A node selection method for cooperative location estimation in high density wireless networks

Abstract

In a cooperative location estimation system, all nodes share their location information in the network. Each node itself can use this shared inaccurate location information to estimate or improve its location. In a network with a high number of nodes, using all available data in location estimation of a specific node requires considerable amount of data processing. In order to reduce the complexity of data processing for location estimation, here we address an algorithm to select some fixed number of nodes for participating their data in location estimation of the target node. The assumed distributed cooperative location estimation method, requires an estimate of the distance between the target node and the other participating nodes. We have obtained a closed form formula for Bayesian Cramer–Rao lower bound (BCRLB) in the assumed network. Based on the BCRLB, a node selection algorithm is proposed which decreases the required computations without a significant reduction in the location estimation accuracy compared with processing all the available data. Furthermore, using this algorithm reduces the number of distance measurements. Computer simulations and field experiment are used to study the performance of the proposed node selection algorithm on the location estimation.

Appendix A Approximation of \({\varvec{ \Psi }}_k\) using the second order Taylor series

The matrix ${\varvec{ \Psi }}_k$ in (5) can be written as ${\varvec{ \Psi }}_k = \frac{1}{\sigma _{n_k}^2 } \mathrm{E}_{\mathbf{p}_{0k}} \left\{ \mathbf{G}_k \right\}$ where $\mathbf{G}_k \triangleq \mathbf{q}_k \mathbf{q}_k ^T$, $\mathbf{q}_k \triangleq \frac{\mathbf{p}_{0k} }{ \Vert \mathbf{p}_{0k} \Vert }$ and $\mathbf{p}_{0k} \triangleq \mathbf{p}_0-\mathbf{p}_k$. The Taylor expansion of (i, j)th element of $\mathbf{G}_k(\mathbf{p}_{0k})$ (i.e., $[\mathbf{G}_k(\mathbf{p}_{0k})]_{i,j}=\mathbf{e}_i^T \mathbf{G}_k(\mathbf{p}_{0k}) \mathbf{e}_j$) around $\overline{\mathbf{p}_{0k}} \triangleq \mathrm{E}\{ \mathbf{p}_{0k} \}$ is

$$\begin{aligned}&[\mathbf{G}_k(\mathbf{p}_{0k})]_{i,j} \approx [\mathbf{G}_k(\overline{\mathbf{p}_{0k}})]_{i,j} \nonumber \\&\quad + (\mathbf{p}_{0k} - \overline{\mathbf{p}_{0k}})^T \bigtriangledown _{\mathbf{p}_{0k}}[\mathbf{G}_k(\mathbf{p}_{0k})]_{i,j}\Big |_{\mathbf{p}_{0k}=\overline{\mathbf{p}_{0k}}} \nonumber \\&\quad + \frac{1}{2!} (\mathbf{p}_{0k}-\overline{\mathbf{p}_{0k}})^T \mathbf{H}(\mathbf{p}_{0k}) \Big |_{\mathbf{p}_{0k}=\overline{\mathbf{p}_{0k}}} (\mathbf{p}_{0k} - \overline{\mathbf{p}_{0k}})\\&\quad + \ldots \end{aligned}$$

(28)

where $\mathbf{H}(\mathbf{p}_{0k})$ is the gradient of $\bigtriangledown _{\mathbf{p}_{0k}}[\mathbf{G}_k(\mathbf{p}_{0k})]_{i,j}$.

With some mathematical manipulation, $\bigtriangledown _{\mathbf{p}_{0k}}[\mathbf{G}_k(\mathbf{p}_{0k})]_{i,j}$ and $\mathbf{H}(\mathbf{p}_{0k})$ can be written as

$$\begin{aligned} \bigtriangledown _{\mathbf{p}_{0k}}[\mathbf{G}_k(\mathbf{p}_{0k})]_{i,j} = \frac{1}{\Vert \mathbf{p}_{0k} \Vert } (\mathbf{I}-\mathbf{q}_k\mathbf{q}_k^T) \mathbf{A}_{ij} \mathbf{q}_k , \end{aligned}$$

(29)

and

$$\begin{aligned}&\mathbf{H}(\mathbf{p}_{0k}) = \frac{1}{\Vert \mathbf{p}_{0k} \Vert ^2} \Big ( \frac{1}{2}\mathbf{A}_{ij}(\mathbf{I}-4\mathbf{q}_k\mathbf{q}_k^T) + \frac{1}{2}(\mathbf{I}-4\mathbf{q}_k\mathbf{q}_k^T)\mathbf{A}_{ij} \nonumber \\&\qquad \qquad \qquad - (\mathbf{q}_k^T\mathbf{A}_{ij}\mathbf{q}_k)(\mathbf{I}-4\mathbf{q}_k\mathbf{q}_k^T) \Big ) , \end{aligned}$$

(30)

where $\mathbf{A}_{ij} \triangleq \mathbf{e}_i \mathbf{e}_j^T + \mathbf{e}_j \mathbf{e}_i^T$. Hence, the (i, j)th element of ${\varvec{ \Psi }}_k$ can be written as

$$\begin{aligned}&[{\varvec{ \Psi }}_k ]_{i,j} \approx \frac{1}{\sigma _{n_k}^2 } \mathbf{e}_i^T \overline{\mathbf{q}_k}\; \overline{\mathbf{q}_k}^T \mathbf{e}_j \nonumber \\&\qquad + \frac{1}{2\sigma _{n_k}^2 } \mathrm{E}_{\mathbf{p}_{0k}} \left\{ (\mathbf{p}_{0k}-\overline{\mathbf{p}_{0k}})^T \mathbf{H}(\overline{\mathbf{p}_{0k}}) (\mathbf{p}_{0k} - \overline{\mathbf{p}_{0k}}) \right\} , \end{aligned}$$

(31)

where $\overline{\mathbf{q}_k}\triangleq \frac{\overline{\mathbf{p}_{0k}}}{ \Vert \overline{\mathbf{p}_{0k}} \Vert }$.

Using the equality of $\mathbf{x}^T\mathbf{y}= \mathrm{tr}(\mathbf{y}\mathbf{x}^T)$ for any $\mathbf{x}$ and $\mathbf{y}$ vectors, we have

$$\begin{aligned}&[{\varvec{ \Psi }}_k ]_{i,j} \approx \frac{1}{\sigma _{n_k}^2 } \mathbf{e}_i^T \overline{\mathbf{q}_k}\; \overline{\mathbf{q}_k}^T \mathbf{e}_j \nonumber \\&\quad + \frac{1}{2\sigma _{n_k}^2 } \mathrm{tr}\Big ( \mathrm{E}_{\mathbf{p}_{0k}} \left\{ (\mathbf{p}_{0k} - \overline{\mathbf{p}_{0k}}) (\mathbf{p}_{0k}-\overline{\mathbf{p}_{0k}})^T \right\} \mathbf{H}(\overline{\mathbf{p}_{0k}}) \Big ). \end{aligned}$$

(32)

By defining $\mathbf{C}_k \triangleq \mathrm{E}_{\mathbf{p}_{0k}} \left\{ (\mathbf{p}_{0k} - \overline{\mathbf{p}_{0k}}) (\mathbf{p}_{0k}-\overline{\mathbf{p}_{0k}})^T \right\} =\mathbf{R}_0+\mathbf{R}_k$, we can rewrite $[{\varvec{ \Psi }}_k ]_{i,j}$ as

$$\begin{aligned}{}[{\varvec{ \Psi }}_k ]_{i,j} \approx \frac{1}{\sigma _{n_k}^2 } \mathbf{e}_i^T \overline{\mathbf{q}_k}\; \overline{\mathbf{q}_k}^T \mathbf{e}_j + \frac{1}{2\sigma _{n_k}^2 } \mathrm{tr}\Big ( \mathbf{C}_k \mathbf{H}(\overline{\mathbf{p}_{0k}}) \Big ). \end{aligned}$$

(33)

Replacing (30) in (33), and with some mathematical manipulation ${\varvec{ \Psi }}_k$ can be written as

$$\begin{aligned}&{\varvec{ \Psi }}_k \approx \frac{1}{\sigma _{n_k}^2 } \Bigg ( \overline{\mathbf{q}_k}\; \overline{\mathbf{q}_k}^T + \frac{1}{\Vert \overline{\mathbf{p}_{0k}} \Vert ^2 } \Big (\mathbf{C}_k -2\mathbf{C}_k\overline{\mathbf{q}_k}\; \overline{\mathbf{q}_k}^T\nonumber \\&\qquad \; -2\overline{\mathbf{q}_k}\; \overline{\mathbf{q}_k}^T \mathbf{C}_k- \mathrm{tr}(\mathbf{C}_k) \overline{\mathbf{q}_k}\; \overline{\mathbf{q}_k}^T + 4(\overline{\mathbf{q}_k}^T\mathbf{C}_k\overline{\mathbf{q}_k})\overline{\mathbf{q}_k}\; \overline{\mathbf{q}_k}^T \Big ) \Bigg ). \end{aligned}$$

(34)