Quantized lower bounds on grid-based localization algorithm for wireless sensor networks

Abstract

In this paper, we introduce a Quantized Cramer Rao Bound (Q-CRB) method, which adapts the use of the CRB to handle grid-based localization algorithms with certain constraints, such as localization boundaries. In addition, we derive a threshold granularity level which identifies where the CRB can be appropriately applied to this type of algorithm. Moreover, the derived threshold value allows the users of grid-based LSE techniques to probably avoid some unnecessary complexities associated with using high grid resolutions. To examine the feasibility of the new proposed bound, the grid-based least square estimation (LSE) technique was implemented. The Q-CRB was used to evaluate the performance of the LSE method under extensive simulation scenarios. The results show that the Q-CRB provided a tight bound in the sense that the Q-CRB can characterize the behaviour of location errors of the LSE technique at various system parameters, e.g. granularity levels, measurement accuracies, and in the presence or absence of localization boundaries.

Appendix A: Derivative of the threshold granularity level

In order to obtain the granularity point at which the Q-CRB is asymptotically approaches the CRB, we assume that Δ represents a vector of changes of the eigenvalues $\hat {\lambda }_{x},\hat {\lambda }_{y}$ over λ _x , λ _y , as

$$\begin{array}{@{}rcl@{}} \bigtriangleup_{j} = \Vert \hat{\textbf{e}}_{j} - \textbf{u} \Vert^{2} - \Vert {\textbf{e}}_{j} - \textbf{u} \Vert^{2} \end{array} $$

(26)

where $j = 0,\frac {Z}{4},\frac {Z}{2},\frac {3Z}{4}$. △ _j is mostly attributed by the shift $\textbf {h}_{j}=\textbf {e}_{j}-\hat {\textbf {e}}_{j}$ of the grid location $\hat {\textbf {e}}_{j}$ from the corresponding point e _j . By combining h _j with Eq. 21 and substituting the result into Eq. 26, we have

$$\begin{array}{@{}rcl@{}} \bigtriangleup_{j} = \Vert\textbf{h}_{j}\Vert^{2} - 2\left( \textbf{e}_{j}-\textbf{u}\right)^{T}\left( \textbf{h}_{j}\right). \end{array} $$

(27)

We focus on computing the granularity level which is required to make Δ ≈ 0. Hence, one could consider the maximum absolute change of Δ, as

$$\begin{array}{@{}rcl@{}} \delta &=& \max\limits_{j}\left\lbrace\vert\bigtriangleup_{j}\vert\right\rbrace, \\ &=& \left\vert\Vert\textbf{h}_{m}\Vert^{2} - 2\left( \textbf{e}_{m}-\textbf{u}\right)^{T}\left( \textbf{h}_{m}\right)\right\vert, \end{array} $$

(28)

where the index m is the jth row of Δ at which |△ _j | is maximum. Note, from Eq. 28, that obtaining the lowest granularity point at which δ ≈ 0 will make Δ ≈ 0. Applying the sub-additivity rule on Eq. 28, we obtain

$$ \delta \geq \left\vert\Vert\textbf{h}_{m}\Vert^{2} - 2\left\vert \textbf{e}_{m}-\textbf{u}\right\vert^{T}\left\vert \textbf{h}_{m}\right\vert\right\vert. $$

(29)

Suppose, without loss of generality, that h _j has uniform distribution over π, in other words $h_{m,\nu }\sim \mathcal {U}(a_{\nu },b_{\nu })$, for ν = x, y. Then the maximum absolute value |h _m | represents the worst case scenario, as shown in Fig. 1, such that $\vert \textbf {h}_{m}\vert = \frac {\textbf {b}-\textbf {a}}{2}\gamma$. Accordingly,

$$\begin{array}{@{}rcl@{}} \delta \geq \left\vert\frac{\Vert\textbf{}-\textbf{a}\Vert^{2}}{4}\gamma^{2} - \left\vert \textbf{e}_{m}-\textbf{u}\right\vert^{T}\left( \textbf{b}-\textbf{a}\right)\gamma\right\vert. \end{array} $$

(30)

Using inequalities properties for δ> 0, Eq. 31 is rewritten as

$$\begin{array}{@{}rcl@{}} -\delta \leq \frac{\Vert\textbf{b}-\textbf{a}\Vert^{2}}{4}\gamma^{2} - \left\vert \textbf{e}_{m}-\textbf{u}\right\vert^{T}\left( \textbf{b}-\textbf{a}\right)\gamma \leq \delta. \end{array} $$

(31)

Next, by computing the derivative of δ w.r.s.t γ and set the result equal zero, we get

$$ 0 \leq \frac{\Vert\textbf{b}-\textbf{a}\Vert^{2}}{2}\gamma - \left\vert \textbf{e}_{m}-\textbf{u}\right\vert^{T}\left( \textbf{b}-\textbf{a}\right) \leq 0. $$

(32)

Therefore,

$$ 0 = \frac{\Vert\textbf{b}-\textbf{a}\Vert^{2}}{2}\gamma - \left\vert \textbf{e}_{m}-\textbf{u}\right\vert^{T}\left( \textbf{b}-\textbf{a}\right). $$

(33)

Thus, solving Eq. 33 for γ = γ _{t h} yields

$$ \gamma_{th} = \frac{2\left\vert \textbf{e}_{m}-\textbf{u}\right\vert^{T}\left( \textbf{b}-\textbf{a}\right)}{\Vert\textbf{b} - \textbf{a}\Vert^{2}}. $$

(34)