Attention-Aided MMSE with Ridge Denoising: How to Train under Noisy Channel Samples

Deep neural channel estimators are typically trained with clean channel state information (CSI), which is unavailable in practical orthogonal frequency-division multiplexing (OFDM) systems. In pilot-based OFDM, naive noisy-target training is structurally biased because the pilot input and noisy full-grid target share the same noise realization, driving the estimator toward identity copying. To address this for Attention-aided MMSE (A-MMSE), we propose a ridge-regularized objective that penalizes the generated filter directly. In a stylized fixed-filter model, this penalty induces scalar shrinkage and recovers the scalar MMSE gain at an explicit penalty value. We further construct surrogate training targets by estimating the channel covariance via eigenvalue clipping of the noisy empirical second-moment matrix, without requiring clean CSI labels. On COST 2100 channels, the proposed Ridge-A-MMSE consistently outperforms the Noise2Noise (N2N) baselines considered in this paper, and the combination of ridge regularization and covariance shrinkage approaches the same network trained with clean CSI labels at high signal-to-noise ratios (SNRs).

State Estimation with 1-Bit Observations and Imperfect Models: Bussgang Meets Kalman in Neural Networks

State estimation from noisy observations is a fundamental problem in many applications of signal processing. Traditional methods, such as the extended Kalman filter, work well under fully-known Gaussian models, while recent hybrid deep learning frameworks, combining model-based and data-driven approaches, can also handle partially known models and non-Gaussian noise. However, existing studies commonly assume the absence of quantization distortion, which is inevitable, especially with non-ideal analog-to-digital converters. In this work, we consider a state estimation problem with 1-bit quantization. 1-bit quantization causes significant quantization distortion and severe information loss, rendering conventional state estimation strategies unsuitable. To address this, inspired by the Bussgang decomposition technique, we first develop the Bussgang-aided Kalman filter by assuming perfectly known models. The proposed method suitably captures quantization distortion into the state estimation process. In addition, we propose a computationally efficient variant, referred to as the reduced Bussgang-aided Kalman filter and, building upon it, introduce a deep learning-based approach for handling partially known models, termed the Bussgang-aided KalmanNet. In particular, the Bussgang-aided KalmanNet jointly uses a dithering technique and a gated recurrent unit (GRU) architecture to effectively mitigate the effects of 1-bit quantization and model mismatch. Through simulations on the Lorenz-Attractor model and the Michigan NCLT dataset, we demonstrate that our proposed methods achieve accurate state estimation performance even under highly nonlinear, mismatched models and 1-bit observations.