## 3 Spatial Decoupling Method for 3D Electric Field Measurement Based on the CRLB
### Decoupling Method Based on the CRLB  
After obtaining the sensitivity coefficient matrix \(\mathbf{K}\), it is necessary to further establish the relationship between the measured electric field and the sensor's output voltage to achieve decoupling. According to the linear coupling relationship in Equation (6), the optimal estimation of the measured electric field \(\mathbf{E}^{\prime}\) can be determined by calculating the CRLB. The CRLB is a theoretical lower bound used to quantify the estimation accuracy. It provides the minimum achievable variance for an unbiased estimator under ideal conditions. In simple terms, the CRLB gives the smallest possible estimation error, serving as an important tool for evaluating the performance of an estimator. In this decoupling model, the CRLB is used to calculate the minimum achievable estimation error of the sensor output under a given model assumption. This lower bound allows us to assess the theoretical minimum error when using a specific decoupling method, thereby providing a theoretical basis for optimizing algorithms. Suppose we want to estimate a vector \(\mathbf{x}=\left[x_{1},x_{2},\ldots,x_{n}\right]\), and its measurement are obtained through a random process. The CRLB is defined based on the likelihood function \(\mathbf{p}(\mathbf{x})\), and the formula is as follows:  
\[CRLB(\mathbf{x})=\left[\mathbb{E}\Big{(}\frac{\partial\ln\mathbf{p}(\mathbf{x })}{\partial\mathbf{x}}\Big{)}\Big{(}\frac{\partial\ln\mathbf{p}(\mathbf{x}) }{\partial\mathbf{x}}\Big{)}^{T}\right]^{-1} \tag{16}\]  
In Equation (16), \(\mathbf{p}(\mathbf{x})\) is the likelihood function of the parameter \(\mathbf{x}\), and \(\frac{\partial\ln\mathbf{p}(\mathbf{x})}{\partial\mathbf{x}}\) is the gradient of the log-likelihood function. In this coupling model, the sensor's bias-corrected output voltage \(\mathbf{U}^{\prime}\) follows a Gaussian distribution, with its likelihood function \(\mathbf{p}(\mathbf{U}^{\prime};\mathbf{E}^{\prime})\) expressed as  
\[\mathbf{p}\big{(}\mathbf{U}^{\prime};\mathbf{E}^{\prime}\big{)}=\frac{1}{4 \pi^{2}|\mathbf{\Sigma}|^{\frac{1}{2}}}\exp\biggl{(}-\frac{1}{2}\big{(}\mathbf{ U}^{\prime}-\mathbf{K}\mathbf{E}^{\prime}\big{)}^{T}\mathbf{\Sigma}^{-1}\big{(} \mathbf{U}^{\prime}-\mathbf{K}\mathbf{E}^{\prime}\big{)}\biggr{)} \tag{17}\]  
taking the logarithm of the likelihood function and computing its second derivative with respect to \(\mathbf{E}^{\prime}\) yields  
\[\frac{\partial^{2}\ln\mathbf{p}(\mathbf{U}^{\prime}_{j}\mathbf{E}^{\prime})}{ \partial\mathbf{E}^{\prime}\partial\mathbf{E}^{\prime T}}=-\mathbf{K}^{T} \mathbf{W}\mathbf{K} \tag{18}\]  
Thus, the Fisher information matrix \(\mathcal{I}(\mathbf{E}^{\prime})\) is obtained as  
\[\mathcal{I}\big{(}\mathbf{E}^{\prime}\big{)}=-\mathbb{E}\biggl{[}\frac{ \partial^{2}\ln\mathbf{p}(\mathbf{U}^{\prime};\mathbf{E}^{\prime})}{ \partial\mathbf{E}^{\prime}\partial\mathbf{E}^{\prime T}}\biggr{]}=\mathbf{K}^ {T}\mathbf{W}\mathbf{K} \tag{19}\]According to the CRLB theory, the covariance matrix of any estimator \(\widetilde{\mathbf{E}}^{\prime}\) of the measured electric field satisfies the following inequality:  
\[Cov\Big{[}\widetilde{\mathbf{E}}^{\prime}\Big{]}\geq\mathcal{I}^{-1}\big{(} \mathbf{E}^{\prime}\big{)} \tag{20}\]  
where \(\mathcal{I}^{-1}(\mathbf{E}^{\prime})\) is the inverse of the Fisher information matrix. Thus, the CRLB can be expressed as  
\[CRLB=\mathcal{I}^{-1}\big{(}\mathbf{E}^{\prime}\big{)}=\Big{(}\mathbf{K}^{T} \mathbf{W}\mathbf{K}\Big{)}^{-1} \tag{21}\]  
From Equation (21), the optimal linear unbiased estimation of the measured electric field \(\mathbf{E}^{\prime}\) should be based on \(\big{(}\mathbf{K}^{T}\mathbf{W}\mathbf{K}\big{)}^{-1}\), which corresponds to the weighted least squares (WLS) estimation.