This survey paper examines recent advancements in low-resolution signal processing, emphasizing quantized compressed sensing. Rising costs and power demands of high-sampling-rate data acquisition drive the interest in quantized signal processing, particularly in wireless communication systems and Internet of Things sensor networks, as 6G aims to integrate sensing and communication within cost-effective hardware. Motivated by this urgency, this paper covers novel signal processing algorithms designed to address practical challenges arising from quantization and modulo operations, as well as their impact on system performance. We begin by introducing the framework of one-bit compressed sensing and discuss relevant theories and algorithms. We explore the application of quantized compressed sensing algorithms to sensor networks, radar, cognitive radio, and wireless channel estimation. We highlight how generic methods can be tailored to an application using specific examples from wireless channel estimation. Additionally, we review other low-resolution techniques beyond one-bit compressed sensing along with their applications. We also provide a brief overview of the emerging concept of unlimited sampling. While this paper does not aim to be exhaustive, it selectively highlights results to inspire readers to appreciate the diverse algorithmic tools (convex optimization, greedy methods, and Bayesian approaches) and sampling techniques (task-based quantization and unlimited sampling).

We study the sparse recovery problem with an underdetermined linear system characterized by a Kronecker-structured dictionary and a Kronecker-supported sparse vector. We cast this problem into the sparse Bayesian learning (SBL) framework and rely on the expectation-maximization method for a solution. To this end, we model the Kronecker-structured support with a hierarchical Gaussian prior distribution parameterized by a Kronecker-structured hyperparameter, leading to a non-convex optimization problem. The optimization problem is solved using the alternating minimization (AM) method and a singular value decomposition (SVD)-based method, resulting in two algorithms. Further, we analytically guarantee that the AM-based method converges to the stationary point of the SBL cost function. The SVD-based method, though it adopts approximations, is empirically shown to be more efficient and accurate. We then apply our algorithm to estimate the uplink wireless channel in an intelligent reflecting surface-aided MIMO system and extend the AM-based algorithm to address block sparsity in the channel. We also study the SBL cost to show that the minima of the cost function are achieved at sparse solutions and that incorporating the Kronecker structure reduces the number of local minima of the SBL cost function. Our numerical results demonstrate the effectiveness of our algorithms compared to the state-of-the-art.

This work studies the problem of jointly estimating unknown parameters from Kronecker-structured multidimensional signals, which arises in applications like intelligent reflecting surface (IRS)-aided channel estimation. Exploiting the Kronecker structure, we decompose the estimation problem into smaller, independent subproblems across each dimension. Each subproblem is posed as a sparse recovery problem using basis expansion and solved using a novel off-grid sparse Bayesian learning (SBL)-based algorithm. Additionally, we derive probabilistic error bounds for the decomposition, quantify its denoising effect, and provide convergence analysis for off-grid SBL. Our simulations show that applying the algorithm to IRS-aided channel estimation improves accuracy and runtime compared to state-of-the-art methods through the low-complexity and denoising benefits of the decomposition step and the high-resolution estimation capabilities of off-grid SBL.

We tackle the problem of estimating a binary occupancy grid map by fusing point cloud data from LiDAR and radar sensors for automotive driving perception. To this end, we introduce two sparsity measurement models for fusion, formulating occupancy mapping as a sparse binary vector reconstruction problem. The first model jointly estimates a common map from all measurements, while the second assumes a shared map and an innovation component for each modality's measurements. We use the pattern-coupled sparse Bayesian learning algorithm to recover maps, leveraging the inherent sparsity and spatial dependencies in automotive occupancy maps. Numerical experiments on the RADIATE public dataset show that our fusion-based approach improves mapping accuracy compared to single-modality and high-level fusion mapping algorithms.

Millimeter-wave radar is a common sensor modality used in automotive driving for target detection and perception. These radars can benefit from side information on the environment being sensed, such as lane topologies or data from other sensors. Existing radars do not leverage this information to adapt waveforms or perform prior-aware inference. In this paper, we model the side information as an occupancy map and design transmit beamformers that are customized to the map. Our method maximizes the probability of detection in regions with a higher uncertainty on the presence of a target. Simulation results on the nuScenes dataset show that the designed beamformer achieves substantially higher detection rates than a conventional omnidirectional beamformer for the same transmitted power.

The convergence of expectation-maximization (EM)-based algorithms typically requires continuity of the likelihood function with respect to all the unknown parameters (optimization variables). The requirement is not met when parameters comprise both discrete and continuous variables, making the convergence analysis nontrivial. This paper introduces a set of conditions that ensure the convergence of a specific class of EM algorithms that estimate a mixture of discrete and continuous parameters. Our results offer a new analysis technique for iterative algorithms that solve mixed-integer non-linear optimization problems. As a concrete example, we prove the convergence of an existing EM-based sparse Bayesian learning algorithm that estimates the state of a linear dynamical system with jointly sparse inputs and bursty missing observations. Our results establish that the algorithm converges to the set of stationary points of the maximum likelihood cost with respect to the continuous optimization variables.

This monograph presents some exciting and new results on the analysis and design of control of discrete-time linear dynamical systems using sparse actuator control. Sparsity constraints arise naturally in the inputs of several linear systems due to limited resources or the underlying physics. The monograph deals with two types of sparsity constraints: time-varying and time-invariant supported sparse control inputs. It first provides a detailed theoretical discussion on controllability under sparsity constraints, including algebraic necessary and sufficient conditions for ensuring controllability. Several related formulations, covering stabilizability, output controllability, and nonnegative controllability under sparsity constraints, are also presented. Further, for sparsely controllable systems, the monograph describes two efficient, systematic, and rigorous approaches to designing sparse control inputs: compressed sensing algorithms and spare actuator scheduling algorithms. Overall, the concepts covered in the monograph provide various sparsity models, algorithms, and analysis tools that are readily accessible to systems and control, signal processing, and applied mathematics readers.

This letter considers the design of sparse actuator schedules for linear time-invariant systems. An actuator schedule selects, for each time instant, which control inputs act on the system in that instant. We address the optimal scheduling of control inputs under a hard constraint on the number of inputs that can be used at each time. For a sparsely controllable system, we characterize sparse actuator schedules that make the system controllable, and then devise a greedy selection algorithm that guarantees controllability while heuristically providing low control effort. We further show how to enhance our greedy algorithm via Markov chain Monte Carlo-based randomized optimization.

Occupancy grid maps provide information about obstacles and available free space in the environment and are crucial in automotive driving applications. An occupancy map is constructed using point cloud data from sensor modalities such as light detection and ranging (LiDAR) and radar used for automotive perception. In this article, we formulate the problem of estimating the occupancy grid map using sensor point cloud data as a sparse binary occupancy value reconstruction problem. Our proposed occupancy grid estimation method, based on pattern-coupled sparse Bayesian learning (PC-SBL), leverages the sparsity and spatial dependencies inherent in occupancy maps typically encountered in automotive scenarios. The proposed method shows enhanced detection capabilities compared to two benchmark methods based on performance evaluation with scenes from the nuScenes and RADIal public datasets.

We consider the control of discrete-time linear dynamical systems using sparse inputs where we limit the number of active actuators at every time step. We develop an algorithm for determining a sparse actuator schedule that ensures the existence of a sparse control input sequence, following the schedule, that takes the system from any given initial state to any desired final state. Since such an actuator schedule is not unique, we look for a schedule that minimizes the energy of sparse inputs. For this, we optimize the trace of the inverse of the resulting controllability Gramian, which is an approximate measure of the average energy of the inputs. We present a greedy algorithm along with its theoretical guarantees. Finally, we empirically show that our greedy algorithm ensures the controllability of the linear system with a small number of active actuators per time step without a significant average energy expenditure compared to the fully actuated system.

We consider the problem of jointly estimating the states and sparse inputs of a linear dynamical system using noisy low-dimensional observations. We exploit the underlying sparsity in the inputs using fictitious sparsity-promoting Gaussian priors with unknown variances (as hyperparameters). We develop two Bayesian learning-based techniques to estimate states and inputs: sparse Bayesian learning and variational Bayesian inference. Through numerical simulations, we illustrate that our algorithms outperform the conventional Kalman filtering based algorithm and other state-of-the-art sparsity-driven algorithms, especially in the low-dimensional measurement regime.

This paper studies the problem of modifying the input matrix of a structured system to make the system strongly structurally controllable. We focus on the generalized structured systems that rely on zero/nonzero/arbitrary structure, i.e., some entries of system matrices are zeros, some are nonzero, and the remaining entries can be zero or nonzero (arbitrary). We derive the feasibility conditions of the problem, and if it is feasible, we reformulate it into another equivalent problem. This new formulation leads to a greedy heuristic algorithm. However, we also show that the greedy algorithm can give arbitrarily poor solutions for some special systems. Our alternative approach is a randomized Markov chain Monte Carlo-based algorithm. Unlike the greedy algorithm, this algorithm is guaranteed to converge to an optimal solution with high probability. Finally, we numerically evaluate the algorithms on random graphs to show that the algorithms perform well.

This paper develops a channel estimation technique for millimeter wave (mmWave) communication systems. Our method exploits the sparse structure in mmWave channels for low training overhead and accounts for the phase errors in the channel measurements due to phase noise at the oscillator. Specifically, in IEEE 802.11ad/ay-based mmWave systems, the phase errors within a beam refinement protocol packet are almost the same, while the errors across different packets are substantially different. Consequently, standard sparsity-aware algorithms, which ignore phase errors, fail when channel measurements are acquired over multiple beam refinement protocol packets. We present a novel algorithm called partially coherent matching pursuit for sparse channel estimation under practical phase noise perturbations. Our method iteratively detects the support of sparse signal and employs alternating minimization to jointly estimate the signal and the phase errors. We numerically show that our algorithm can reconstruct the channel accurately at a lower complexity than the benchmarks.

We tackle the anomaly detection problem within a given set of binary processes through a learning-based controlled sensing approach. This problem is particularly pertinent to applications related to the Internet of Things (IoT) that monitor multiple related processes. Each process is defined by a binary random variable indicating its anomalous status. To pinpoint anomalies, a decision-making agent observes a subset of processes at each time point, with each observation incurring an associated cost. We formulate a sequential selection policy that dynamically determines which processes to observe at each moment, aiming to minimize both decision delay and sensing cost. The conventional solution using model-based active hypothesis testing algorithms overlooks the joint statistics of the processes and fails to consider unequal sensing costs or errors in observations. To solve this problem, we, for the first time, pose it as a sequential hypothesis testing problem within the framework of Markov decision processes (MDPs), leveraging both a Bayesian log-likelihood ratio-based reward and an entropy-based reward. We address this problem through two approaches: 1) a deep reinforcement learning-based method involving the design of both deep Q-learning (DQN) and policy gradient actor-critic (AC) algorithms and 2) a deep active inference (AI)-based approach. Our model-based posterior updates to tackle the uncertainties in the observations, combined with the data-driven neural networks to handle the underlying statistical dependence between the processes, strike a balance between the model-based and the data-driven approaches. Our numerical experiments showcase the effectiveness of our algorithms, illustrating their capability to adapt to any unknown statistical dependence among the processes.

We address the problem of sequentially selecting and observing processes from a given set to find the anomalies among them. The decision-maker observes a subset of the processes at any given time instant and obtains a noisy binary indicator of whether or not the corresponding process is anomalous. We develop an anomaly detection algorithm that chooses the processes to be observed at a given time instant, decides when to stop taking observations, and declares the decision on anomalous processes. The objective of the detection algorithm is to identify the anomalies with an accuracy exceeding the desired value while minimizing the delay in decision making. We devise a centralized algorithm where the processes are jointly selected by a common agent as well as a decentralized algorithm where the decision of whether to select a process is made independently for each process. Our algorithms rely on a Markov decision process defined using the marginal probability of each process being normal or anomalous, conditioned on the observations. We implement the detection algorithms using the deep actor-critic reinforcement learning framework. Unlike prior work on this topic that has exponential complexity in the number of processes, our algorithms have computational and memory requirements that are both polynomial in the number of processes. We demonstrate the efficacy of these algorithms using numerical experiments by comparing them with state-of-the-art methods.

In this article, we study the conditions to be satisfied by a discrete-time linear system to ensure output controllability using sparse control inputs. A set of necessary and sufficient conditions can be directly obtained by extending the Kalman rank test for output controllability. However, the verification of these conditions is computationally heavy due to their combinatorial nature. Therefore, we derive noncombinatorial conditions for output sparse controllability that can be verified with polynomial time complexity. Our results also provide bounds on the minimum sparsity level required to ensure output controllability of the system. This additional insight is useful for designing sparse control input that drives the system to any desired output.

The problem of estimating occupancy grids to support automotive driving applications using LiDAR sensor point clouds is considered. We formulate the problem as a sparse binary occupancy value reconstruction problem. Our proposed occupancy grid estimation method is based on pattern-coupled sparse Bayesian learning and exploits the inherent sparsity and spatial occupancy dependencies in LiDAR sensor measurements. The proposed method demonstrates enhanced detection capabilities compared to commonly used benchmark methods, as observed through testing on scenes from the nuScenes dataset.

This paper presents novel cascaded channel estimation techniques for an intelligent reflecting surface-aided multiple-input multiple-output system. Motivated by the channel angular sparsity at higher frequency bands, the channel estimation problem is formulated as a sparse vector recovery problem with an inherent Kronecker structure. We solve the problem using the sparse Bayesian learning framework which leads to a non-convex optimization problem. We offer two solution techniques to the problem based on alternating minimization and singular value decomposition. Our simulation results illustrate the superior performance of our methods in terms of accuracy and run time compared with the existing works.

The emergence of compressive sensing and the associated ℓ₁ recovery algorithms and theory have generated considerable excitement and interest in their applications. This chapter will examine recent developments and a complementary set of tools based on a Bayesian framework to address the general problem of sparse signal recovery and the challenges associated with it. Bayesian methods offer superior performance compared to convex optimization-based methods and are parameter tuning-free. They also have the flexibility necessary to deal with a diverse range of measurement modalities and structured sparsity in signals than hitherto possible. Parsimonious signal representation using overcomplete dictionaries for compression, estimation of sparse communication channels with large delay spread as in underwater acoustics, low-dimensional representation of MIMO wireless channels, brain imaging techniques, such as MEG and EEG, are a few examples. We provide a mathematically rigorous and in-depth overview of this fascinating area within sparse signal recovery. We highlight the generality and flexibility of Bayesian approaches and show how it greatly facilitates their deployment in communications-related applications, even though they generally lead to nonconvex optimization problems. Further, we show that, by reinterpreting the Bayesian cost function as a technique to perform covariance matching, one can develop new, ultrafast Bayesian algorithms for sparse signal recovery. As an example application, we discuss the utility of these algorithms in the context of 5G communications with several case studies including wideband time-varying channel estimation and low-resolution analog-to-digital conversion-based signal recovery.

In this paper, we consider the problem of estimating the states of a linear dynamical system whose inputs are jointly sparse and outputs at a few unknown time instants are missing. We model the missing data mechanism using a Markov chain with two states representing the missing and non-missing data. This mechanism with memory governed by the Markov chain models intermittent outages due to communication channels and occlusions corresponding to moving objects. We rely on the sparse Bayesian learning framework to derive an estimation algorithm that uses Kalman smoothing to handle temporal correlation and the Viterbi algorithm to handle missing data. Further, we demonstrate the utility of our algorithm by applying it to the frequency division duplexed multiple input multiple output downlink channel estimation problem.