Theoretical Limits for Time Delay Based Location Estimation in Cooperative Relay Networks

Abstract

Node localization in wireless networks is crucial for supporting advanced location-based services and improving the performance of network algorithms such as routing schemes. In this paper, we study the fundamental limits for time delay based location estimation in cooperative relay networks. The theoretical limits are investigated by obtaining Cramer–Rao Lower Bound (CRLB) expressions for the unknown source location under different relaying strategies when the location of the destination is known and unknown. More specifically, the effects of amplify-and-forward and decode-and-forward relaying strategies on the location estimation accuracy are studied. Furthermore, the CRLB expressions are derived for the cases where the location of only source as well as both source and destination nodes are unknown considering the relays as reference nodes. In addition, the effects of the node topology on the location estimation accuracy of the source node are investigated. The results reveal that the relaying strategy at relay nodes, the number of relays, and the node topology can have significant impacts on the location accuracy of the source node. Additionally, knowing the location of the destination node is crucial for achieving accurate source localization in cooperative relay networks.

Appendix

We can represent the log-likelihood function of $\tau _{i} \varLambda (\tau _{i})$ for the signal at the D node as follows,

$$\begin{aligned} \varLambda (\tau _{i})=-\frac{1}{2\sigma ^2_{Ti}}\int \limits _0^T \left[ y_{di}(t)-Gs(t-\tau _i) \right] ^2 \text {d}t, \end{aligned}$$

(72)

where $\tau _i=\tau _{1i}+\tau _{2i}+\tau _{proc},\quad i=1,2,3,\dots ,M, G=\alpha _ih_{1i}h_{2i}\sqrt{P}$ when the AF relaying strategy is employed and $G=\sqrt{P}h_{2i}$ when the DF relaying strategy is used. Assume that $\alpha _i,h_{1i}$ and $h_{2i}$ are known parameters. Then, the joint log-likelihood function at the D node is given by,

$$\begin{aligned} \varLambda (\tau )=\displaystyle \sum \limits _{i=1}^M \varLambda (\tau _i). \end{aligned}$$

(73)

In this case, the unknown parameters at the D node are $\varvec{\theta }=\left[ \tau _{11},\tau _{12},\dots ,\tau _{1M},\tau _{21},\tau _{22},\dots ,\right.$ $\left. \tau _{2M} \right]$. In order to calculate the CRLBs for $\varvec{\theta }$ unknown parameters, the FIM can be obtained as follows,

$$\begin{aligned} I_{\varvec{\theta }}=E\left[ \frac{\partial }{\partial \varvec{\theta }}\varLambda (\tau /\varvec{\theta })\left( \frac{\partial }{\partial \varvec{\theta }}\varLambda (\tau /\varvec{\theta })\right) ^T \right] . \end{aligned}$$

(74)

Using (74), the 2Mx2M FIM matrix is calculated, which is given by,

$$\begin{aligned} I_{\varvec{\theta }}=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{11}^2} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{11}\partial \tau _{12}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{11}\partial \tau _{1M}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{11}\partial \tau _{21}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{11}\partial \tau _{22}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{11}\partial \tau _{2M}} \\ \frac{\partial ^2 \varLambda (\tau )}{\partial \partial \tau _{12}\partial \tau _{11}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{12}^2} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{12}\partial \tau _{1M}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{12}\partial \tau _{21}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{12}\partial \tau _{22}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{12}\partial \tau _{2M}} \\ \vdots &{} \vdots &{} &{} \vdots &{} \vdots &{} \vdots &{} &{} \vdots \\ \frac{\partial ^2 \varLambda (\tau )}{\partial \partial \tau _{1M}\partial \tau _{11}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{12}\partial \tau _{1M}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{1M}^2} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{1M}\partial \tau _{21}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{1M}\partial \tau _{22}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{1M}\partial \tau _{2M}} \\ \frac{\partial ^2 \varLambda (\tau )}{\partial \partial \tau _{21}\partial \tau _{11}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{21}\partial \tau _{12}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{21}\partial \tau _{1M}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{21}^2} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{21}\partial \tau _{22}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{21}\partial \tau _{2M}} \\ \frac{\partial ^2 \varLambda (\tau )}{\partial \partial \tau _{22}\partial \tau _{11}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{22}\partial \tau _{12}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{22}\partial \tau _{1M}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{22}\partial \tau _{21}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{22}\partial \tau _{22}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{22}\partial \tau _{2M}} \\ \vdots &{} \vdots &{} &{} \vdots &{} \vdots &{} \vdots &{} &{} \vdots \\ \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{2M}\partial \tau _{11}}&{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{2M}\partial \tau _{12}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{2M}\partial \tau _{1M}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{2M}\partial \tau _{21}} &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{2M}\partial \tau _{22}} &{} \cdots &{} \frac{\partial ^2 \varLambda (\tau )}{\partial \tau _{2M}^2} \\ \end{array} \right] \end{aligned}$$

(75)

The elements of $I_{\varvec{\theta }}$ matrix can be obtained after straightforward manipulations as follows [14],

$$\begin{aligned} I_{\varvec{\theta }}= \begin{bmatrix} \gamma _1\tilde{E}_1&\quad \cdots&\quad 0&\quad \gamma _1\tilde{E}_1&\quad \cdots&\quad 0 \\ \vdots&\quad \ddots&\quad \vdots&\quad \vdots&\quad \ddots&\quad \vdots \\ 0&\quad \cdots&\quad \gamma _M\tilde{E}_M&\quad 0&\quad \cdots&\quad \gamma _M\tilde{E}_M\\ \gamma _1\tilde{E}_1&\quad \cdots&\quad 0&\quad \gamma _1\tilde{E}_1&\quad \cdots&\quad 0 \\ \vdots&\quad \ddots&\quad \vdots&\quad \vdots&\quad \ddots&\quad \vdots \\ 0&\quad \cdots&\quad \gamma _M\tilde{E}_M&\quad 0&\quad \cdots&\quad \gamma _M\tilde{E}_M\\ \end{bmatrix} \end{aligned}$$

(76)

where $\gamma _{i}=\frac{G^2}{\sigma ^2_{Ti}}$ and $\tilde{E}_{i}=\int _0^T \left[ s^{'}(t-\tau _{i})\right] ^2\text {d}t$. We can divide $I_{\varvec{\theta }}$ matrix into MxM blocks such as,

$$\begin{aligned} I_{\varvec{\theta }}= \begin{bmatrix} A&A\\ A&A \end{bmatrix}, \end{aligned}$$

(77)

where

$$\begin{aligned} A= \begin{bmatrix} \gamma _1\tilde{E}_1&\quad \cdots&\quad 0\\ \vdots&\quad \ddots&\quad \vdots \\ 0&\quad \cdots&\quad \gamma _M\tilde{E}_M \\ \end{bmatrix}. \end{aligned}$$

(78)

If we apply the block matrix inversion to $I_{\varvec{\theta }}$ in order to find the CRLBs for $\varvec{\theta }$ parameters, the first MxM block of $I^{-1}_{\varvec{\theta }}$ matrix can be obtained as follows,

$$\begin{aligned} I^{-1}_{\varvec{\theta }}=\left( A-AA^{-1}A\right) ^{-1}. \end{aligned}$$

(79)

By using (79), the elements of $I^{-1}_{\varvec{\theta }}$ get $\infty$ values. Therefore, the CRLBs for $\varvec{\theta }$ parameters are $\infty$, namely, $\xi ^2_{1i}=\xi ^2_{2i}=\infty ,~i=1,2,3,\dots ,M$.