# The Analysis and Verification of Unbiased Estimator for Multilateral Positioning
## 4 Unbiased Estimator of Multilateration Model  
Equation (5) is a classical and simple estimator. However, in the actual positioning process, ranging errors are inevitable due to the influence of various environments. It is assumed that the mean square error of the ranging is  
\[\sigma_{\hat{d}}^{2}=E\Big{[}(\hat{d_{i}}-d_{i})^{2}\Big{]} \tag{6}\]  
Figure 1: Principle of two-dimensional plane multilateral positioning.  
When the ranging error model satisfies \(\hat{d}_{i}=d_{i}+e\), the mean square error \(\sigma_{\hat{d}}^{2}\) is the variance of the environmental noise \(e\) and is not affected by the actual distance \(d_{i}\). This type of error model satisfies the unbiased estimation, and can use the Chan algorithm, [PERSON] algorithm, etc. to solve the coordinates of the target node. In actual positioning, the error model is often more complicated. As an illustration, the variance of SX1280 meets [25]:  
\[\sigma_{\hat{d}}^{2}=c\cdot\left(\frac{1}{16\cdot\pi^{2}\cdot SNR\cdot\sqrt{N} \cdot BW^{2}\cdot 2^{SF}}\right) \tag{7}\]  
where \(N\) is the configurable number of CSS (Chirp Spread Spectrum) symbols used in the ranging data, \(BW\) (bandwidth) = 403 kHz, SF (Spreading Factor) and a Signal to Noise Ratio (\(SNR\)), which decreases with distance. It can be concluded that \(\sigma_{\hat{d}}^{2}\) will increase with the increase of \(d_{i}\). In order to illustrate the impact of unbiased estimators, the error model is simplified in this paper as a linear one, in which the relationship between the mean square error \(\sigma_{\hat{d}}^{2}\) and \(d_{i}\) is constructed to satisfy:  
\[\hat{d}_{i}=d_{i}(1+e) \tag{8}\]  
where the random distribution \(e\) follows the normal distribution \((0,\sigma^{2})\).  
Therefore, \(E(\hat{d}^{2}_{i})=E(d_{i}^{2}(1+e)^{2})\) can be obtained from Equation (8). Since the expectation of a random distribution is 0, we get  
\[E(\hat{d}_{i}^{2})=d_{i}^{2}E\left(1+e^{2}\right) \tag{9}\]  
It can be seen that \(\hat{d}_{i}^{2}\) is not an unbiased estimator of \(d_{i}^{2}\). If it is directly substituted into the multilateral positioning formula, the solution value will deviate from the target node. By deforming Equation (9), it can be found that \(\frac{\hat{d}^{2}}{1+E(e^{2})}\) is the unbiased estimate of \(d_{i}^{2}\), namely:  
\[d_{i}^{2}=\frac{\hat{d}_{i}^{2}}{1+E(e^{2})} \tag{10}\]  
Substitute this expression into Formula (2), where B is improved as:  
\[B=\begin{bmatrix}x_{1}{}^{2}-x_{n}{}^{2}+y_{1}{}^{2}-y_{n}{}^{2}+\frac{\hat{d} _{n}{}^{2}}{1+e^{2}}-\frac{\hat{d}_{n}{}^{2}}{1+e^{2}}\\ x_{2}{}^{2}-x_{n}{}^{2}+y_{2}{}^{2}-y_{n}{}^{2}+\frac{\hat{d}_{n}{}^{2}}{1+e^{ 2}}-\frac{\hat{d}_{n}{}^{2}}{1+e^{2}}\\ \ldots\\ x_{1}{}^{2}-x_{n}{}^{2}+y_{1}{}^{2}-y_{n}{}^{2}+\frac{\hat{d}_{n}{}^{2}}{1+e^{ 2}}-\frac{\hat{d}_{n}{}^{2}}{1+e^{2}}\end{bmatrix} \tag{11}\]  
When the ranging error model obeys Equation (9), and the statistical variance value in the ranging error model is known, the squared value of ranging can be corrected by Equation (10) to obtain an unbiased estimate of the true distance. When the ranging information used by the positioning algorithm is the squared value of the ranging, such as \(d_{n}{}^{2}\) in (4), the corrected ranging value can be substituted into the formula to locate the target node, as shown in (11).