# A Novel Time Delay Estimation Algorithm for 5G Vehicle Positioning in Urban Canyon Environments
## 3 Methodology
### Unsupervised Multipath Estimation  
The unsupervised multipath estimation part is used to estimate the number of multipath time delay to provide more accurate estimates. Eigenvalues could theoretically be gathered for two types of noise and signal eigenvalue, because the noise in the process of transmission is more stable, and it is so different between each signal path delay and signal loss. So, the noise subspace corresponding eigenvalues can be gathered for a cluster, the characteristics of the signal subspace corresponding values cannot be together for a cluster. The clustering results can be estimated in reverse. In other words, the outlier that cannot be clustered into a cluster can be taken as the eigenvalues corresponding to the signal subspace. The number of these outliers is the number of multipaths. The number of clusters of noise eigenvalues is not a constant, so we must adopt a clustering method which doesn't need the number of clusters. Moreover, all of the process is happening during the positioning, so a high complexity method shouldn't be adopted. In this paper, the density-based spatial clustering of applications with noise (DBSCAN) [33] algorithm is adopted to cluster the eigenvalue \(\mathbf{\lambda}\) because it doesn't need to provide the number of clusters and has a low complexity. By reversely estimating the clustering results, the outliers that cannot be clustered as a class are regarded as the eigenvalues corresponding to the signal subspace, and the number of these outliers is the number of multipaths. Unsupervised multipath estimation is detailed below. DBSCAN is a density-based clustering method. Its main parameters are clustering radius (_radius_) and the minimum number of elements in a class (_Pts_). Clustering radius affects the range of neighborhood density judgment. With the influence of clustering radius and the minimum number of elements in a class, the different values of the two parameters affect the threshold of the data density in the clustering. The data density above this threshold can be clustered into a class, while the data density below this threshold cannot be clustered into a class. In this paper, the data are the eigenvalues of the received signal covariance matrix, and the eigenvalues of the noise subspace have high similarity, in order to cluster the noise subspace eigenvalues with high similarity into one class and filter out the signal subspace eigenvalues with low similarity by clustering. We experimentally analyzed the values of _radius_ and _Pts_, and found that signal subspace eigenvalues will be treated as noise subspace eigenvalues if the data density threshold is high, and noise subspace eigenvalues will be treated as signal subspace eigenvalues if the data density threshold is low, both of these circumstances will affect the accuracy of the final delay estimation results. So, we choose the appropriate radius and Pts to cluster the eigenvalues. In this paper, _Pts_ = 4 and _radius_ is as follows:  
\[radius=\frac{\max(\mathbf{\lambda})-\min(\mathbf{\lambda})}{5\times 10^{4}} \tag{35}\]  
The algorithm process of unsupervised multipath estimation is described as Algorithm 1. The estimated value of the output multipath number \(N_{NC}\) is taken as the number of eigenvalues corresponding to the noise subspace in Equation (14), and the noise subspace \(\mathbf{U_{e}}\) is obtained.