LOS/NLOS channel identification for improved localization in wireless ultra-wideband networks

Abstract

The paper presents techniques for line-of-sight (LOS) and non-line-of-sight (NLOS) link identification in ultra-wideband wireless networks for the purpose of improving mobile positioning reliability in indoor and outdoor environments. Statistical hypothesis testing is applied by using specific parameters of the channel impulse response (CIR) related to skewness, kurtosis, root mean square delay and mean excess delay. Analytical multi-variate lognormal statistical models are developed for the joint probability densities of the CIR parameters which are found to have distinct features under LOS and NLOS conditions, and this is exploited in determining the nature of the propagation channels. Simulation results demonstrate that link identification with accuracy rates exceeding 95% are achievable for most types of environments, particularly when using combined amplitude and delay parameters. Reliable link identification is then integrated with time-of-arrival positioning, first using a low-complexity iterative maximum likelihood algorithm for localizing mobile nodes with LOS/NLOS links to a network of fixed access nodes. Numerical results show that reliable identification can greatly enhance mobile positioning accuracy, bringing the localization error to within 2 m with 90% probability. In the case of NLOS-dominated dynamically changing environments, another approach is developed based on the use of the unscented Kalman filter with integrated least-squares bias estimation and mitigation, and it is shown that accurate mobile tracking can be achieved, with performance approaching that of NLOS-free operating conditions.

Appendix A: Unscented Kalman filter processing steps

The UKF uses a set of 2n + 1 sample vectors (known as sigma points) that are propagated through the nonlinear system transformation and used to recursively estimate the state vector [28]. Following initialization, the UKF estimator alternates between two successive steps: prediction and filtering.

The system is defined for an n-dimensional state vector $\varvec{X}_{k}$ at instant k, with transition model: $\varvec{X}_{k + 1} = \varvec{F}\left( {\varvec{X}_{k} } \right) + \varvec{v}_{k} ,$ and measurement model: $\varvec{Y}_{k} = \varvec{h}\left( {\varvec{X}_{k} } \right) + \varvec{n}_{k}$ where $\varvec{F}\left( \cdot \right)$ and $\varvec{h}$(·) are the state transition and measurement functions, and $\varvec{v}_{k}$ and $\varvec{n}_{k}$ denote state and measurement noise terms, respectively.

Prediction Step:

At each sample time, a matrix is formed with a concatenation of a set of (2n + 1) sample vectors (sigma points) $\varvec{\xi}_{k|k} = \left[ {\begin{array}{*{20}c} {\varvec{\xi}_{1, k|k} } & {\varvec{\xi}_{2, k|k} } & {\varvec{\xi}_{3, k|k} } & \ldots & {\varvec{\xi}_{2n + 1, k|k} } \\ \end{array} } \right]$ calculated around the filtered state estimate $\hat{\varvec{X}}_{k|k}$ according to:

$$ \varvec{\xi}_{k|k} = \left[ {\begin{array}{*{20}c} {\hat{\varvec{X}}_{k|k} } & {\hat{\varvec{X}}_{k|k} +\varvec{\zeta}_{1} } \\ \end{array} \begin{array}{*{20}c} {\hat{\varvec{X}}_{k|k} -\varvec{\zeta}_{1} } & \ldots & {\hat{\varvec{X}}_{k|k} +\varvec{\zeta}_{n} \hat{\varvec{X}}_{k|k} -\varvec{\zeta}_{n} } \\ \end{array} } \right] $$

(A.1)

where $\varvec{\zeta}_{i}$ is the ith column $\left( {i = 1,2, \ldots ,n} \right)$ of the matrix ${\mathbf{B}}$ given by:

$$ {\mathbf{B}} = \sqrt {\left( {n + \lambda } \right){\mathbf{P}}_{XX,k|k} } $$

(A.2)

with $\lambda = n \times \left( {\alpha^{2} - 1} \right)$ for a parameter $0 < \alpha < 1$, and ${\mathbf{P}}_{XX,k|k}$ denoting the current filtered state covariance matrix.

The matrix ${\mathbf{B}}$ can be computed by many techniques such as Cholesky factorization, thus giving:

$$ {\mathbf{BB}}^{\text{T}} = \sqrt {\left( {n + \lambda } \right){\mathbf{P}}_{XX,k|k} } \left( {\sqrt {\left( {n + \lambda } \right){\mathbf{P}}_{XX,k|k} } } \right)^{T} = \left( {n + \lambda } \right){\mathbf{P}}_{XX,k|k} $$

(A.3)

Substituting the sigma points in the transformation $\varvec{F}$(·), we get:

$$ \varvec{\xi}_{k + 1|k} = \left[ {\begin{array}{*{20}c} {\varvec{F}(\hat{\varvec{X}}_{k|k} )} & {\varvec{F}\left( {\hat{\varvec{X}}_{k|k} +\varvec{\zeta}_{1} } \right) } \\ \end{array} \begin{array}{*{20}c} {\varvec{F}(\hat{\varvec{X}}_{k|k} -\varvec{\zeta}_{1} )} & \ldots & {\varvec{F}(\hat{\varvec{X}}_{k|k} +\varvec{\zeta}_{n} ) \varvec{F}(\hat{\varvec{X}}_{k|k} -\varvec{\zeta}_{n} )} \\ \end{array} } \right] $$

(A.4)

The predicted state estimate is then found by taking a weighted average of the elements in $\varvec{\chi}_{k + 1|k}$ as follows:

$$ \hat{\varvec{X}}_{k + 1|k} = \mathop \sum \limits_{j = 1}^{2n + 1} w_{j}^{m}\varvec{\xi}_{j, k + 1|k} $$

(A.5)

The weights used in computing $\hat{\varvec{X}}_{k + 1|k}$ are given by:

$$ w_{j}^{m} = \left\{ {\begin{array}{*{20}l} {\frac{\lambda }{n + \lambda }} \hfill & {j = 1} \hfill \\ {\frac{1}{{2\left( {n + \lambda } \right)}}} \hfill & {j = 2, 3, \ldots ,2n + 1} \hfill \\ \end{array} } \right. $$

(A.6)

Afterwards, the covariance matrix ${\mathbf{P}}_{XX,k + 1|k}$ is calculated as follows:

$$ {\mathbf{P}}_{XX,k + 1|k} = {\varvec{\Gamma}}_{k} {\mathbf{Q\varGamma }}_{k}^{T} + \mathop \sum \limits_{j = 1}^{2n + 1} w_{j}^{c} \left( {\varvec{\xi}_{j, k + 1|k} - \hat{\varvec{X}}_{k + 1|k} } \right)\left( {\varvec{\xi}_{j, k + 1|k} - \hat{\varvec{X}}_{k + 1|k} } \right)^{T} $$

(A.7)

The weights used in computing ${\mathbf{P}}_{XX,k + 1|k}$ are given by:

$$ w_{j}^{c} = \left\{ {\begin{array}{*{20}l} {\frac{\lambda }{n + \lambda } + \left( {1 + \beta - \alpha^{2} } \right)} \hfill & {j = 1} \hfill \\ {\frac{1}{{2\left( {n + \lambda } \right)}}} \hfill & {j = 2, 3, \ldots ,2n + 1} \hfill \\ \end{array} } \right. $$

(A.8)

where $\beta$ is a parameter (set around 2). Similarly, the elements of $\varvec{\xi}_{k + 1|k}$ are substituted in the function $\varvec{h}$(·), giving the matrix ${\varvec{\uppsi}}_{k + 1|k}$ specified as:

$$ {\varvec{\uppsi}}_{k + 1|k} = \left[ {\begin{array}{*{20}c} {\varvec{\psi}_{1, k + 1|k} } & {\varvec{\psi}_{2, k + 1|k} } & {\varvec{\psi}_{3, k + 1|k} } & \ldots & {\varvec{\psi}_{2n + 1, k + 1|k} } \\ \end{array} } \right] $$

(A.9)

where:

$$ \varvec{\psi}_{j, k + 1|k} = \varvec{h}\left( {\varvec{\xi}_{j, k + 1|k} } \right), j = 1,2, \ldots , 2n + 1 $$

(A.10)

The predicted output is then found by taking the weighted average of the elements in $\varvec{\psi}_{k + 1|k}$ as follows:

$$ \hat{\varvec{Y}}_{k + 1|k} = \mathop \sum \limits_{j = 1}^{2n + 1} w_{j}^{m}\varvec{\psi}_{j, k + 1|k} $$

(A.11)

Afterwards, the predicted output covariance matrix ${\mathbf{P}}_{YY,k + 1|k}$ is found as follows:

$$ {\mathbf{P}}_{YY,k + 1|k} = {\mathbf{R}} + \mathop \sum \limits_{j = 1}^{2n + 1} w_{j}^{c} \left( {\varvec{\psi}_{j, k + 1|k} - \hat{\varvec{Y}}_{k + 1|k} } \right)\left( {\varvec{\psi}_{j, k + 1|k} - \hat{\varvec{Y}}_{k + 1|k} } \right)^{T} $$

(A.12)

On the other hand, the cross-covariance matrix ${\mathbf{P}}_{XY,k + 1|k}$ is obtained by:

$$ {\mathbf{P}}_{XY,k + 1|k} = \mathop \sum \limits_{j = 1}^{2n + 1} w_{j}^{c} \left( {\varvec{\xi}_{j, k + 1|k} - \hat{\varvec{X}}_{k + 1|k} } \right)\left( {\varvec{\psi}_{j, k + 1|k} - \hat{\varvec{Y}}_{k + 1|k} } \right)^{T} $$

(A.13)

Filtering Step:

The filtered estimate can be obtained as:

$$ \hat{\varvec{X}}_{k + 1|k + 1} = \hat{\varvec{X}}_{k + 1|k} + \varvec{K}_{k + 1} \left( {\varvec{Y}_{k + 1} - \hat{\varvec{Y}}_{k + 1|k} } \right) $$

(A.14)

where the Kalman gain $\varvec{K}_{k + 1}$ is given by:

$$ \varvec{K}_{k + 1} = {\mathbf{P}}_{XY,k + 1|k} {\mathbf{P}}_{YY,k + 1|k}^{ - 1} $$

(A.15)

Finally, the updated filtered covariance matrix is obtained as:

$$ {\mathbf{P}}_{XX,k + 1|k + 1} = {\mathbf{P}}_{XX,k + 1|k} - \varvec{K}_{k + 1} {\mathbf{P}}_{YY,k + 1|k} \varvec{K}_{k + 1}^{T} . $$

(A.16)