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.

Learning the principal eigenfunctions of a kernel is at the core of many machine-learning problems. Common methods usually deal with symmetric kernels based on Mercer’s Theorem. However, in the communication systems, the channel kernel is usually asymmetric due to the inconsistencies between the uplink and the downlink propagation environment. In this paper, we propose an explainable Neural Network for extracting eigenfunctions from generic multi-dimensional asymmetric channel kernels based on a recent method called High Order Generalized Mercer’s Theorem (HOGMT), by decomposing it into jointly orthogonal eigenfunctions. The proposed neural network based approach is efficient and can be easily implemented compared to the conventional SVD based solutions used for eigen decomposition. We also discuss the effect of different hyperparameters on the training time, constraint satisfaction, and overall performance. Finally, we show that multiplexing using these eigenfunctions mitigates interference across all the available Degrees of Freedom (DoF), both mathematically as well as via neural network based system-level simulations.

Eigenfunctions are widely used to characterize ker-nels in many data-driven analyses. In machine learning, eigen- function decomposition is primarily based on Mercer's theorem, which requires the kernel to be symmetric. This is difficult to satisfy in communication systems as the channel kernel is usually asymmetric due to the different downlink and uplink propagation environments. High Order Generalized Mercer's Theorem (HOGMT) provides a principled way to decompose any multi-dimensional asymmetric kernel into eigenfunctions. To manage the complexity of the eigen-decomposition, we propose an equivalent Neural Network (NN) for decomposing a gen-eral channel kernel. This is further improved by applying the Augmented Lagrangian Method (ALM) to reduce the training time and parameter tuning, which avoids additional tuning rounds when the size of the kernel or the number of eigen- components change depending on the wireless environment. We validate the adaptability of the proposed NN and its accu-racy using simulations in PyTorch. The code is available at https://github.com/ZBZou/HOGMT-ALM/tree/main.