# MIMO Over-the-Air Computation for Distributed Estimation
## 3 The Proposed MIMO-AirComp Framework  
Figure 1 illustrates the detailed operations of the proposed MIMO-AirComp framework to compute various target functions over the fading channel with additive noise. The target function is computed by exploiting the collaboration between multiple nodes, the wireless channel, and the FC. Each node preprocesses the raw measurement to input to the aggregation function of the wireless channel, whereas the FC applies the postprocessing function on the superimposed received signal. * Preprocessing at the node: The local preprocessing function of the nodes converts the raw data \(\mathbf{y}_{k}\in\mathds{R}^{L_{y}}\) to the multimodal symbol vector \(\mathbf{d}_{k}\in\mathds{R}^{L_{d}}\) in order to input the MIMO-AirComp. The application-dependent function \(\boldsymbol{\phi}_{k}:\mathds{R}^{L_{y}}\rightarrow\mathds{R}^{L_{d}}\) is applied to the raw data \(\mathbf{y}_{k}\) and generates \(\mathbf{s}_{k}\) to distribute the computation load of the target function. In addition, the source encoder \(\boldsymbol{\eta}:\mathds{R}^{L_{d}}\rightarrow\mathds{R}^{L_{d}}\) is the element-wise mapping from data \(\mathbf{s}_{k}\) to the transmit symbol \(\mathbf{d}_{k}\), such that the symbol is used to compute the target function through the MIMO-AirComp. Thus, the input symbol vector to the MIMO-AirComp with the preprocessing becomes \[\mathbf{d}_{k}=\boldsymbol{\eta}(\boldsymbol{\phi}_{k}(\mathbf{y}_{k}))\,.\] (3)  
Figure 1: System diagram of the proposed MIMO-AirComp framework consisting of the local preprocessing of nodes, the aggregation of the wireless channel, and the postprocessing at the FC. MIMO-AirComp: Let us denote \(\mathbf{d}=\sum_{k=1}^{K}\mathbf{d}_{k}\) as the desired vector of the MIMO-AirComp for ease of exposition. The overall operation of the MIMO-AirComp is to apply the transmit beamforming matrix \(\boldsymbol{\mu}_{k}\in\mathds{C}^{N_{\mathrm{Rx}}\times L_{d}}\) to each preprocessed signal \(\mathbf{d}_{k}\) and the receive beamforming matrix \(\boldsymbol{\
u}\in\mathds{C}^{N_{\mathrm{Rx}}\times L_{d}}\) to the received signals [15]. The fading channel model consists of the MIMO channel matrix \(\mathbf{H}_{k}\in\mathds{C}^{N_{\mathrm{Rx}}\times N_{\mathrm{Rx}}}\) for the link from node \(k\) to the FC and the additive white Gaussian noise vector with independent identically distributed elements \(\mathbf{n}\in\mathcal{CN}(0,\sigma_{\mathrm{cn}}^{2}\mathbf{I})\). Each MIMO channel is subject to independent identically distributed Rayleigh fading, modeled as independent identically distributed complex Gaussian random variables with zero mean and unit variance. Then, the received symbol vector by the FC is \[\hat{\mathbf{d}}=\boldsymbol{\
u}^{H}\Bigg{(}\sum_{k=1}^{K}\mathbf{H}_{k} \boldsymbol{\mu}_{k}\mathbf{d}_{k}+\mathbf{n}\Bigg{)},\] (4) which is the superimposed received signal sent by each node over the fading channel. Thus, the MIMO-AirComp essentially converts the set of the transmitted symbol of all nodes \(\{\mathbf{d}_{k}\}_{k=1}^{K}\) to \(\hat{\mathbf{d}}\in\mathds{R}^{L_{d}}\). * Postprocessing at the FC: The postprocessing function of the FC estimates the target vector \(\mathbf{f}\in\mathds{R}^{L_{f}}\) by using the aggregated received vector \(\hat{\mathbf{d}}\in\mathds{R}^{L_{d}}\). It first applies the source decoder \(\boldsymbol{\eta}^{-1}(\cdot)\) in order to extract \(\hat{\mathbf{s}}\approx\sum_{k=1}^{K}\mathbf{s}_{k}\), where \(\hat{\mathbf{s}}\) is the vector required at the application layer. Then, the application-dependent function \(\boldsymbol{\psi}(\cdot)\) is applied to \(\hat{\mathbf{s}}\in\mathds{R}^{L_{d}}\) to obtain the estimated target vector \(\hat{\mathbf{f}}\in\mathds{R}^{L_{f}}\). Thus, the postprocessing function of the FC is \[\hat{\mathbf{f}}=\boldsymbol{\psi}(\boldsymbol{\eta}^{-1}(\hat{\mathbf{d}})),\] (5) where \(\boldsymbol{\psi}:\mathds{R}^{L_{d}}\rightarrow\mathds{R}^{L_{f}}\) can be a highly nonlinear function dependent on the application.