Explainable Neural Network for Joint Orthogonal Bases of Doubly Selective Channels

In this paper, we propose an explainable neural network for decomposing channel kernels into Eigenwaves and implement practical Multi-dimensional Eigenwave Multiplexing (MEM) over doubly selective channels. The quality of Eigenwave decomposition is evaluated using three key metrics: 1) eigenvalue, 2) orthogonality, and 3) duality. The eigenvalue determines the subchannel gains in the eigen domain, while orthogonality and duality impact the interference from other symbols and the distortion of the target symbol, respectively. We prove that maximizing the sum of eigenvalues is equivalent to minimizing the MSE loss function and demonstrate that the duality and orthogonality constraints not only minimize interference for multiplexing but also guide the convergence for NN. Furthermore, we show that these duality and orthogonality constraints are equivalent, allowing them to be combined for model simplification. To further enhance the adaptability of the proposed method, we introduce a second NN architecture that incorporates the Augmented Lagrangian Method (ALM). This approach eliminates the need for parameter tuning under different MIMO scales. We evaluate the proposed methods under two scenarios: 1) 2D doubly selective channels, and 2) 4D doubly selective MIMO channels with both perfect imperfect Channel State Information (CSI) and imperfect CSI.