Outlier-Insensitive Kalman Filtering: Theory and Applications

State estimation of dynamical systems from noisy observations is a fundamental task in many applications. It is commonly addressed using the linear Kalman filter (KF), whose performance can significantly degrade in the presence of outliers in the observations, due to the sensitivity of its convex quadratic objective function. To mitigate such behavior, outlier detection algorithms can be applied. In this work, we propose a parameter-free algorithm which mitigates the harmful effect of outliers while requiring only a short iterative process of the standard update step of the KF. To that end, we model each potential outlier as a normal process with unknown variance and apply online estimation through either expectation maximization or alternating maximization algorithms. Simulations and field experiment evaluations demonstrate competitive performance of our method, showcasing its robustness to outliers in filtering scenarios compared to alternative algorithms.

Adaptive KalmanNet: Data-Driven Kalman Filter with Fast Adaptation

Combining the classical Kalman filter (KF) with a deep neural network (DNN) enables tracking in partially known state space (SS) models. A major limitation of current DNN-aided designs stems from the need to train them to filter data originating from a specific distribution and underlying SS model. Consequently, changes in the model parameters may require lengthy retraining. While the KF adapts through parameter tuning, the black-box nature of DNNs makes identifying tunable components difficult. Hence, we propose Adaptive KalmanNet (AKNet), a DNN-aided KF that can adapt to changes in the SS model without retraining. Inspired by recent advances in large language model fine-tuning paradigms, AKNet uses a compact hypernetwork to generate context-dependent modulation weights. Numerical evaluation shows that AKNet provides consistent state estimation performance across a continuous range of noise distributions, even when trained using data from limited noise settings.

NUV-DoA: NUV Prior-based Bayesian Sparse Reconstruction with Spatial Filtering for Super-Resolution DoA Estimation

Achieving high-resolution Direction of Arrival (DoA) recovery typically requires high Signal to Noise Ratio (SNR) and a sufficiently large number of snapshots. This paper presents NUV-DoA algorithm, that augments Bayesian sparse reconstruction with spatial filtering for super-resolution DoA estimation. By modeling each direction on the azimuth's grid with the sparsity-promoting normal with unknown variance (NUV) prior, the non-convex optimization problem is reduced to iteratively reweighted least-squares under Gaussian distribution, where the mean of the snapshots is a sufficient statistic. This approach not only simplifies our solution but also accurately detects the DoAs. We utilize a hierarchical approach for interference cancellation in multi-source scenarios. Empirical evaluations show the superiority of NUV-DoA, especially in low SNRs, compared to alternative DoA estimators.

Deep Learning-Aided Subspace-Based DOA Recovery for Sparse Arrays

Sparse arrays enable resolving more direction of arrivals (DoAs) than antenna elements using non-uniform arrays. This is typically achieved by reconstructing the covariance of a virtual large uniform linear array (ULA), which is then processed by subspace DoA estimators. However, these method assume that the signals are non-coherent and the array is calibrated; the latter often challenging to achieve in sparse arrays, where one cannot access the virtual array elements. In this work, we propose Sparse-SubspaceNet, which leverages deep learning to enable subspace-based DoA recovery from sparse miscallibrated arrays with coherent sources. Sparse- SubspaceNet utilizes a dedicated deep network to learn from data how to compute a surrogate virtual array covariance that is divisible into distinguishable subspaces. By doing so, we learn to cope with coherent sources and miscalibrated sparse arrays, while preserving the interpretability and the suitability of model-based subspace DoA estimators.

Uncertainty Quantification in Deep Learning Based Kalman Filters

Various algorithms combine deep neural networks (DNNs) and Kalman filters (KFs) to learn from data to track in complex dynamics. Unlike classic KFs, DNN-based systems do not naturally provide the error covariance alongside their estimate, which is of great importance in some applications, e.g., navigation. To bridge this gap, in this work we study error covariance extraction in DNN-aided KFs. We examine three main approaches that are distinguished by the ability to associate internal features with meaningful KF quantities such as the Kalman gain (KG) and prior covariance. We identify the differences between these approaches in their requirements and their effect on the training of the system. Our numerical study demonstrates that the above approaches allow DNN-aided KFs to extract error covariance, with most accurate error prediction provided by model-based/data-driven designs.

Joint Communications and Sensing Hybrid Beamforming Design via Deep Unfolding

Joint communications and sensing (JCAS) is envisioned as a key feature in future wireless communications networks. In massive MIMO-JCAS systems, hybrid beamforming (HBF) is typically employed to achieve satisfactory beamforming gains with reasonable hardware cost and power consumption. Due to the coupling of the analog and digital precoders in HBF and the dual objective in JCAS, JCAS-HBF design problems are very challenging and usually require highly complex algorithms. In this paper, we propose a fast HBF design for JCAS based on deep unfolding to optimize a tradeoff between the communications rate and sensing accuracy. We first derive closed-form expressions for the gradients of the communications and sensing objectives with respect to the precoders and demonstrate that the magnitudes of the gradients pertaining to the analog precoder are typically smaller than those associated with the digital precoder. Based on this observation, we propose a modified projected gradient ascent (PGA) method with significantly improved convergence. We then develop a deep unfolded PGA scheme that efficiently optimizes the communications-sensing performance tradeoff with fast convergence thanks to the well-trained hyperparameters. In doing so, we preserve the interpretability and flexibility of the optimizer while leveraging data to improve performance. Finally, our simulations demonstrate the potential of the proposed deep unfolded method, which achieves up to 33.5% higher communications sum rate and 2.5 dB lower beampattern error compared with the conventional design based on successive convex approximation and Riemannian manifold optimization. Furthermore, it attains up to a 65% reduction in run time and computational complexity with respect to the PGA procedure without unfolding.

Joint Communications and Sensing Design for Multi-Carrier MIMO Systems

In conventional joint communications and sensing (JCAS) designs for multi-carrier multiple-input multiple-output (MIMO) systems, the dual-functional waveforms are often optimized for the whole frequency band, resulting in limited communications--sensing performance tradeoff. To overcome the limitation, we propose employing a subset of subcarriers for JCAS, while the communications function is performed over all the subcarriers. This offers more degrees of freedom to enhance the communications performance under a given sensing accuracy. We first formulate the rate maximization under the sensing accuracy constraint to optimize the beamformers and JCAS subcarriers. The problem is solved via Riemannian manifold optimization and closed-form solutions. Numerical results for an 8x4 MIMO system with 64 subcarriers show that compared to the conventional subcarrier sharing scheme, the proposed scheme employing 16 JCAS subcarriers offers 60% improvement in the achievable communications rate at the signal-to-noise ratio of 10 dB. Meanwhile, this scheme generates the sensing beampattern with the same quality as the conventional JCAS design.

Model-Based Deep Learning

Signal processing traditionally relies on classical statistical modeling techniques. Such model-based methods utilize mathematical formulations that represent the underlying physics, prior information and additional domain knowledge. Simple classical models are useful but sensitive to inaccuracies and may lead to poor performance when real systems display complex or dynamic behavior. More recently, deep learning approaches that use deep neural networks are becoming increasingly popular. Deep learning systems do not rely on mathematical modeling, and learn their mapping from data, which allows them to operate in complex environments. However, they lack the interpretability and reliability of model-based methods, typically require large training sets to obtain good performance, and tend to be computationally complex. Model-based signal processing methods and data-centric deep learning each have their pros and cons. These paradigms can be characterized as edges of a continuous spectrum varying in specificity and parameterization. The methodologies that lie in the middle ground of this spectrum, thus integrating model-based signal processing with deep learning, are referred to as model-based deep learning, and are the focus here. This monograph provides a tutorial style presentation of model-based deep learning methodologies. These are families of algorithms that combine principled mathematical models with data-driven systems to benefit from the advantages of both approaches. Such model-based deep learning methods exploit both partial domain knowledge, via mathematical structures designed for specific problems, as well as learning from limited data. We accompany our presentation with running examples, in super-resolution, dynamic systems, and array processing. We show how they are expressed using the provided characterization and specialized in each of the detailed methodologies.

SubspaceNet: Deep Learning-Aided Subspace Methods for DoA Estimation

Direction of arrival (DoA) estimation is a fundamental task in array processing. A popular family of DoA estimation algorithms are subspace methods, which operate by dividing the measurements into distinct signal and noise subspaces. Subspace methods, such as Multiple Signal Classification (MUSIC) and Root-MUSIC, rely on several restrictive assumptions, including narrowband non-coherent sources and fully calibrated arrays, and their performance is considerably degraded when these do not hold. In this work we propose SubspaceNet; a data-driven DoA estimator which learns how to divide the observations into distinguishable subspaces. This is achieved by utilizing a dedicated deep neural network to learn the empirical autocorrelation of the input, by training it as part of the Root-MUSIC method, leveraging the inherent differentiability of this specific DoA estimator, while removing the need to provide a ground-truth decomposable autocorrelation matrix. Once trained, the resulting SubspaceNet serves as a universal surrogate covariance estimator that can be applied in combination with any subspace-based DoA estimation method, allowing its successful application in challenging setups. SubspaceNet is shown to enable various DoA estimation algorithms to cope with coherent sources, wideband signals, low SNR, array mismatches, and limited snapshots, while preserving the interpretability and the suitability of classic subspace methods.

Latent-KalmanNet: Learned Kalman Filtering for Tracking from High-Dimensional Signals

The Kalman filter (KF) is a widely-used algorithm for tracking dynamic systems that are captured by state space (SS) models. The need to fully describe a SS model limits its applicability under complex settings, e.g., when tracking based on visual data, and the processing of high-dimensional signals often induces notable latency. These challenges can be treated by mapping the measurements into latent features obeying some postulated closed-form SS model, and applying the KF in the latent space. However, the validity of this approximated SS model may constitute a limiting factor. In this work, we study tracking from high-dimensional measurements under complex settings using a hybrid model-based/data-driven approach. By gradually tackling the challenges in handling the observations model and the task, we develop Latent-KalmanNet, which implements tracking from high-dimensional measurements by leveraging data to jointly learn the KF along with the latent space mapping. Latent-KalmanNet combines a learned encoder with data-driven tracking in the latent space using the recently proposed-KalmanNet, while identifying the ability of each of these trainable modules to assist its counterpart via providing a suitable prior (by KalmanNet) and by learning a latent representation that facilitates data-aided tracking (by the encoder). Our empirical results demonstrate that the proposed Latent-KalmanNet achieves improved accuracy and run-time performance over both model-based and data-driven techniques by learning a surrogate latent representation that most facilitates tracking, while operating with limited complexity and latency.