM-SLE method. 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 \(P_{\min}(\varphi)\) 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 \(\mathbf{c}=\left[c_{1},c_{2},\ldots,c_{L}\right]^{T}\), and likelihood ratio test results can be computed. This process can be equivalently interpreted as evaluating the corresponding error probability \(P_{\min}(\varphi,\mathbf{c})\) for a binary discrimination system. \[ \begin{split} H_{1}:& r(t-\tau,\mathbf{c})\\ H_{2}:& r(t-\tau-\varphi,\mathbf{c})\end{split}  \tag{13}\]  
where \(\mathbf{c}\) is known.