Cramer-Rao Research Articles

Theoretical comparison and optimization of cadmium telluride and gallium arsenide photon-counting detectors for contrast-enhanced spectral mammography.

Most photon-counting detectors (PCDs) being developed use cadmium telluride (CdTe), which has nonoptimal characteristic x-ray emission with energies in the range used for breast imaging. New PCD using a gallium arsenide (GaAs) has been developed. Since GaAs has characteristic x-rays lower in energy than those of CdTe, it is hypothesized that this new PCD might be beneficial for spectral x-ray breast imaging. We performed simulations using realistic mammography x-ray spectra with both CdTe and GaAs PCDs. Five different experiments were conducted, each comparing the performance of CdTe and GaAs: (1) sensitivity of iodine quantification to charge cloud size and electronic noise, (2) effect of photon spectrum on iodine quantification, (3) effect of varying the number of energy bins, (4) a dose analysis to assess any possible dose reduction from using either detector, and (5) spectral performance of ideal CdTe and GaAs PCDs. For each study, 3 sets of 5000 noise realizations were used to calculate the Cramer-Rao lower bound (CRLB) of iodine quantification. For all spectra studied, GaAs gave a lower CRLB for iodine quantification, with 10 of the 12 spectra showing a statistically significant difference ( ). The photon energy spectrum that optimized iodine detection for both detector materials was the 40 kVp beam with 2-mm Al filtration, which produced CRLBs of and for CdTe and GaAs, respectively, when using five energy bins. GaAs is a promising detector material for contrast-enhanced spectral mammography that offers better spectral performance than CdTe.

CRB Weighted Source Localization Method Based on Deep Neural Networks in Multi-UAV Network

With the advent of the Internet of Things (IoT) era, the multi-unmanned aerial vehicle (UAV) networks have attracted great attention in the fields of source detection and localization. However, as a real-time signal processing performance of the UAV is limited by the computing speed and accuracy of the embedded hardware, the effectiveness of source localization is greatly reduced. Aiming at improving the accuracy and computational efficiency of source localization, a Cramer-Rao bound (CRB) weighted multi-UAV network source localization method is proposed based on the deep neural networks (DNN) and spatial spectrum fitting (SSF). The proposed source localization system is composed of UAVs equipped with a radar array. The source location can be achieved using the DOA of the sources signals of UAVs, but the accuracy and real-time performance of the conventional DOA estimation algorithms are not satisfactory, and the data fusion strategy of the conventional cross-location framework needs further improvement. In the proposed method, a DNN-based SSF denoted as the deep SSF (DeepSSF), is designed to achieve accurate DOA estimation. In the DeepSSF, the DOA estimation performance is guaranteed by the DNN’s strong nonlinear fitting ability and highly parallel structure. In addition, based on the obtained DOA information, the source is located once by every two UAVs. Finally, the source localization is realized based on the weighted CRB according to the principle that the more the DOA distribution deviates from zero, the lower the estimation accuracy. The simulation results verify the efficiency of the proposed method.

STAR‐RIS‐enabled simultaneous indoor and outdoor 3D localisation: Theoretical analysis and algorithmic design

AbstractRecent research and development interests deal with metasurfaces for wireless systems beyond their consideration as intelligent tunable reflectors. Among the latest proposals is the simultaneously transmitting (a.k.a. refracting) and reflecting reconfigurable intelligent surface (STAR‐RIS) which intends to enable bidirectional indoor‐to‐outdoor, and vice versa communications thanks to its additional refraction capability. This double functionality provides increased flexibility in concurrently satisfying the quality‐of‐service requirements of users located at both sides of the metasurfaces, for example, the achievable data rate and localisation accuracy. The authors focus on STAR‐RIS‐empowered simultaneous indoor and outdoor three‐dimensional (3D) localisation, and study the fundamental performance limits via Fisher information analyses and Cramér Rao lower bounds (CRLBs). The authors also devise an efficient localisation algorithm based on an off‐grid compressive sensing (CS) technique relying on atomic norm minimisation (ANM). The impact of the training overhead, the power splitting at the STAR‐RIS, the power allocation between the users, the STAR‐RIS size, the imperfections of the STAR‐RIS‐to‐BS channel, as well as the role of the multi‐path components on the positioning performance are assessed via extensive computer simulations. It is theoretically demonstrated that high‐accuracy, up to centimetre level, 3D localisation can be simultaneously achieved for indoor and outdoor users, which is also accomplished via the proposed ANM‐based estimation algorithm.

Over-the-Air Phase Calibration of Reconfigurable Intelligent Surfaces

The reconfigurable intelligent surface (RIS) has emerged as a promising technology to enhance the capacity of wireless systems, as it can configure the wireless channel using low-cost passive devices, i.e., printed dipoles and phase shifters. But the real phases of the RIS elements usually deviate from the nominal values in practice, and the phase deviations need to be precisely estimated and calibrated for some applications. This letter proposes a novel over-the-air (OTA) approach that can estimate the real phases of the RIS elements at each gear. We also derive the Cramér Rao Bounds (CRBs) for the RIS calibration as a performance benchmark. The simulation results verify the effectiveness of the proposed algorithm as the root mean square errors (RMSEs) of the phase estimates are close to the CRBs.

Ranging Code Design for UAV Swarm Self-Positioning in Green Aerial IoT

Utilizing the unmanned aerial vehicle (UAV) swarm to realize location awareness of ground users (GUs) is a promising technology in green aerial Internet of Things (IoT) systems. However, a typical UAV-based positioning system is frequently applied in dense urban areas, where the traditional satellite positioning systems are severely impaired. Due to the co-frequency interference and the inevitable defect of limited resources of the positioning anchors, the UAVs positioning ability and system resource utilization suffer serious challenges. In this article, we propose a novel ranging code design for UAV swarm self-positioning, which consists of the code truncation algorithm and Greedy-based code group optimization algorithm, aiming to obtain the code group with the optimal correlation characteristics and shorter code length based on the pseudo noise (PN) code. In particular, the design can flexibly change the truncated length, so as to ensure the self-positioning accuracy with less positioning resource. To evaluate the influence of the proposed design for the self-positioning of UAVs on the aerial IoT, the root-mean-square error (RMSE) of UAVs self-positioning is provided, and based on this, the Cramer–Rao lower bound (CRLB) of GUs location estimate is derived. Numerical results demonstrate that the ranging code design is a superior way for future green aerial IoT in both positioning and resource utilization, as compared with a state-of-the-art approach.

GNSS Spoofer Localization for Vehicles Based on Doppler and Clock Drift Double Difference

Global navigation satellite systems (GNSS) have been widely used in our daily lives. Meanwhile, GNSS are facing a severe threat of spoofing attacks, which can mislead users by false position, velocity and time (PVT) solutions, resulting in serious consequences. Especially, with the development of vehicle navigation, autonomous driving and unmanned aerial vehicles (UAV), increasing spoofing attacks have emerged towards vehicles, where anti-spoofing techniques are desperately needed. Among these techniques, spoofer localization is a necessary step to fundamentally address the spoofing attacks. Most of the conventional spoofer localization methods are based on time of arrival (TOA) or time difference of arrival (TDOA). These methods require measurements observed by multiple receivers at the same time, where time synchronization is indispensable. However, it is hard to apply these methods to the existing vehicles equipped with independent commercial receivers. To cope with this problem, in this paper we develop a flexible method for vehicles to locate a stationary spoofer based on Doppler and clock drift double difference (DCDD). DCDD is calculated using Doppler measurements and clock drift solutions of one vehicular receiver and another reference receiver. Generally, Doppler frequency of GNSS signals changes slowly with time, which releases the restriction for high-precision time synchronization. DCDD represents the Doppler frequency caused by the vehicle motion, from which the spoofer position can be estimated. In addition, with the elimination of irrelevant components in Doppler measurements, DCDD minimizes the user complexity required for localization, where at least two receivers and one vehicle is needed. Utilizing DCDD, an iterative weighted least squares (WLS) algorithm is used to solve the spoofer position, and a closed-form initial value is given to improve the convergence rate. We evaluate the localization performance by the derived Cramér-Rao lower bound (CRLB) in the presence of vehicle position and velocity errors. Simulations and field vehicle experiments are conducted to verify the feasibility and effective of the proposed method. The experimental results show that with an average vehicle velocity of 8 m/s, the horizontal localization error can be less than 40 m and 20 m, using DCDD observations of 4 epochs and 8 epochs respectively.

High-Performance Channel Estimation for mmWave Wideband Systems With Hybrid Structures

In this paper, a channel estimation problem for millimeter-wave (mmWave) multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems with hybrid structures is studied. Firstly, a beamspace multiple signal classification (MUSIC) algorithm for mmWave wideband channels is proposed to simultaneously estimate the angles of arrival (AOAs), angles of departure (AODs) and transmission delays. Since the traditional spectral peak search method has high complexity, a multi-spectral peak search method is skillfully designed to search for multiple spectral peaks on the MUSIC spatial spectrum more quickly and accurately. Then, the proposed channel estimator is extended to more actual systems equipped with uniform planar arrays (UPAs). Finally, the Cramér–Rao bound (CRB) results of these channel parameters are derived for evaluating the performance of the proposed channel parameter estimator. Simulation results demonstrate that the proposed channel estimator has greatly high channel estimation accuracy.

DOA and subarray-interval estimation for arbitrarily distributed UAV swarm system

In this paper, DOA and subarray-interval estimation are considered and applied to arbitrarily distributed unmanned aerial vehicle (UAV) swarm system, in which multiple small UAVs containing uniform linear array (ULA) are divided by unknown intervals because of dynamic moving. Three parameters are taken to indicate the steering vector, namely, the direction of arrivals (DOAs) of target users, the intervals of UAVs, and the orientation angles of UAVs. The orientation angles are first estimated with an auxiliary user and the DOAs are obtained through a search free rooting method, despite the intervals among the UAVs. Afterwards, the intervals among UAVs can also be calculated via exhaustive when the number of target users are no less than three. We further develop a low-complex method to reduce the computational complexity during subarray-interval estimation. The deterministic Cramer-Rao bound (CRB) of the DOA, orientation angle and subarray-interval can be inferred in a closed form. Eventually, numerical instances are cited to verify the research results.

Hybrid Transceiver Design for Tera-Hertz MIMO Systems Relying on Bayesian Learning Aided Sparse Channel Estimation

Hybrid transceiver design in multiple-input multiple-output (MIMO) Tera-Hertz (THz) systems relying on sparse channel state information (CSI) estimation techniques is conceived. To begin with, a practical MIMO channel model is developed for the THz band that incorporates its molecular absorption and reflection losses, as well as its non-line-of-sight (NLoS) rays associated with its diffused components. Subsequently, a novel CSI estimation model is derived by exploiting the angular-sparsity of the THz MIMO channel. This is followed by designing a sophisticated Bayesian learning (BL)-based approach for efficient estimation of the sparse THz MIMO channel. The Bayesian Cramer-Rao Lower Bound (BCRLB) is also determined for benchmarking the performance of the CSI estimation techniques developed. Finally, an optimal hybrid transmit precoder and receiver combiner pair is designed, which directly relies on the beamspace domain CSI estimates and only requires limited feedback. Finally, simulation results are provided for quantifying the improved mean square error (MSE), spectral-efficiency (SE) and bit-error rate (BER) performance for transmission on practical THz MIMO channel obtained from the HIgh resolution TRANsmission (HITRAN)-database.

Localization of mixed coherently and incoherently distributed sources based on generalized array manifold

In practice, signals received by an array may be a mixture of coherently distributed (CD) and incoherently distributed (ID) sources, due to some complicated multipath/scattering effects. To localize mixed CD and ID sources, a method based on the generalized array manifold (GAM) is developed in this work. Firstly, source enumeration and source type classification are achieved by jointly employing the minimum description length (MDL) criterion, the covariance matrix difference technique and the second-order statistics of associated eigenvalues. Secondly, with the source enumeration result, the nominal direction of arrival (DOA) of ID sources is estimated by the rank-reduction principle based one-dimensional (1-D) spectral search, where shift invariance of the GAM associated with ID sources is considered. Thirdly, the oblique projection technique is adopted to separate the CD sources from ID ones, and then the nominal DOA of CD sources is estimated via the sparsity-cognizant total least-squares (S-TLS) method. Finally, with aid of the nominal DOA estimates, the corresponding angular spreads are calculated. An analysis of the proposed method is discussed and the approximate Cramér-Rao bound (CRB) is derived. Simulations are provided to demonstrate the performance of the proposed method.

An auxiliary source-based near field source localization method with sensor position error

In order to avoid the influence of errors brought by simplified model and sensor position error on near-field source location, a non-uniform linear array source location algorithm using precise model and based on auxiliary source (AS) is proposed under the condition of sensor position error. The array model used in this algorithm has position errors except for reference sensor. AS method is used to accurately estimate the position of the special sensor. Then, the generalized eigenvalue including the estimated position of the special sensor is obtained by using the fourth-order cumulant and the idea of Least Squares-Estimating Signal Parameter via Rotational Invariance Techniques (LS-ESPRIT). Finally, the estimation results of the source location can be obtained by using the correlation formula. In addition, the Cramér Rao Lower Bound (CRLB) is proposed for the parameter estimation problem, which determines a lower bound for the variance of any unbiased estimator. Two cases of near - field source and mixed source are simulated respectively. The results show that the proposed algorithm is applicable to both cases and has very accurate source location estimation results.

Accurate Broadband Gradient Estimates Enable Local Sensitivity Analysis of Ocean Acoustic Models

Sensitivity analysis is a powerful tool for analyzing multi-parameter models. For example, the Fisher information matrix (FIM) and the Cramér–Rao bound (CRB) involve derivatives of a forward model with respect to parameters. However, these derivatives are difficult to estimate in ocean acoustic models. This work presents a frequency-agnostic methodology for accurately estimating numerical derivatives using physics-based parameter preconditioning and Richardson extrapolation. The methodology is validated on a case study of transmission loss in the 50–400[Formula: see text]Hz band from a range-independent normal mode model for parameters of the sediment. Results demonstrate the utility of this methodology for obtaining Cramér–Rao bound (CRB) related to both model sensitivities and parameter uncertainties, which reveal parameter correlation in the model. This methodology is a general tool that can inform model selection and experimental design for inverse problems in different applications.

Efficient cooperative localization method with node selection based on position error bound

Cooperative Localization (CL) has become a promising solution that circumvents the needs for high-density anchor deployment, and offers additional localization accuracy in Wireless Sensor Networks (WSNs). However, it suffers from high computational complexity and increased network traffic problem. To alleviate this problem, we propose an iterative greedy node selection approach based on Position Error Bound (PEB) for Angle-of-arrival (AOA) based CL. We introduce a Boolean vector to formulate the node selection optimization problem that maximizes the Decrease of PEB (DPEB) from agents’ cooperation. Since the problem is NP-hard and difficult to solve, the iterative greedy algorithm is proposed to approach the sub-optimal solutions. Then, in order to improve the accuracy of localization phase, we derive the efficient closed-form solution of AOA CL problem and propose an iterative constrained Least Squares (LS) method. Moreover, the theoretical analysis shows that the localization estimator is very close to the Cramer Rao Lower Bound (CRLB) performance. Finally, simulation results show that the proposed node selection approach can greatly decrease the number of agents participating in the CL without significant degradation in localization accuracy. Besides, the proposed CL method outperforms the existing ones in terms of localization accuracy.

Efficient semidefinite solutions for TDOA-based source localization under unknown PS

Two efficient solutions via Semi-Definite Programming (SDP) are proposed for source localization problems using time difference of arrival (TDOA)-based ranging measurements when the propagation speed (PS) is unavailable and considered as a variable. For this problem, we propose a relaxed SDP (RSDP) solution, the performance of which is suboptimal. Accordingly, we propose a two-stage SDP method to improve the performance by applying the rank-reduction method. Besides, we also propose a penalty function-based SDP (PF-SDP) by introducing the penalty term. By doing so, the cost function becomes tighter so that the solution performs better. The simulated results show that the performance of two-stage SDP is sufficiently close to the Cramér-Rao Lower Bound (CRLB) accuracy at high noise levels. The PF-SDP outperforms the two-stage SDP in the presence of low noise levels.

Experimental Basis Sets of Quantification of Brain 1H-Magnetic Resonance Spectroscopy at 3.0 T.

In vivo short echo time (TE) proton magnetic resonance spectroscopy (1H-MRS) is a useful method for the quantification of human brain metabolites. The purpose of this study was to evaluate the performance of an in-house, experimentally measured basis set and compare it with the performance of a vendor-provided basis set. A 3T clinical scanner with 32-channel receive-only phased array head coil was used to generate 16 brain metabolites for the metabolite basis set. For voxel localization, point-resolved spin-echo sequence (PRESS) was used with volume of interest (VOI) positioned at the center of the phantoms. Two different basis sets were subjected to linear combination of model spectra of metabolite solutions in vitro (LCModel) analysis to evaluate the in-house acquired in vivo 1H-MR spectra from the left prefrontal cortex of 22 healthy subjects. To evaluate the performance of the two basis sets, the Cramer-Rao lower bounds (CRLBs) of each basis set were compared. The LCModel quantified the following metabolites and macromolecules: alanine (Ala), aspartate (Asp), γ-amino butyric acid (GABA), glucose (Glc), glutamine (Gln), glutamate (Glu), glutathione (GHS), Ins (myo-Inositol), lactate (Lac), N-acetylaspartate (NAA), N-acetylaspartylglutamate (NAAG), taurine (Tau), phosphoryl-choline + glycerol-phosphoryl-choline (tCho), N-acetylaspartate + N-acetylaspartylglutamate (tNA), creatine + phosphocreatine (tCr), Glu + Gln (Glx) and Lip13a, Lip13b, Lip09, MM09, Lip20, MM20, MM12, MM14, MM17, Lip13a + Lip13b, MM14 + Lip13a + Lip13b + MM12, MM09 + Lip09, MM20 + Lip20. Statistical analysis showed significantly different CRLBs: Asp, GABA, Gln, GSH, Ins, Lac, NAA, NAAG, Tau, tCho, tNA, Glx, MM20, MM20 + Lip20 (p < 0.001), tCr, MM12, MM17 (p < 0.01), and Lip20 (p < 0.05). The estimated ratio of cerebrospinal fluid (CSF) in the region of interest was calculated to be about 5%. Fitting performances are better, for the most part, with the in-house basis set, which is more precise than the vendor-provided basis set. In particular, Asp is expected to have reliable CRLB (<30%) at high field (e.g., 3T) in the left prefrontal cortex of human brain. The quantification of Asp was difficult, due to the inaccuracy of Asp fitting with the vendor-provided basis set.

A New Subsample Time Delay Estimation Algorithm for LFM-Based Detection

This paper investigated a Subsample Time delay Estimation (STE) algorithm based on the amplitude of cross-correlation function to improve the estimation accuracy. In this paper, a rough time delay estimation is applied based on traditional cross correlator, and a fine estimation is achieved by approximating the sampled cross-correlation sequence to the amplitude of the theoretical cross-correlation function for linear frequency modulation (LFM) signal. Simulation results show that the proposed algorithm outperforms existing methods and can effectively improve time delay estimation accuracy with the complexity comparable to the traditional cross-correlation method. The theoretical Cramér-Rao Bound (CRB) is derived, and simulations demonstrate that the performance of STE can approach the boundary. Eventually, four important parameters discussed in the simulation to explore the impact on Mean Squared Error (MSE).

Quantum-Enhanced Transmittance Sensing

We consider the problem of estimating unknown transmittance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta$</tex-math></inline-formula> of a target bathed in thermal background light. As quantum estimation theory yields the fundamental limits, we employ the lossy thermal-noise bosonic channel model, which describes sensor-target interaction quantum mechanically in many practical active-illumination systems (e.g., using emissions at optical, microwave, or radio frequencies). We prove that quantum illumination using two-mode squeezed vacuum (TMSV) states asymptotically achieves minimal quantum Cramér-Rao bound (CRB) over all quantum states (not necessarily Gaussian) in the limit of low transmitted power. We characterize the optimal receiver structure for TMSV input, and show its advantage over other receivers using both analysis and Monte Carlo simulation.

Finite rate of innovation sampling of Gaussian pulse streams with variable shape

The existing Finite Rate of Innovation (FRI) sampling theory can realize the efficient measurement of Gaussian pulse streams with fixed modal functions. However, it is hard to sample that signal with variable pulse width factors by a sub-Nyquist rate. To get around this challenge, we propose an extended FRI sampling method based on Taylor series expansion in this paper, where the signal is first filtered by the exponential reproducing kernel, then each pulse can be reconstructed sequentially. For a single pulse, we first prove that the pulse width factor does not interfere with the estimation process of the time delay parameter, and we also present its Cramer-Rao lower Bound (CRB). The signal Fourier coefficient can then be reformulated by the Taylor series expansion of order N−1 to realize the decoupling of the pulse width factor before the final estimation. We derive the order limit of expansion from the perspective of numerical stability, and we also give the upper bound on the estimation error of the pulse width factor for N=2. Simulation results show that, compared with the classical methods, the proposed state-of-the-art algorithm can achieve a higher precision reconstruction of the Gaussian pulse stream signal.

A coarse to accurate noise-tolerant positioning evaluation for mobile target based on modified genetic algorithm

The location information of mobile target can be mapped between physical space and signal space, which is one of the key technologies for real-time tracking and multi-machine cooperation. Benefiting from the distributed perception and ubiquitous communication capabilities, wireless sensor networks (WSNs) can be used for target real-time positioning indoor and outdoor. However, various sensor noises and environmental interferences bring uncertainty to target positioning, resulting in inconsistent positioning performance. Hence, this paper proposes a coarse to accurate noise-tolerant positioning evaluation for mobile target based on modified genetic algorithm. Considering the uncertainty of various measurements in WSNs, the preliminary positioning results are estimated by total least squares. Taking the preliminary results as the initial search values, the modified genetic algorithm is used to refine the target positioning accuracy with use of adaptively adjusted crossover probability and improved mutation operation. In addition, the theoretical accuracy model is derived by the Cramer–Rao Lower Bound (CRLB) as a benchmark. Comprehensive experiments are conducted through the simulation and testing platform. The experimental results of the proposed algorithm have a 0.25 m average error and 0.152 error variance. The result trends of simulation are consistent with the result of experimental platform, which can validate the superior accuracy of the proposed algorithm compared with relevant positioning algorithms.

Analysis of asymptotic and exact Cramér-Rao lower bounds on the parameters of a noisy periodic signals

In this paper expressions for the Cramér-Rao Lower Bound (CRLB) and the Asymptotic (high values for both number of cycles and number of samples) CRLB (ACRLB) on the parameters of a periodic signal affected by additive white Gaussian noise are derived and compared with each other. The comparison is performed as a function of the number of acquired signal cycles when the signal frequency is assumed either known or unknown. It is shown that, when the frequency is known and the sampling process is coherent (i.e., the number of acquired signal cycles is an integer), the ACRLBs coincide with the related CRLBs. The presented results are useful for the assessment of the statistical efficiency of any unbiased parameter estimator of a periodic signal embedded in wideband Gaussian noise when a few signal cycles are observed.