## 3 Spatial Decoupling Method for 3D Electric Field Measurement Based on the CRLB  
### Decoupling Model for the MEMS 3D Electric Field Sensor  
Based on Equations (6) and (7), the accurate measurement of the target electric field requires sensor decoupling calibration using standard electric fields. Assuming that \(n\) sets of calibration data are available, the relationship between the output voltage \(\mathbf{U}_{j}\) of the 3D electric field sensor and the input electric field \(\mathbf{E}_{j}\) in the \(j\)-th calibration set can be expressed as  
\[\mathbf{U}_{j}-\mathbf{U}_{0}=\mathbf{K}\mathbf{E}^{\prime}_{j}+\mathbf{N}_{j} \tag{8}\]  
where \(\mathbf{N}_{j}\) represents the sensor noise in the \(j\)-th dataset, assumed to follow a zero-mean Gaussian distribution \(\mathbf{N}_{j}\sim N(0,\mathbf{\Sigma}_{j})\), with \(\mathbf{\Sigma}_{j}\) being the covariance matrix of the sensor noise, given by \(\mathbf{\Sigma}_{j}=diag(\sigma_{1,j}^{2},\sigma_{2,j}^{2},\sigma_{3,j}^{2}, \sigma_{4,j}^{2})\). To eliminate bias effects, the bias-corrected output voltage \(\mathbf{U}^{\prime}_{j}\) is introduced as \(\mathbf{U}^{\prime}_{j}=\mathbf{U}_{j}-\mathbf{U}_{0}\). To obtain the optimal sensitivity coefficient matrix \(\mathbf{K}\), the residual function between the observed and theoretical voltage values is minimized:  
\[\min_{\mathbf{K}}\sum_{j=1}^{n}\left\|\mathbf{U}^{\prime}_{j}-\mathbf{K} \mathbf{E}^{\prime}_{j}\right\|_{\mathbf{\Sigma}_{j}}^{2} \tag{9}\]  
Since noise characteristics may vary among different sensing chips, a weighted linear least squares method is applied to optimize the residual function. The optimization objective function \(\mathcal{J}_{1}(\mathbf{K})\) is defined as  
\[\mathcal{J}_{1}(\mathbf{K})=\sum_{j=1}^{n}\left(\mathbf{U}^{\prime}_{j}- \mathbf{K}\mathbf{E}^{\prime}_{j}\right)^{T}\mathbf{W}_{j}\left(\mathbf{U}^{ \prime}_{j}-\mathbf{K}\mathbf{E}^{\prime}_{j}\right) \tag{10}\]  
where \(\mathbf{W}_{j}\) is the noise weight matrix, defined as the inverse of the sensor noise covariance matrix: \(\mathbf{W}_{j}=\mathbf{\Sigma}_{j}^{-1}\). Taking the derivative of \(\mathcal{J}_{1}(\mathbf{K})\) with respect to \(\mathbf{K}\), and equating it to zero, yields  
\[\frac{\partial\mathcal{J}_{1}(\mathbf{K})}{\partial\mathbf{K}}=-2\sum_{j=1}^{ n}\mathbf{W}_{j}\mathbf{U}^{\prime}_{j}\mathbf{E}^{\prime}_{j}{}^{T}+2\sum_{j=1}^{ n}\mathbf{W}_{j}\mathbf{K}\mathbf{E}^{\prime}_{j}{}^{T}=0 \tag{11}\]  
Rearranging the equation gives  
\[\sum_{j=1}^{n}\mathbf{W}_{j}\mathbf{U}^{\prime}_{j}\mathbf{E}^{\prime}_{j}{}^ {T}=\mathbf{K}\sum_{j=1}^{n}\mathbf{W}_{j}\mathbf{E}^{\prime}_{j}\mathbf{E}^{ \prime}_{j}{}^{T} \tag{12}\]  
Thus, the sensitivity coefficient matrix \(\mathbf{K}\) can be expressed as  
\[\mathbf{K}=\left(\sum_{j=1}^{n}\mathbf{W}_{j}\mathbf{U}^{\prime}\mathbf{E}^{ \prime}_{j}{}^{T}\right)\left(\sum_{j=1}^{n}\mathbf{W}_{j}\mathbf{E}^{\prime} _{j}{}^{T}\right)^{-1} \tag{13}\]During standard field calibration, if the covariance matrix of the sensor noise remains approximately constant across different calibration sets (\(\Sigma_{1}=\Sigma_{2}=\cdots=\Sigma_{n}\)), Equation (13) can be further simplified as  
\[\mathbf{K}=\left(\sum_{j=1}^{n}\mathbf{U}^{\prime}_{j}\mathbf{E}^{\prime}_{j} \right)\left(\sum_{j=1}^{n}\mathbf{E}^{\prime}_{j}\mathbf{E}^{\prime}_{j} \right^{T}\right)^{-1} \tag{14}\]  
By defining \(\mathbf{U}^{\prime}=[\mathbf{U}^{\prime}_{1},\ldots,\mathbf{U}^{\prime}_{n}]\), \(\mathbf{E}^{\prime}=[\mathbf{E}^{\prime}_{1},\ldots,\mathbf{E}^{\prime}_{n}]\), the sensitivity coefficient matrix \(\mathbf{K}\) can be expressed as  
\[\mathbf{K}=\left(\mathbf{U}^{\prime}\mathbf{E}^{\prime T}\right)\left( \mathbf{E}^{\prime}\mathbf{E}^{\prime T}\right)^{-1} \tag{15}\]  
During the calibration process, standard electric fields are applied sequentially in an orthogonal manner to ensure that \(\mathbf{E}^{\prime}\mathbf{E}^{\prime T}\) is invertible. Thus, Equation (15) uniquely determines the sensitivity coefficient matrix, enabling the high-precision decoupling calibration of the sensor.