This paper investigates the effects of coarse quantization with mixed precision on measurements obtained from sparse linear arrays, synthesized by a collaborative automotive radar sensing strategy. The mixed quantization precision significantly reduces the data amount that needs to be shared from radar nodes to the fusion center for coherent processing. We utilize the low-rank properties inherent in the constructed Hankel matrix of the mixed-precision array, to recover azimuth angles from quantized measurements. Our proposed approach addresses the challenge of mixed-quantized Hankel matrix completion, allowing for accurate estimation of the azimuth angles of interest. To evaluate the recovery performance of the proposed scheme, we establish a quasi-isometric embedding with a high probability for mixed-precision quantization. The effectiveness of our proposed scheme is demonstrated through numerical results, highlighting successful reconstruction.

Recently, one-bit modulo sampling, also known as unlimited one-bit (UNO), has been proposed as a bridge between modulo sampling and coarse quantization. This approach successfully combines the benefits of both techniques by providing efficient, low-cost quantization for modulo sampling while also offering a natural method for designing dithers that are uniformly distributed across the signal’s dynamic range (DR). However, the scheme faces challenges due to the need for multiple dithering sequences, which complicates their generation and implementation. In this paper, we address this issue by applying quantization to the discrete cosine transform (DCT) coefficients of the modulo samples rather than to the samples themselves. This allows us to achieve successful signal reconstruction using only a single dithering sequence, as demonstrated through numerical results.

We explore the impact of coarse quantization on low-rank matrix sensing in the extreme scenario of dithered one-bit sampling, where the high-resolution measurements are compared with random time-varying threshold levels. To recover the low-rank matrix of interest from the highly-quantized collected data, we offer an enhanced randomized Kaczmarz algorithm that efficiently solves the emerging highly-overdetermined feasibility problem. Additionally, we provide theoretical guarantees in terms of the convergence and sample size requirements. Our numerical results demonstrate the effectiveness of the proposed methodology.

One-bit quantization with time-varying sampling thresholds (also known as random dithering) has recently found significant utilization potential in statistical signal processing applications due to its relatively low power consumption and low implementation cost. In addition to such advantages, an attractive feature of one-bit analog-to-digital converters (ADCs) is their superior sampling rates as compared to their conventional multi-bit counterparts. This characteristic endows one-bit signal processing frameworks with what one may refer to assample abundance. We show that sample abundance plays a pivotal role in many signal recovery and optimization problems that are formulated as (possibly non-convex) quadratic programs with linear feasibility constraints. Of particular interest to our work are low-rank matrix recovery and compressed sensing applications that take advantage of one-bit quantization. We demonstrate that the sample abundance paradigm allows for the transformation of such problems to merely linear feasibility problems by forming large-scale overdetermined linear systems—thus removing the need for handling costly optimization constraints and objectives. To make the proposed computational cost savings achievable, we offer enhanced randomized Kaczmarz algorithms to solve these highly overdetermined feasibility problems and provide theoretical guarantees in terms of their convergence, sample size requirements, and overall performance. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.

To facilitate target localization, active radar signals or sequences are designed to have low auto-correlation. This goal is typically achieved by the minimization of the auto-correlation integrated side-lobe level (ISL) metric, or the weighted more general version of ISL, known as the WISL metric. In this work, we propose an efficient approach to WISL minimization for unimodular sequence design that takes advantage of the low-cost and easily implementable power method-like iterations. Several numerical results are presented to illustrate the effectiveness of the proposed method.

One-bit quantization with time-varying sampling thresholds has recently found significant utilization potential in statistical signal processing applications due to its relatively low power consumption and low implementation cost. In addition to such advantages, an attractive feature of one-bit analog-to-digital converters (ADCs) is their superior sampling rates as compared to their conventional multi-bit counterparts. This characteristic endows one-bit signal processing frameworks with what we refer to as sample abundance. On the other hand, many signal recovery and optimization problems are formulated as (possibly non-convex) quadratic programs with linear feasibility constraints in the one-bit sampling regime. We demonstrate, with a particular focus on quadratic compressed sensing, that the sample abundance paradigm allows for the transformation of such quadratic problems to merely a linear feasibility problem by forming a large-scale overdetermined linear system; thus removing the need for costly optimization constraints and objectives. To efficiently tackle the emerging overdetermined linear feasibility problem, we further propose an enhanced randomized Kaczmarz algorithm, called Block SKM. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.

A longstanding problem in statistics pertains to the estimation of probability density functions of continuous random variables from a finite set of their samples. In this paper, we propose a new parametric probability density function estimator based on convex programming. Our formulation decomposes the unknown distribution as a Gaussian penalty function plus an error function, which is then expanded by multi-scale wavelet functions (specifically frames) such as B-Spline and Mexican Hat wavelets. To recover the wavelet coefficients in the error function, a convex quadratic program is formulated which takes into account the positivity of the probability density function-through a linear constraint. The proposed decomposition model is shown to facilitate an accurate estimation of the probability density functions of interest.

Understanding the mathematics and the innate machinery of sparsity measures is instrumental in the proper usage of such information measures in various application arenas, ranging from information collection and sensing, to communications and signal processing. In this paper, the structure of sparsity measures is investigated. Specifically, it is shown that sparsity measures satisfying proper sparsity axioms may only be constructed by vector norms. Moreover, the asymptotic behavior of sparsity measures is studied. Owing to their mathematical structure, our numerical results illustrate a convergence of sparsity measures, as the number of input samples grows large.

One-bit quantization has attracted considerable attention in signal processing for communications and sensing. The arcsine law is a useful relation often used to estimate the normalized covariance matrix of zero-mean stationary input signals when they are sampled by one-bit analog-to-digital converters (ADCs)—practically comparing the signals with a given threshold level. This relation, however, only considers a zero threshold which can cause a remarkable information loss. For the first time in the literature, this paper introduces an approach to extending the arcsine law to the case where one-bit ADCs apply time-varying thresholds. In particular, the proposed method is shown to accurately recover the variance and autocorrelation of the stationary signals of interest.