# MIMO Over-the-Air Computation for Distributed Estimation
## 4 Distributed Estimation using MIMO-AirComp  
This section discusses the integration of the proposed MIMO-AirComp framework to estimate the system state from spatially distributed sensor readings [12; 21]. We consider multiple sensors that measure a time-invariant variable and transmit it to the FC. The FC is responsible for processing the sensors' data to extract an interesting parameter. We assume a random variable \(\mathbf{x}\in\mathds{R}^{L_{x}}\) being observed by \(K\) sensors that generate measurements \(\mathbf{y}_{k}\in\mathds{R}^{L_{y}}\) of the form  
\[\mathbf{y}_{k}=\mathbf{C}_{k}\mathbf{x}+\mathbf{v}_{k},\quad\forall k=1,\dots,K, \tag{10}\]  
where \(\mathbf{C}_{k}\in\mathds{R}^{L_{y}\times L_{x}}\) is the observation matrix, and \(\mathbf{v}_{k}\in\mathcal{N}(0,\mathbf{R}_{v,k})\) is a white Gaussian noise with covariance matrix \(\mathbf{R}_{v,k}\in\mathds{R}^{L_{y}\times L_{y}}\) of node \(k\). The objective of the state estimation problem is to obtain an estimate of \(\mathbf{x}\) given all the measurements \(\mathbf{y}=[\mathbf{y}_{1}^{\top},\dots,\mathbf{y}_{K}^{\top}]^{\top}\in \mathds{R}^{KL_{y}}\). Note that the length of the desired vector is \(L_{d}=L_{x}\). Let us denote the estimate of \(\mathbf{x}\) based on all measurements \(\{\mathbf{y}_{k}\}_{k=1}^{K}\) (respectively, only the measurement \(\mathbf{y}_{k}\)) by \(\hat{\mathbf{x}}\) (respectively, \(\hat{\mathbf{x}}_{k}\)). The conventional minimum MSE estimation problem uses the linear estimator \(\hat{\mathbf{x}}=\mathbf{G}\mathbf{y}\), where the estimator is linear with the state measurements [21]. The linear estimation of \(\mathbf{x}\) becomes  
\[\mathbf{P}^{-1}\hat{\mathbf{x}}=\sum_{k=1}^{K}\mathbf{P}_{k}^{-1}\hat{\mathbf{ x}}_{k}, \tag{11}\]  
where \(\mathbf{P}\in\mathds{R}^{L_{x}\times L_{x}}\) (resp. \(\mathbf{P}_{k}\in\mathds{R}^{L_{x}\times L_{x}}\)) is the error covariance corresponding to \(\hat{\mathbf{x}}\) (resp. \(\hat{\mathbf{x}}_{k}\)). The error covariance satisfies  
\[\mathbf{P}^{-1}=\sum_{k=1}^{K}\mathbf{P}_{k}^{-1}-(K-1)\mathbf{R}_{x}^{-1}, \tag{12}\]where \(\mathbf{R}_{x}\in\mathbb{R}^{L_{x}\times L_{x}}\) is the covariance of \(\mathbf{x}\)[22]. In Equation (11), each estimate \(\hat{\mathbf{x}}_{k}\) is weighted by the inverse of the error covariance matrix. The higher the confidence we have in a particular sensor, the higher the trust we place in its measurement. In Equation (11), the weighted estimation \(\mathbf{P}_{k}^{-1}\hat{\mathbf{x}}_{k}\) of node \(k\) is  
\[\mathbf{P}_{k}^{-1}\hat{\mathbf{x}}_{k}=\mathbf{C}_{k}^{\top}\mathbf{R}_{v,k}^ {-1}\mathbf{y}_{k}, \tag{13}\]  
where \(\hat{\mathbf{x}}_{k}\) only depends on the local parameters of \(\mathbf{P}_{k},\mathbf{C}_{k}\), and \(\mathbf{R}_{v,k}\) and the local measurement \(\mathbf{y}_{k}\)[22]. Thus, the form of the global estimator of Equation (11) follows the linear form with the measurement \(\mathbf{y}_{k}\). The estimation problem is solved if the FC has access to all local measurements of sensors at every time step. However, this conventional separated communication and computation approach is not the preferred implementation due to the high communication and computation loads. In particular, the FC needs to handle matrix operations whose size grows in proportion to the number of sensors in the global estimator of Equation (11). Thus, we attempt to transmit the sensing reading after some local processing rather than transmitting raw measurements to reduce the computational burden. Equation (11) essentially shows that what we need at the FC is a weighted mean of the local estimates \(\mathbf{P}_{k}^{-1}\hat{\mathbf{x}}_{k}\). By inputting Equation (13) into Equation (11), the target estimated state \(\hat{\mathbf{x}}\) forms the nomographic function.