Cramer-Rao Research Articles

Cramer-Rao bound investigation of a novel virtually extended nested arc array for 2D DOA estimation

This paper exploits a sparse array representation, a novel nested arc array, to generate a virtual non-uniform circular arrays that have been presented and investigated. To increase the array aperture, the virtual array has been developed using higher-order statistics, namely, the Forth-Order Cumulants (FOC). The extended array contains a combination of physical as well as virtual sensors. The increased degrees of freedom (DOFs) in the resultant virtual array make it suitable for over-and under-determined direction of arrival (DOA) estimation. The Cramer-Rao bound (CRB) and the (DOA) estimation for such an array are developed and investigated for various arrays. The proposed array performance is superior to that of the uniform one. Further, the offered array is suitable for two-dimensional (2D) DOA estimation problems.

AUV-Aided joint time synchronization and localization of underwater target with propagation speed uncertainties

The accurate sensor localization is one of the most basic requirements for the Underwater Acoustic Sensor Networks (UASNs) in many applications. However, in harsh underwater environment, the existence of propagation speed variation with depth (i.e., stratification effect) and asynchronous clock generally affects the ranging estimation, and in turn significantly decreases localization accuracy. Taking into account these specific characteristics of UASNs, we first propose an efficient Autonomous Underwater Vehicle (AUV) aided joint localization and time synchronization solution for UASNs, subjected to stratification effect constraint, in which AUV acts as a mobile anchor and broadcasts beacon signals periodically to provide localization reference for unknown target sensors. Next, since the estimation accuracy of localization and synchronization is very sensitive to the accuracy of the AUV locations, the joint localization and synchronization method in the case of inaccurate AVU locations is considered. Then, Cramer Rao Lower Bound (CRLB) analyses are presented for both accurate and inaccurate AVU locations cases. Finally, numerical simulation results show that the proposed method in this paper outperforms the existing ones in terms of estimation errors.

DOA Estimation Assisted by Reconfigurable Intelligent Surfaces

A reconfigurable intelligent surface (RIS) assisted scheme is proposed for direction-of-arrival (DOA) estimation in this work. In the scheme, the RIS is utilized to provide a reflected path, and the RIS-reflected signal is received by array antennas together with the direct-path signal. The model consisting of the direct-path signal and the RIS-reflected signal is proposed, and the Cramer-Rao Bound (CRB) is derived based on the signal model to assess the ultimate estimation performance. In order to enhance the estimation performance of RIS-assisted scheme, two possible strategies for the phases design of RIS are presented. The first possibility considers RIS phase shifts that maximize the coherence of signals from the direct path and the RIS-reflected path. The other possibility considers to obtain the phase shifts that minimize the CRB of DOA estimation in the proposed scheme through a manifold optimization method. Moreover, DOA estimation methods are proposed for the RIS-assisted scheme to validate the scheme feasibility. Target tracking is proposed as a practical application for the RIS-assisted scheme. In the end, both theoretical analysis and simulation results illustrate that the accuracy of DOA estimation and target tracking has been improved with the aid of RIS.

Optimal linear array orientation design for 3D direct position determination via semi-Definite relaxation

This paper presents an optimization strategy for array orientations in a three-dimensional (3D) direct position determination (DPD) system. Specifically, we consider a scenario in which uniform linear arrays (ULAs) are used to locate emitters, and we seek to optimize the array orientations in terms of localization accuracy. The E-optimality criterion, which minimizes the spectral norm of the Cramér-Rao lower bound (CRLB), is exploited to formulate this problem. As the objective function is non-convex with bilinear structures, we leverage semi-definite relaxation (SDR) to transform it into a convex semi-definite programming (SDP) problem by substituting the bilinear terms with a matrix variable. In addition, we tackle the array orientation configuration problem in the context of multi-emitter scenarios and develop an SDR solution for large-scale antenna array systems. Simulation results validate that the ULAs with array orientations designed by the proposed SDR-based method have near-optimal localization performance.

Direction-of-Arrival (DoA) Estimation Performance for Satellite Applications in a Multipath Environment with Rician Fading and Spatial Correlation

Direction-of-arrival (DoA) estimation methods are highly versatile and find extensive applications in satellite communication. DoA methods are employed across a range of orbits, from low Earth orbits (LEO) to geostationary Earth orbits (GEO). They serve multiple applications, including altitude determination, geolocation and estimation accuracy, target localization, and relative and collaborative positioning. This paper provides a framework for modeling the DoA angle in satellite communications with respect to the elevation angle. The proposed approach employs a closed-form expression that incorporates various factors, such as the antenna boresight angle, satellite and Earth station positions, and the altitude parameters of the satellite stations. By leveraging this formulation, the work accurately calculates the Earth station’s elevation angle and effectively models the DoA angle. To the authors’ knowledge, this contribution is unique and has not been previously addressed in the available literature. Furthermore, this paper studies the impact of spatial correlation in the channel on well-known DoA estimation techniques. As a significant part of this contribution, the authors introduce a signal model incorporating correlation in satellite communication. Although selected studies have presented spatial signal correlation models in satellite communications to analyze the performance metrics, such as the bit error or symbol error probability, outage probability, and ergodic capacity, this work stands out by presenting and adapting a correlation model in the signal specifically for studying DoA estimations. Accordingly, this paper evaluates DoA estimation performance using root mean square error (RMSE) measurements for different satellite communication link conditions (uplink and downlink) through extensive Monte Carlo simulations. The simulation’s performance is evaluated by comparing it with the Cramer–Rao lower bound (CRLB) performance metric under additive white Gaussian noise (AWGN) conditions, i.e., thermal noise. The simulation results demonstrate that incorporating a spatial signal correlation model for DoA estimations significantly improves RMSE performance in satellite systems.

A novel design of multi-user sequence set for joint vehicular radar-communication based on Oppermann family

In this paper, a novel multi-user sequence set which are to be potentially used as communication preambles for joint vehicular radar and communication using Oppermann sequences is proposed. Herein, a set of Oppermann sequences with suitable auto-ambiguity functions (AF) and cross-ambiguity functions (AF) properties have been proposed. Moreover, a channel estimation algorithm using the designed Oppermann sequences is also proposed. The designed preamble, composed of optimized Oppermann sequences is an appropriate candidate for wireless standards, where both channel estimation and multi-user vehicular radar sensing are jointly pursued. The simulation results demonstrate that the designed sequences provide a significant performance improvement for vehicular radar sensing and don't degrade the channel estimation accuracy. The proposed optimized Oppermann sequences have better auto- and cross-AF compared existing Golay complementary sequences (used in IEEE 802.11ad as preamble), Zadoff-Chu sequences (used in 5G to generate preambles for the Physical Random Access Channel (PRACH)) and the non-optimized Oppermann sequences. Specifically, compared to the preamble composed of Golay complementary sequences in IEEE 802.11ad, the proposed sequences reduce the maximum sidelobe level of the auto-AF and maximum value level of cross-AF by at least 2 and 2.85 times, respectively. The target can still be accurately detected in mutual interference scenario caused by multiple radars in the vicinity. Both the sequences, when used for channel estimation achieve the CRLB limit. Both the proposed sequences and Golay complementary sequences, when used for channel estimation achieve the Cramer-Rao Lower Bound.

Real-Time CRLB Based Antenna Selection in Planar Antenna Arrays

Estimation of User Terminals’ (UTs’) Angle of Arrival (AoA) plays a significant role in the next generation of wireless systems. Due to high demands, energy efficiency concerns, and scarcity of available resources, it is pivotal how these resources are used. Installed antennas and their corresponding hardware at the Base Station (BS) are of these resources. In this paper, we address the problem of antenna selection to minimize Cramer-Rao Lower Bound (CRLB) of a planar antenna array when fewer antennas than total available antennas have to be used for a UT. First, the optimal antenna selection strategy to minimize the expected CRLB is proposed. Then, using this strategy as a preliminary step, we present a two-stage greedy antenna selection method whose goal is to minimize the instantaneous CRLB. The optimal start point of the greedy algorithm is presented alongside some methods to reduce the algorithm’s computational complexity. Numerical results confirm the accuracy of proposed solutions. They demonstrate that the proposed antenna selection method only requires a small proportion of the total available antennas to accomplish a significant amount of the total performance, enhancing hardware utilization efficiency. Also, it is shown that the presented algorithm has a high error tolerance.

Cramér–Rao Lower Bound Analysis of Differential Signal Strength Fingerprinting for Crowdsourced IoT Localization

Crowdsourcing is considered an efficient and promising paradigm for constructing large-scale signal fingerprint radio maps due to the proliferation of Wi-Fi-enabled devices. However, a crowdsourced Indoor Positioning System (IPS) has to handle diverse devices and the inherent heterogeneity in Received Signal Strength (RSS) measurements. To address the device heterogeneity problem, differential fingerprinting methods have been explored, which mitigate the device characteristics that cause RSS from different commercial devices to report differently. In this paper, we focus on Mean Differential Fingerprinting (MDF) that produces the differential fingerprints by subtracting the mean RSS value of all APs from the original RSS fingerprints. We study the localization performance of the MDF method by means of the Cramér-Rao Lower Bound (CRLB) and show analytically that it outperforms another method that addresses device diversity. Furthermore, we evaluate the localization accuracy of existing solutions using real-life Wi-Fi RSS datasets collected by multiple consumer devices. The experimental results confirm our analytical findings and demonstrate the effectiveness of the MDF method to mitigate device diversity, as well as other factors that affect the RSS readings including the device carrying mode and power control schemes of the Wi-Fi infrastructure, thus contributing to the wider adoption of crowdsourced IPS.

Trade-Off Between Positioning and Communication for Millimeter Wave Systems With Ziv-Zakai Bound

In this paper, we investigate the trade-off between positioning and communication for an integrated positioning and communication (IPAC) millimeter wave system. First, in terms of the positioning in the IPAC system, the Cramér-Rao bound (CRB) is commonly used as a performance metric. Unfortunately, the CRB is only tight in a certain region for the high signal-to-noise ratio (SNR). To compensate for this deficiency, we derive the Ziv-Zakai bound (ZZB) for the IPAC system by exploiting the a priori delay information extracted from both the time delay parameter and the channel amplitude attenuation. Further, we derive the expected CRB (ECRB) and the weighted CRB (WCRB) of the system for comparisons. Second, we analyze the trade-off between positioning and communication for this IPAC system based on the derived ZZB. Specifically, we aim to optimize the power allocation to maximize the achievable data rate subject to the ZZB and total transmit power constraints. Numerical results show that the ZZB provides a tighter and more reasonable bound for the minimum mean square error (MMSE) estimator over the wide range of SNRs compared to the ECRB and WCRB. Moreover, the trade-off between positioning and communication is revealed via changing critical parameters by simulations.

On Absoluteness and Stationary Condition of WMDS for Range-Based Localization

Weighted multidimensional scaling (WMDS), an algorithm extending multidimensional scaling (MDS), has been utilized in a broad spectrum of localization. However, there are still two unsolved theoretical questions. (i) The estimator provided by MDS depicts a relative placement of targets which requires further Procrustes analysis to recover the actual placement, referred to as the absolute placement. This fact produces a question: Does the estimator of WMDS is still relative? If not, what underlying mechanisms assert the absoluteness? (ii) It has been proved that WMDS attains the Cramér-Rao lower bound (CRLB) when the ranging distribution is Gaussian. Does it hold for a general distribution? If not, what restrictions on distributions are required? Motivated by such theoretical incompleteness, this paper offers an in-depth theoretical analysis on WMDS in the scheme of range-based localization. With regard to question (i), we reveal the mechanisms that assert the estimator of WMDS always represents exactly the actual locations, and prove that the absoluteness is introduced at the moment of variable separation and all the subsequent matrix equation transformations preserve the absoluteness. As for question (ii), the functional behaviors and maximum condition of CRLB are examined under a general ranging distribution. Then, via the comparison to the variance of WMDS estimator, the stationary condition on ranging distributions for WMDS to attain the CRLB is provided. Extensive simulations are performed to validate the theoretical conclusions numerically. And there exists conformity between numerical and theoretical results.

Quantification of Cerebral Glucose Concentrations via Detection of the H1-α-Glucose Peak in 1 H MRS at 7 T.

Sensitive detection and quantification of cerebral glucose is desired. To quantify cerebral glucose by detecting the H1-α-glucose peak at 5.23 ppm in 1 H magnetic resonance spectroscopy at 7 T. Prospective. Twenty-eight non-fasted healthy subjects (aged 20-28 years). Short echo time stimulated echo acquisition mode (short-TE STEAM) and semi-localized by adiabatic selective refocusing (semi-LASER) at 7 T. Single voxel spectra were obtained from the posterior cingulate cortex (27-mL) using a 32-channel head coil. The H1-α-glucose peak in the spectrum with retrospective removal of the residual water peak was fitted using LCModel with a glucose basis set of only the H1-α-glucose peak. Conventional spectral analysis was performed with a glucose basis set of a full spectral pattern of glucose, also. Fitting precision was evaluated with Cramér-Rao lower bounds (CRLBs). The repeatability of glucose quantification via the semi-LASER sequence was tested. Paired or Welch's t-test were used for normally distributed values. A P value of <0.05 was considered significant. The repeatability of measures was analyzed using coefficient of variation (CV). Removal of the residual water peak improved the flatness and stability of baselines around the H1-α-glucose peak and reduced CRLBs for fitting the H1-α-glucose peak. The semi-LASER sequence was superior to the short-TE STEAM in the higher signal-to-noise ratio of the H1-α-glucose peak (mean ± SD 7.9 ± 2.5, P < 0.001). The conventional analysis overfitted the H1-α-glucose peak. The individual CVs of glucose quantification by detecting the H1-α-glucose peak were smaller than the corresponding CRLBs. Cerebral glucose concentration is quantitated to be 1.07 mM by detecting the H1-α-glucose peak in the semi-LASER spectra. Despite requiring long scan times, detecting the H1-α-glucose peak allows true glucose quantification free from the influence of overlapping taurine and macromolecule signals. 2 TECHNICAL EFFICACY STAGE: 1.

Source Localization Using TDOA Measurements From Underwater Acoustic Sensor Networks

In this letter, we investigate the problem of TDOA localization with unknown signal propagation speed in underwater acoustic sensor networks (UASNs). Although the state-of-the-art semidefinite programming (SDP)-based algorithm has good accuracy, its computational complexity is huge. We develop a two-step algorithm to estimate the source position and signal propagation speed. It not only has better accuracy than the SDP-based algorithm but also has much less computational complexity. The first step of the proposed algorithm is a least squares (LS) estimator, and the second step of the proposed algorithm is a fixed point iteration (FPI) estimator. Numerical simulations validate that the proposed algorithm is superior to the SDP algorithm and reaches the Cramér-Rao lower bounds (CRLB).

Parameter estimation of LFM signals based on time reversal

In this paper, parameter estimation of linear frequency modulation (LFM) signals containing additive white Gaussian noise is studied. Because the center frequency estimation of an LFM signal is affected by the error propagation effect, resulting in a higher signal to noise ratio (SNR) threshold, a parameter estimation method for LFM signals based on time reversal is proposed. The proposed method avoids SNR loss in the process of estimating the frequency, thus reducing the SNR threshold. The simulation results show that the threshold is reduced by 5 dB compared with the discrete polynomial transform (DPT) method, and the root-mean-square error (RMSE) of the proposed estimator is close to the Cramer-Rao lower bound (CRLB).

A High Precision Joint Time Synchronization and Location Algorithm for Moving Target

Time synchronization is the guarantee of normal operation of wireless sensor networks, and it is the premise of ranging and location. At present, most researches focus on joint time synchronization and location of non-moving targets, while there are few types of research on joint time synchronization and location of moving targets. Therefore in this paper, under the condition that the time synchronization of the moving target is not carried out, and the information measurement means such as the speed and position of the moving target are not available. We established the system model of joint time synchronization and location with the help of the anchor nodes whose clock has been synchronized and location is known, and derived the Cramer-Rao lower bound (CRLB) of the system. Then we proposed the two-step least square method and iterative method. The former has lower computation and suboptimal performance, while the latter has higher time synchronization accuracy and positioning accuracy, which can attain CRLB. The results are verified by simulation.

An Information-Theoretic Branch-and-Prune Algorithm for Discrete Phase Optimization of RIS in Massive MIMO

In this paper, we consider passive RIS-assisted multi-user communication between wireless nodes to improve the blocked line-of-sight (LOS) link performance. The wireless nodes are assumed to be equipped with Massive Multiple-Input Multiple-Output antennas, hybrid precoder, combiner, and low-resolution analog-to-digital converters (ADCs). We first derive the expression for the Cramer-Rao lower bound (CRLB) of the Mean Squared Error (MSE) of the received and combined signal at the intended receiver under interference. By appropriate design of the hybrid precoder, combiner, and RIS phase settings, it can be shown that the MSE achieves the CRLB. We further show that minimizing the MSE w.r.t. the phase settings of the RIS is equivalent to maximizing the throughput and energy efficiency of the system. We then propose a novel Information-Directed Branch-and-Prune (IDBP) algorithm to derive the phase settings of the RIS. We, for the first time in the literature, use an information-theoretic measure to decide on the pruning rules in a tree-search algorithm to arrive at the RIS phase-setting solution, which is vastly different compared to the traditional branch-and-bound algorithm that uses bounds of the cost function to define the pruning rules. In addition, we provide the theoretical guarantees of the near-optimality of the RIS phase-setting solution thus obtained using the Asymptotic Equipartition property. This also ensures near-optimal throughput and MSE performance.

Sensing RISs: Enabling Dimension-Independent CSI Acquisition for Beamforming

Reconfigurable intelligent surfaces (RISs) are envisioned as a potentially transformative technology for future wireless communications. However, RISs' inability to process signals and the attendant increased channel dimension have brought new challenges to RIS-assisted systems, including significantly increased pilot overhead required for channel estimation. To address these problems, several prior contributions that enhance the hardware architecture of RISs or develop algorithms to exploit the channels' mathematical properties have been made, where the required pilot overhead is reduced to be proportional to the number of RIS elements. In this paper, we propose a dimension-independent channel state information (CSI) acquisition approach in which the required pilot overhead is independent of the number of RIS elements. Specifically, in contrast to traditional signal transmission methods, where signals from the base station (BS) and the users are transmitted in different time slots, we propose a novel method in which signals are transmitted from the BS and the user simultaneously during CSI acquisition. With this method, an electromagnetic interference random field (IRF) will be induced on the RIS, and we propose the structure of sensing RIS to capture its features. Moreover, we develop three algorithms for parameter estimation in this system, in which one of the proposed vM-EM algorithm is analyzed with the fixed-point perturbation method to obtain an asymptotic achievable bound. In addition, we also derive the Cram\'er-Rao lower bound (CRLB) and an asymptotic expression for characterizing the best possible performance of the proposed algorithms. Simulation results verify that our proposed signal transmission method and the corresponding algorithms can achieve dimension-independent CSI acquisition for beamforming.

Distributed Localization of Target for MIMO Radar With Widely Separated Directional Transmitters and Omnidirectional Receivers

Conventional target localization algorithms for the multiple-input multiple-output (MIMO) radar with widely separated antennas are mainly based on the centralized localization framework, which suffer from huge burdens of computation and communication as well as robustness defect in complex environments. To overcome these issues, this paper first considers the MIMO radar with widely separated directional transmitters and omnidirectional receivers generating the bistatic range (BR) and angle of incidence (AOI) measurements for target localization, which is proven that the position of the target can be uniquely determined with one transmitter and one receiver. Then, the distributed localization framework using the hybrid BR and AOI measurements is developed, where each receiver acts as a fusion center extracting the BR and AOI measurements to perform the localization process, and the localization problem is formulated as solving a linear matrix equation. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed constrained total least squares</i> (DCTLS) algorithm considering errors in both data matrix and observation vector is proposed for each receiver for target localization. The localization performance regarding the Cramér-Rao lower bound (CRLB) is derived for theoretical analysis. Numerical simulations are performed to validate the efficacy and superiority of the proposed DCTLS algorithm over other typical localization algorithms.

Near-Field Channel Estimation in Mixed LoS/NLoS Environments for Extremely Large-Scale MIMO Systems

Accurate channel model and channel estimation are essential to empower extremely large-scale MIMO (XL-MIMO) in 6G networks with ultra-high spectral efficiency. With the sharp increase in the antenna array aperture of the XL-MIMO scenario, the electromagnetic propagation field will change from far-field to near-field. Unfortunately, due to the near-field effect, most of the existing XL-MIMO channel models fail to describe mixed line-of-sight (LoS) and non-line-of-sight (NLoS) path components simultaneously. In this paper, a mixed LoS/NLoS near-field XL-MIMO channel model is proposed to match the practical near-field XL-MIMO scenario, where the LoS path component is modeled by the geometric free space propagation assumption while NLoS path components are modeled by the near-field array response vectors. Then, to define the range of near-field for XL-MIMO, the MIMO Rayleigh distance (MIMO-RD) and MIMO advanced RD (MIMO-ARD) is derived. Next, a two stage channel estimation algorithm is proposed, where the LoS path component and NLoS path components are estimated separately. Moreover, the Cramér-Rao lower bound (CRLB) of the proposed algorithm is derived in this paper. Numerical simulation results demonstrate that, the proposed two stage scheme is able to outperform the existing methods in both the theoretical channel model and the QuaDRiGa channel emulation platform.

Two Rapid Power Iterative DOA Estimators for UAV Emitter Using Massive/Ultra-Massive Receive Array

To provide rapid direction finding (DF) for unmanned aerial vehicle (UAV) emitters in future wireless networks, a low-complexity direction of arrival (DOA) estimation architecture for massive multiple-input multiple-output (MIMO) receiver arrays is constructed. In this paper, we propose two strategies to address the extremely high complexity caused by eigenvalue decomposition of the received signal covariance matrix. Firstly, a rapid power iterative rotational invariance (RPI-RI) method is proposed, which adopts the signal subspace generated by power iteration to obtain the final direction estimation through rotational invariance between subarrays. RPI-RI causes a significant complexity reduction at the cost of a substantial performance loss. In order to further reduce the complexity and provide good directional measurement results, the rapid power iterative polynomial rooting (RPI-PR) method is proposed, which utilizes the noise subspace combined with the polynomial solution method to obtain the optimal direction estimation. In addition, the influence of initial vector selection on convergence in the power iteration is analyzed, especially when the initial vector is orthogonal to the incident wave. Simulation results show that the two proposed methods outperform the conventional DOA estimation methods in terms of computational complexity. In particular, the RPI-PR method achieves more than two orders of magnitude lower complexity than conventional methods and achieves performance close to the Cramér–Rao Lower Bound (CRLB). Moreover, it is verified that the initial vector and the relative error have a significant impact on the performance with respect to the computational complexity.

Micromotion Feature Extraction with VEMW Radar Based on Rotational Doppler Effect

Micro-Doppler (m-D) analysis is the most effective mechanism for detecting rotating targets or components; however, it fails when the target rotation plane is perpendicular to the radar line of sight (LOS). The vortex electromagnetic wave (VEMW) provides a unconventional structure of wavefront phase modulation on the cross-plane of the radar LOS, on which the radial m-D vanishes while the rotational Doppler (RD) appears. In the absence of the position of rotation center, this paper focuses on the micromotion parameters estimation based on RD effect for rotating target, and then proposes an estimation procedure, referred to as the two-step method. The micromotion parameters of the rotating target include the rotation attitude, the rotation radius and the position of the rotation center while the latter is coupled to the former two. Firstly, the micromotion parameters are roughly estimated based on the RD curve parameters obtained from the time-frequency (TF) spectrum of the received signal. Secondly, the maximum likelihood estimation (MLE) is used to accurately estimate the micromotion parameters. In addition, the Cramér–Rao bound (CRB) of parameter estimation is derived. The simulation studies the influencing factors of estimation performance and verifies that the proposed estimation method can provide excellent estimation accuracy of the micromotion parameters.