Cramer-Rao Research Articles

Enhanced Automotive Sensing Assisted by Joint Communication and Cognitive Sparse MIMO Radar

Automotive radar is mandated to provide high accuracy direction of arrival (DOA) estimation for safe driving, while remaining a low cost device for feasible mass production. Sparse multiple input multiple output (MIMO) arrays emerge as a primary candidate to meet these requirements. As DOA estimation accuracy is a main indicator of tracking performance, Cramer Rao Bound (CRB) is chosen as the goodness measurement for sparse MIMO array optimization, but its application requires prior information of the road environment. We propose a cognitive sparse MIMO array automotive radar, which “perceives” the road environment via automotive sensing supplemented by co-existed communication from roadside unit to vehicle (R2V). This information is used for co-designing a sparse MIMO array for enhanced automotive sensing and vehicle to roadside unit (V2R) communication. Note that both static RSU and dynamic RSU are usually deployed in Internet of Vehicles (IoV), which can provide continuous transmission coverage and permanent connectivity. The bi-directional communications are integrated into the automotive radar. This is achieved by joint transmit waveform design with spectral nulls for communications and with shared sparse MIMO array co-design for both sensing and high quality V2R communications. Simulation results validate the enhanced automotive sensing performance assisted by the integrated bi-directional communications in the cycle of cognitive-driven optimization.

Hybrid Reconfigurable Intelligent Surfaces-Assisted Near-Field Localization

Reconfigurable intelligent metasurface (RIS) assisted user localization has been a hot topic along with the development of the sixth generation wireless communications. The employment of large-scale antennas goes far beyond the improvement of spectral efficiency, resulting in near-field communications between users and the RIS. In this letter, we consider to employ the hybrid RIS (HRIS) which is capable of simultaneous reflection and reception in near-field user localization for the first time. With such a new architecture, we derive the Cramer-Rao lower bound (CRLB) for assessing its localization performance, and subsequently propose an automatic differentiation-based gradient descent approach to optimize the HRIS configuration. The theoretical analysis is further supported by numerical simulations, indicating the superiority of the HRIS over the conventional purely reflective RIS in near-field user localization.

Bistatic MIMO DFRC System Waveform Design via Fractional Programming

The Cramér-Rao lower bound (CRLB) and sum-rate (SR) are widely-used metrics in the radar parameter estimation performance evaluation and communication quality-of-service (QoS) assessment, respectively. In this paper, we adopt these two metrics in waveform design for bistatic multiple-input multiple-output dual-function radar-communication (MIMO DFRC) systems, which is different from existing methods where only suboptimal optimization criteria are employed. However, both of the design formulations result in nonconvex and nonlinear fractional programming (FP) problems with complex fractional constraint and objective function. Especially, the CRLB for direction-of-departure (DOD) and direction-of-arrival (DOA) in bistatic DFRC MIMO system is a fraction with quadratic polynomial in the numerator and quartic polynomial in the denominator; whereas the SR expression corresponds to a summation-log-fraction function. By fractional function transformation and convexification, complex function design and simplification, and summation-log-fraction function separation, this paper derives an FP framework to solve these complicated problems. Numerical results demonstrate the excellent performance of our design solutions.

Low-Rank Room Impulse Response Estimation

In this paper we consider low-rank estimation of room impulse responses (RIRs). Inspired by a physics-driven room-acoustical model, we propose an estimator of RIRs that promotes a low-rank structure for a matricization, or reshaping, of the estimated RIR. This low-rank prior acts as a regularizer for the inverse problem of estimating an RIR from input-output observations, preventing overfitting and improving estimation accuracy. As directly enforcing a low rank of the estimate results is an NP-hard problem, we consider two different relaxations, one using the nuclear norm, and one using the recently introduced concept of quadratic envelopes. Both relaxations allow for implementing the proposed estimator using a first-order algorithm with convergence guarantees. When evaluated on both synthetic and recorded RIRs, it is shown that under noisy output conditions, or when the spectral excitation of the input signal is poor, the proposed estimator outperforms comparable existing methods. The performance of the two low-rank relaxations methods is similar, but the quadratic envelope has the benefit of superior robustness to the choice of regularization hyperparameter in the case when the signal-to-noise ratio is unknown. The performance of the proposed method is compared to that of ordinary least squares, Tikhonov least squares, as well as the Cramér-Rao lower bound (CRLB).

Reconfigurable Intelligent Surface Aided Position and Orientation Estimation Based on Joint Beamforming With Limited Feedback

The sparsity of millimeter wave (mmWave) channels in angular and temporal domains has been exploited for channel estimation, while the associated channel parameters can be utilized for localization. However, line-of-sight (LoS) blockage makes localization highly challenging, which may lead to big positioning inaccuracy. One promising solution is to employ reconfigurable intelligent surfaces (RIS) to generate virtual line-of-sight (VLoS) paths. Hence, it is essential to investigate the wireless positioning in RIS-aided mmWave systems. In this paper, an adaptive joint LoS and VLoS localization scheme is proposed, where the VLoS is constructed by a beamforming protocol operated between RIS and mobile station (MS). More specifically, to sense the location and orientation of a MS, a novel interlaced scanning beam sweeping algorithm is proposed to acquire the optimal beams. In this algorithm, LoS and VLoS paths are separately estimated and the optimal beams are selected according to the received signal strength so as to mitigate the LoS blockage problem. Then, based on the selected beams and received signal strength, angle of arrival (AoA), angle of departure (AoD), angle of reflection (AoR) and time of arrival (ToA) are estimated. Finally, with the aid of these estimated parameters, the location and orientation of the MS are estimated. We derive the Cramer-Rao lower bounds (CRLBs) for both location estimation and orientation estimation, and compare them with the corresponding results obtained from simulations. We compare the performance of our proposed scheme with that attained by three legacy schemes, when various aspects are considered. The performance results show the superiority of our proposed beam training algorithm, which is capable of achieving a localization error within 15 cm and an orientation error within 0.003 rads. Furthermore, the training overhead is 100 times less than that of the conventional exhaustive search algorithm, while obtaining a 13 dBm power gain when compared with the hierarchical codebook based search algorithm.

Joint Localization and Environment Sensing of Rigid Body With 5G Millimeter Wave MIMO

Accurately localizing a target in three-dimensional (3D) space assisted with the fifth generation (5G) wireless systems in an indoor environment could enable a wide variety of new applications, including precise control for factory automation, self-maneuver of the vehicle and so on. However, controlling a target in 3D space relies on a rigid body modelling with six degrees of freedom, which dramatically increases the localization difficulty and complexity. Furthermore, for radio-based localization methods, the lack of line-of-sight (LOS) and the existence of reflection points in the environment will also influence the rigid body localization process. To improve the rigid body localization accuracy as well as unravel useful environmental information from the received signal, a novel rigid body joint active localization and environment sensing scheme is proposed in this paper. Specifically, the multi-path effect of millimeter wave (mm-wave) signal with a single reflection can be exploited to enhance the rigid body localization accuracy, and it can also be utilized to locate the reflection points, which further enables a new way for environmental sensing. Hence, we first propose a two-step hierarchical compressive sensing algorithm to extract the angular and distance information of the LOS (if available) and single-bounce specular reflections. Then a particle swarm optimization (PSO) based method is derived to recover the posture of the rigid body and the location of reflection points. The Cramér-Rao lower bound (CRLB) on angle, rigid body posture and reflection points position uncertainty is also analyzed. The simulation results demonstrate that the proposed scheme can achieve high accuracy rigid body localization and locate the reflection points around the rigid body even under obstructed-line-of-sight (OLOS) conditions in an indoor scene.

Near-Field Localization for Holographic RIS Assisted mmWave Systems

In this letter, we envision the integration of holographic reconfigurable intelligent surface (HRIS) into a millimeter-wave (mmWave) localization system. The large aperture of HRIS in the mmWave band allows the system to operate in the near-field propagation regime, where the wavefront tends to be spherical and contains distance information. To investigate the fundamental limits of localization accuracy of the proposed system, we obtain the Fisher information matrix (FIM) and Cramér-Rao lower bound (CRLB) taking into account the radiation patterns of antennas. The analytical results reveal that the FIM quadratically increases with the size of HRIS. Then, to explore the potential of HRIS passive beamforming for improving localization quality, we formulate the HRIS phases optimization problem to minimize a worst-case CRLB with respect to an uncertainty set of user equipment (UE) positions. To address this formulated problem, an iterative entropy regularization (IER) based method is proposed. Numerical results evaluate the effect of key parameters on the squared position error bound and verify the effectiveness of the proposed IER-based method.

Accuracy of UWB Path Loss-Based Localization of Wireless Capsule Endoscopy.

Wireless capsule endoscopy (WCE) is used to diagnose lesions in the gastrointestinal (GI) tract. The physicians require to know the exact position of the lesions which can be performed by localizing the WCE in the GI tract. In this paper, we propose ultra-wideband (UWB) path loss-based WCE localization and compute the Cramer-Rao lower bound (CRLB) to evaluate the accuracy bounds of localization in the small intestine. First, we propose the estimation of smoothed path loss by minimizing the path loss deviations caused by shadow fading effects of body tissues. Then, the estimated path loss is used to estimate the degree of path loss and to compute the weight of the sensor's positions. Finally, we propose the smoothed path loss degree-based weighted centroid localization (SPLD-WCL) algorithm to estimate the location of the WCE. We simulate the proposed SPLD-WCL algorithm and verify the accuracy by comparing it to the computed CRLB. The proposed SPLD-WCL localization algorithm shows a significantly high accuracy of localization with a 6.83 mm root mean square error (RMSE) without any advanced knowledge of unknown parameters and bounds.

Multistatic Localization by Differential Time Delays and Time Differences of Arrival in the Absence of Transmitter Position

We can only extract the differential time delay (DTD) measurements between the direct and reflected paths in multistatic localization when there is no synchronization in time between the transmitter and receiver and among the receivers. This paper first addresses the problem of multistatic localization of a fixed object when the transmitter position is not available by using the DTD measurements. We propose a two-step optimization method for jointly estimating the object and transmitter positions. In the first step, we formulate a non-convex constrained weighted least squares (CWLS) problem by transforming the DTD measurement model and introducing nuisance variables. Such a non-convex CWLS problem is then relaxed to a tractable convex semidefinite programming (SDP) problem by applying semidefinite relaxation. In the second step, the error coming from relaxation and approximation in the SDP solution is reduced iteratively through solving a generalized trust region subproblem (GTRS) in each iteration. If the receivers are synchronized such that the time difference of arrival (TDOA) measurements can be acquired in addition to DTD, we formulate a different CWLS problem by utilizing both DTD and TDOA measurements, which is solved by convex relaxation as well. The relaxed SDP problem can achieve the optimal solution of the CWLS problem, and further refinement is no longer needed. We conduct the mean square error (MSE) analysis to validate that both proposed methods are able to achieve the Cramer-Rao Lower Bound (CRLB) performance under small Gaussian noise, which is also validated by simulations.

Direct Target Localization With Quantized Measurements in Noncoherent Distributed MIMO Radar Systems

In this paper, a direct target localization algorithm with quantized measurements for non-coherent MIMO systems is proposed. In this system, each receiver transmits a low-bit quantized version of the echo rather than the full raw data to the fusion center. We construct a joint likelihood function based on the low-bit data of each receiver, from which the unknown target position can be directly determined. The Cramer-Rao lower bound (CRLB) is derived to analyze the localization performance of our proposed algorithm. To maximize the localization performance, a CRLB-based objective function is designed to obtain the optimum quantization thresholds. The formulated problem is a high-dimensional and non-convex optimization problem that when solved, determines two types of coupled parameters for the quantization thresholds and the complex-valued scaling coefficients of the signal. We propose a batch gradient descent embedded particle swarm optimization algorithm to solve this problem effectively. Numerical results show that the proposed algorithm delivers superior performance in terms of maximizing the overall localization performance, and the 3-bit quantized algorithm is able to provide performance that is very close to the unquantized algorithm. Experimental data recorded by three small radars are also provided to demonstrate the effectiveness of the proposed algorithm.

Dynamic and Robust Sensor Selection Strategies for Wireless Positioning With TOA/RSS Measurement

Emerging wireless applications are requiring ever more accurate location-positioning from sensor measurements. In this paper, we develop sensor selection strategies for 3D wireless positioning based on time of arrival (TOA) and received signal strength (RSS) measurements to handle two distinct scenarios: (i) known approximated target location, for which we conduct dynamic sensor selection to minimize the positioning error; and (ii) unknown approximated target location, in which the worst-case positioning error is minimized via robust sensor selection. We derive expressions for the Cramér-Rao lower bound (CRLB) as a performance metric to quantify the positioning accuracy resulted from selected sensors. For dynamic sensor selection, two greedy selection strategies are proposed, each of which exploits properties revealed in the derived CRLB expressions. These selection strategies are shown to strike an efficient balance between computational complexity and performance suboptimality. For robust sensor selection, we show that the conventional convex relaxation approach leads to instability, and then develop three algorithms based on (i) iterative convex optimization (ICO), (ii) difference of convex functions programming (DCP), and (iii) discrete monotonic optimization (DMO). Each of these strategies exhibits a different tradeoff between computational complexity and optimality guarantee. Simulation results show that the proposed sensor selection strategies provide significant improvements in terms of accuracy and/or complexity compared to existing sensor selection methods.

Quantum Fisher information of multi-qubit WV entangled state under Lipkin-Meshkov-Glick model

As an important quantity in the field of parameter estimation theory and quantum precision measurement, quantum Fisher information (QFI) can not only be used to set the theoretical limit of measurement precision in quantum system, but also be exploited to witness metrological useful quantum entanglement. Recently, it has also been broadly used in many aspects of quantum information science, including quantum metrology, multipartite entanglement structure detection, quantum phase transition, quantum chaos, quantum computation and etc. In this work, from the perspective of quantum measurement, we study the quantum Fisher information of an <i>N</i>-qubit WV state (<inline-formula><tex-math id="M4">\begin{document}$\alpha \left\vert W_N \right\rangle +\sqrt{1-\alpha^2}\left\vert 00\cdots0\right\rangle$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231179_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231179_M4.png"/></alternatives></inline-formula>) under local operation and Lipkin-Meshkov-Glick (LMG) model. Furthermore, with the general Cramér-Rao lower bound (CRLB) we analyze its performance in high-precision phase measurement. The results show that, under the local operation, the QFI of an <i>N</i>-qubit WV state becomes larger with the increase of parameter <i>α</i>. This not only means the enhanced quantum entanglement, but also implies the powerful ability in high-precision quantum measurement. In the LMG model, as the increase of interactional strength <i>γ</i> the QFI of <inline-formula><tex-math id="M5">\begin{document}$N=3$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231179_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231179_M5.png"/></alternatives></inline-formula> qubits WV state gradually tends to be stable and almost not be affected by parameter <i>α</i>, which relaxes the requirement in the preparation of target state and indicates a great potential in achieving the relatively stable measurement precision. When the number of qubits from WV state is larger than 3, the QFI of WV state increases with the increase of parameter <i>α</i>. In the case of fixed parameter <i>α</i>, we investigate the QFI of an <i>N</i>-qubit WV state with respect to interaction strength <i>γ</i>. It is found that the QFI of WV state will increase with the increasing interaction strength, which implies that the greater the interaction strength, the stronger the quantum measurement ability of the WV state. Our work will promote the development of high-precision quantum metrology and especially the interaction-enhanced quantum measurement, and further provide new insights in quantum information processing.

Localization of Multiple Directional Transmitters in Cognitive Radio Context

Localization of transmitters has been gaining interest increasingly. Due to the emergence of directional transmitters in future technologies, localization in this manner is particularly considered. However, limited studies consider the localization of multiple directional transmitters. Our proposed system model describes multiple directional emitters by the means of a Uniform Linear Array (ULA). To localize these transmitters, we adapt the classical Direction of Arrival (DoA) localization techniques with our model. The localization accuracy is studied for multiple affecting parameters and compared to the theoretical limit of Cramer Rao Bound (CRB). The results show that a directional transmitter can be localized by the means of its high sidelobe effects. Root MUSIC outperforms other techniques in localizing the users.

A Bias-Reduced Solution for Multistatic Localization Using Differential Delays and Doppler Shifts

In this paper, the multistatic localization problem with unknown propagation speed is investigated using differential delays and Doppler shifts between the signals from direct and indirect paths. A series of pseudo-linear equations are formulated via the transformation of measurement models. A weighted least squares (WLS) formulation is then proposed after ignoring the second-order error terms, which can be rewritten as a non-convex optimization problem with the relationships among variables included as constraints. To deal with the non-convexity of the problem, semidefinite relaxation is applied, resulting in a convex semidefinite program (SDP). Several reasonable second-order cone constraints constructed via basic inequality and Cauchy-Schwarz inequality are added to tighten the relaxed SDP problem. By preserving the second-order error terms in equations, the bias of the estimate from the WLS formulation is also derived and then subtracted to nearly eliminate the bias and reach a bias-reduced solution. Simulation results show that the mean square error (MSE) of the proposed method approaches the Cramer-Rao lower bound (CRLB) and the bias is reduced significantly.

RAPP: A Radio Tomography Localization Method Characterized by Performance Parameterization in Rapid-Moving RFID System

Exploring and developing the perception system is an important means to promote the development of intelligent transportation. In this paper, we propose RAPP, a radio tomography-based method enabling performance parameterization to achieve parameter adjustment of localization accuracy without dense deployment of passive UHF RFID devices. The reader antenna takes a rapid-moving system as the carrier and generates virtual antenna elements to inventory received signal strength (RSS) of the pre-deployed tags. To eliminate the impact of the lack of tag backward signals during the mobile process and supplement the link distribution for each segment of the monitored platform, the Cubic Spline Interpolation algorithm (CSIA) is proposed to homogenize the spacing of the virtual antenna elements by preprocessing the measurements. Then the optimized link information is adopted to perform radio tomographic imaging to recognize the locations of targets. Theoretical analysis on ill-condition is derived to analyze how the number of virtual interpolation elements affects the localization accuracy. Further derivation about the relationship between the Bayesian Cramér-Rao Bound (CRB) and the link density provides a solution to influence the localization performance through parameter adjustment. Extensive simulation and experiment show that RAPP outperforms on multiple targets localization and behaves robustly in a multipath environment.

Monostatic Sensing With OFDM Under Phase Noise: From Mitigation to Exploitation

We consider the problem of monostatic radar sensing with orthogonal frequency-division multiplexing (OFDM) joint radar-communications (JRC) systems in the presence of phase noise (PN) caused by oscillator imperfections. We begin by providing a rigorous statistical characterization of PN in the radar receiver over multiple OFDM symbols for free-running oscillators (FROs) and phase-locked loops (PLLs). Based on the delay-dependent PN covariance matrix, we derive the hybrid maximum-likelihood (ML)/maximum a-posteriori (MAP) estimator of the deterministic delay-Doppler parameters and the random PN, resulting in a challenging high-dimensional nonlinear optimization problem. To circumvent the nonlinearity of PN, we then develop an iterated small angle approximation (ISAA) algorithm that progressively refines delay-Doppler-PN estimates via closed-form updates of PN as a function of delay-Doppler at each iteration. Moreover, unlike existing approaches where PN is considered to be purely an impairment that has to be mitigated, we propose to exploit PN for resolving range ambiguity by capitalizing on its delay-dependent statistics (i.e., the range correlation effect), through the formulation of a parametric Toeplitz-block Toeplitz covariance matrix reconstruction problem. Simulation results indicate quick convergence of ISAA to the hybrid Cramér-Rao bound (CRB), as well as its remarkable performance gains over state-of-the-art benchmarks, for both FROs and PLLs under various operating conditions, while showing that the detrimental effect of PN can be turned into an advantage for sensing.

Resolving Doppler Ambiguity for Fast-moving Targets with FDA-MIMO Radar

Although the problem of range ambiguity is solved with the frequency diverse array (FDA)-multiple-input multiple-output (MIMO) radar by utilizing the degree-of-freedom (DOFs) in the range domain, the phenomenon of Doppler ambiguity occurs for fast-moving target. In this article, an offset frequency alignment (OFA) method is proposed to solve the problem of Doppler ambiguity in FDA-MIMO radar. At the design stage, the Doppler frequency offsets, corresponding to different transmit pulses, are aligned by eliminating the residual phases. In this regard, the point targets can be focused in the transmit spatial frequency domain. Furthermore, the Doppler ambiguity index is estimated after principal velocity compensation resorting to maximum likelihood (ML) criterion, where both the cases of known and unknown range ambiguity indexes are investigated respectively. Moreover, in order to ensure the estimation performance, the frequency offset across the transmit array elements is designed by enlarging the term related to the Doppler frequency offset. At the analysis stage, the performance of Doppler ambiguity index estimation in terms of root mean square error (RMSE) is assessed in comparison with the Cramer-Rao bound (CRB). Numerical results are provided to verify the effectiveness of the proposed method to resolve the Doppler ambiguity.

Advances in UAV-Assisted Localization: Joint Source and UAV Parameter Estimation

This paper presents efficient estimators to address the problem of source localization aided by unmanned aerial vehicles (UAVs) with unknown location parameters. Leveraging semidefinite relaxation technique and successive weighted least squares estimation, the source and UAV parameters are simultaneously estimated in both mixed line-of-sight/non-line-of-sight (NLOS) and NLOS scenarios. The proposed estimators are shown to approximately achieve the Cramer-Rao bound (CRB) under mild Gaussian noise conditions when the measurement noises are small compared to the associated ranges. Numerical simulations demonstrate that localization without prior UAV location information is still possible with a meter-level positioning and sub-meter per seconds velocity estimation accuracy.

CRLB Analysis for Passive Sensor Network Localization Using Intelligent Reconfigurable Surface and Phase Modulation

Passive wireless sensor network (PWSN) requires high positioning for network management. The harvested energy of the passive sensor is modulated as the ranging data and the position is derived accordingly. Thus, the wireless power transfer (WPT) is a dominant factor for such localization. With the help of intelligent reconfigurable surface (IRS), the WPT efficiency can be significantly improved. In this paper, we propose the Fisher information matrix (FIM) and the Cramér–Rao lower bound (CRLB) analyzing model of the PWSN localization. We prove the impacts of phase modulation of IRS on the localization performance. Based on our analysis, we develop an approximation algorithm and a genetic algorithm to control the IRS phases. Then, the localization accuracy of PWSN can be further improved. The simulation results demonstrate that the phase modulation based on GA can achieve high accurate localization for PWSN using IRS.

Cramér–Rao Bounds for DoA Estimation of Sparse Bayesian Learning with the Laplace Prior

In this paper, we derive the Cramér-Rao lower bounds (CRLB) for direction of arrival (DoA) estimation by using sparse Bayesian learning (SBL) and the Laplace prior. CRLB is a lower bound on the variance of the estimator, the change of CRLB can indicate the effect of the specific factor to the DoA estimator, and in this paper a Laplace prior and the three-stage framework are used for the DoA estimation. We derive the CRLBs under different scenarios: (i) if the unknown parameters consist of deterministic and random variables, a hybrid CRLB is derived; (ii) if all the unknown parameters are random, a Bayesian CRLB is derived, and the marginalized Bayesian CRLB is obtained by marginalizing out the nuisance parameter. We also derive the CRLBs of the hyperparameters involved in the three-stage model and explore the effect of multiple snapshots to the CRLBs. We compare the derived CRLBs of SBL, finding that the marginalized Bayesian CRLB is tighter than other CRLBs when SNR is low and the differences between CRLBs become smaller when SNR is high. We also study the relationship between the mean squared error of the source magnitudes and the CRLBs, including numerical simulation results with a variety of antenna configurations such as different numbers of receivers and different noise conditions.