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.

OFDM modulation and OTFS modulation have demonstrated their efficacy in mitigating interference in the time and frequency domains, respectively, caused by path delay and Doppler shifts. However, no established modulation technique exists to address inter-Doppler interference (IDI) resulting from time-varying Doppler shifts. Additionally, both OFDM and OTFS require supplementary precoding techniques to mitigate inter-user interference (IUI) in MU-MIMO channels. To address these limitations, we present a generalized modulation method for any multidimensional channel, based on Higher Order Mercer's Theorem (HOGMT) [1], [2] which has been proposed recently to decompose multi-user non-stationary channels into independent fading subchannels (Eigenwaves). The proposed method, called multidimensional Eigenwaves Multiplexing (MEM) modulation, uses jointly orthogonal eigenwaves decomposed from the multidimensional channel as subcarriers, thereby avoiding interference from other symbols transmitted over multidimensional channels. We show that MEM modulation achieves diversity gain in eigenspace, which in turn achieves the total diversity gain across each degree of freedom(e.g., space (users/antennas), time-frequency and delay-Doppler). The accuracy and generality of MEM modulation are validated through simulation studies on three non-stationary channels.

Dirty Paper Coding (DPC) is considered as the optimal precoding which achieves capacity for the Gaussian Multiple-Input Multiple-Output (MIMO) broadcast channel (BC). However, to find the optimal precoding order, it needs to repeat$N!$times for$N$users as there are$N!$possible precoding orders. This extremely high complexity limits its practical use in modern wireless networks. In this paper, we show the equivalence of DPC and the recently proposed Higher Order Mercer's Theorem (HOGMT) precoding [1], [2] in 2-D (spatial) case, which provides an alternate implementation for DPC. Furthermore, we show that the proposed implementation method is linear over the permutation operator when permuting over multi-user channels. Therefore, we present a low complexity algorithm that optimizes the precoding order for DPC with beamforming, eliminating repeated computation of DPC for each precoding order. Simulations show that our method can achieve the same result as conventional DPC with$\approx 20\text{dB}$lower complexity for$N=5$users.

The high mobility, density and multi-path evident in modern wireless systems makes the channel highly non-stationary. This causes temporal variation in the channel distribution that leads to the existence of time-varying joint interference across multiple degrees of freedom (DoF, e.g., users, antennas, frequency and symbols), which renders conventional precoding sub-optimal in practice. In this work, we derive a High-Order Generalization of Mercer’s Theorem (HOGMT), which decomposes the multi-user non-stationary channel into two (dual) sets of jointly orthogonal subchannels (eigenfunctions), that result in the other set when one set is transmitted through the channel. This duality and joint orthogonality of eigenfuntions ensure transmission over independently flat-fading subchannels. Consequently, transmitting these eigenfunctions with optimally derived coefficients eventually mitigates any interference across its degrees of freedoms and forms the foundation of the proposed joint spatio-temporal precoding. The transferred dual eigenfuntions and coefficients directly reconstruct the data symbols at the receiver upon demodulation, thereby significantly reducing its computational burden, by alleviating the need for any complementary post-coding. Additionally, the eigenfunctions decomposed from the time-frequency delay-Doppler channel kernel are paramount to extracting the second-order channel statistics, and therefore completely characterize the underlying channel. We evaluate this using a realistic non-stationary channel framework built in Matlab and show that our precoding achieves${\geqslant }4$orders of reduction in BER at SNR${\geqslant }15$dB in OFDM systems for higher-order modulations and less complexity compared to the state-of-the-art precoding.

Recently, there has been great interest in facilitating coexistence of active and passive users of the electromagnetic (EM) spectrum, with the primary objective of higher spectral utilization. The major challenge for passive users, such as Radio Astronomy Services (RAS), is the need for extremely quiet skies to make astronomical observations with maximum sensitivity of the radio telescope. This is increasingly difficult to guarantee because of densification of allocated spectrum, exponential growth of ubiquitous wireless communication and out-of-band astronomical observations required to observe fast radio bursts. This requires either bidirectional collaboration between active and passive users or innovative signal processing at the telescope site to cancel any incident Radio Frequency Interference (RFI). In this work, we show the feasibility of such a paradigm, where RFI from airborne sources, e.g., aircraft, LEO satellites, etc., is cancelled at the receiver of a Radio Telescope, by shaping the EM wavefront by an array of Reconfigurable Intelligent Surfaces (RIS). In contrast to conventional beam-nulling applications for RIS, this method requires precise calculation of the phase and the amplitude of the reflected signal by the RIS in order to guarantee complete cancellation of the incident RFI. We simulate this approach in a practical setting to study its error performance and boundary conditions of the system parameters, that will lead to a demonstrable prototype in near future. Our results indicate that an RIS array with 364 elements can fully cancel RFI for ADS-B systems at an elevation of$60^{\circ}$and an altitude of 10000 m.

Modern wireless channels are increasingly dense and mobile making the channel highly non-stationary. The time-varying distribution and the existence of joint interference across multiple degrees of freedom (e.g., users, antennas, frequency and symbols) in such channels render conventional precoding sub-optimal in practice, and have led to historically poor characterization of their statistics. The core of our work is the derivation of a high-order generalization of Mercer’s Theorem to decompose the non-stationary channel into constituent fading sub-channels (2-D eigenfunctions) that are jointly orthogonal across its degrees of freedom. Consequently, transmitting these eigenfunctions with optimally derived coefficients eventually mitigates any interference across these dimensions and forms the foundation of the proposed joint spatio-temporal precoding. The precoded symbols directly reconstruct the data symbols at the receiver upon demodulation, thereby significantly reducing its computational burden, by alleviating the need for any complementary decoding. These eigenfunctions are paramount to extracting the second-order channel statistics, and therefore completely characterize the underlying channel. Theory and simulations show that such precoding leads to >104× BER improvement (at 20dB) over existing methods for non-stationary channels.