# Interference Mitigation for Metro Train Location Based on Karhunen-Loeve Decomposition
## 3 Interference Mitigation Algorithm Based on KL Decomposition  
In this section, the details of the interference mitigation algorithm are introduced. Karhunen-Loeve expansion is a method for representing random processes with a set of uncorrelated random variables [26]. Orthogonal expansion of random process \(r(t)\) in observation time \(T_{s}\) as follows. \[r(t)=\sum_{n=1}^{\infty}Z_{n}\phi_{n}(t),0<t<N_{s}T_{s} \tag{9}\]  
\[Z_{n}=\int_{0}^{N_{s}T_{s}}r(t)\phi_{n}(t)dt,n\geq 1. \tag{10}\]  
The variance of the transform coefficient \(Z_{n}\) is the eigenvalue \(\lambda_{n}\). The eigenvalue \(\lambda_{n}\) corresponds to the eigenfunction \(\phi_{n}\), representing the energy value of the signal at this eigenfunction. Suppose the random process \(r(t)\) has a mean of zero. The characteristic function \(\phi_{n}\) has the following relationship with the autocorrelation function of the random process \(r(t)\). \[\int_{0}^{N_{s}T_{s}}E\{r(t_{1})r(t_{2})\}\phi_{i}(t_{1})dt_{1}=\lambda_{n} \phi_{n}(t_{2}). \tag{11}\]  
where \(E\{r(t_{1})r(t_{2})\}\) is the Toeplitz autocorrelation matrix. Eigenfunctions and corresponding eigenvalues are obtained by decomposing \(E\{r(t_{1})r(t_{2})\}\) matrix. Interfering signals and positioning signals can be distinguished by the distribution of eigenvalues. As mentioned in Section 2, this paper mainly considers the first interference mode, and the localization  
Figure 3: The structure of the train positioning terminal. signal characteristics and the interference signal characteristics are constant. The distribution of the signal \(r_{CPIS}(t)\) is approximate Gaussian white noise, and its energy values on the eigenvectors are equal. The correlation of a deterministic signal is higher than that of noise, so its energy content will be higher, and its eigenvalues will be larger than those of the noise component. Figure 4 shows the processing of the KL expansion algorithm. Figure 5 below shows the distribution of the normalized eigenvalues of the KL expansion of the received signal in the absence of interference and narrowband interference. The magnitude of the eigenvalue represents the strength of the signal energy. When there is interference, as shown by the red curve in Figure 5, the eigenvalues corresponding to the interference signal are concentrated on the main eigenvalues, which is in obvious contrast with the absence of interference signals. We can judge whether there is an interference signal according to the corresponding relationship between the interference and the magnitude of the eigenvalue [27]. The interference signal is reconstructed by inverse KL decomposition using the eigenvalue with a high energy value and the corresponding eigenfunction, and the interference mitigation is realized by subtracting the estimation of the interference signal from the received signal. The interference mitigation algorithm proposed in this paper can be described as having the following steps. Figure 4: The processing of KL decomposition algorithm. Figure 5: The positioning signal is decomposed in KL with narrow-band interference (NBI) and interference-free (NI). (1) Calculation the Toeplitz matrix from \(N=N_{\text{s}}T_{\text{s}}\) samples autocorrelation \(R(n)\) of received signal \(r(t)\). \[R_{\text{Toc}}=\left[ \begin{array}{cccc}R(0)&\cdots&\cdots&R(N)\\ R(1)&R(0)&\cdots&R(N)\\ \cdots&\cdots&R(0)&\cdots\\ R(N)&\cdots&\cdots&R(0)\end{array} \right] \tag{12}\]  
(2) Solve the eigenvalue \(\lambda_{n}\) and eigenfunction \(\phi_{n}\) of Equation (12);  
(3) Calculate the expansion coefficient \(Z_{n}\) according to Formula (10);  
(4) Sort all eigenvalues \(\lambda_{n}\) from largest to smallest, select the first L eigenvalues as the eigenvalues corresponding to the interference signal, and reconstruct the interference signal \(\hat{J}(t)\) according to Formula (9);  
(5) The interference-stripped positioning signal is obtained by subtracting the interference signal from the received signal in Formula (2).