Cramer-Rao Research Articles

Stochastic resonance in Bayesian estimation and CRLB for nonlinear system

In this work, noise enhanced parameter estimation problems are investigated for a general nonlinear system, where an additive noise is added to the nonlinear system input and a Bayesian estimator is developed based on the noise modified output. The optimal probability distribution of the additive noises is formulated successively for minimizing the mean square error (MSE) of the optimal Bayesian estimation and the Cramer–Rao lower bound (CRLB). Then the optimal additive noises for the two different noise enhanced optimization problems are explicitly derived as constant vectors, which implies the randomization of constant vectors is not beneficial to these optimizations. Finally, numerical results are presented to illustrate the theoretical results.

Thermal properties and quantum information theory with the shifted Morse potential

By employing the Nikiforov–Uvarov functional analysis (NUFA) method, we solved the radial Schrödinger equation with the shifted Morse potential model. The analytical expressions of the energy eigenvalues, eigenfunctions and numerical results were determined for selected values of the potential parameters. Variations of different thermodynamic functions with temperature were discussed extensively. Different quantum information-theoretic measures, including Shannon entropy, Fisher information and Fisher–Shannon product of the shifted Morse potential, were investigated numerically and graphically in position and momentum spaces for the ground and the first excited states. The quantum information theories considered satisfied their corresponding inequalities, including Bialynicki–Birula–Mycielski, Stam–Cramer–Rao inequalities and the Fisher–Shannon product relation.

Reconfigurable Intelligent Surface-Aided Emitter Localization

Localization of radio-frequency (RF) transmitters using time difference of arrival (TDOA)-based methods is one of the conventional passive techniques that admits noncooperative source position finding, but suffers from challenging requirements such as precise intersensor synchronization and high-throughput transmission data links. A tradeoff governing the TDOA systems is in the sensor placement configuration. The more distance the sensors are placed, the more accurate localization is carried out, while the cost for the synchronization and data link increases, at the same time. In this work, a novel reconfigurable intelligent surface (RIS)-aided localization system is proposed that enjoys precise location accuracy while reducing the required expenses. The study reveals that the new setup, utilizing the beam-scanning property of the RIS sensors, improves the localization algorithm and outperforms the conventional approach. Also, comparisons with Cramér Rao lower bound (CRLB) are provided, which confirms the efficiency of the proposed method. Several numerical examples indicate the effectiveness of the proposed method, especially in a low signal-to-noise ratio (SNR) regime.

Iterative Joint Estimation Procedure of Channel and PDP for OFDM Systems.

The power-delay profile (PDP) estimation of wireless channels is an important step to generate a channel correlation matrix for channel linear minimum mean square error (LMMSE) estimation. Estimated channel frequency response can be used to obtain time dispersion characteristics that can be exploited by adaptive orthogonal frequency division multiplexing (OFDM) systems. In this paper, a joint estimator for PDP and LMMSE channel estimation is proposed. For LMMSE channel estimation, we apply a candidate set of frequency-domain channel correlation functions (CCF) and select the one that best matches the current channel to construct the channel correlation matrix. The initial candidate set is generated based on the traditional CCF calculation method for different scenarios. Then, the result of channel estimation is used as an input for the PDP estimation whereas the estimated PDP is further used to update the candidate channel correlation matrix. The enhancement of LMMSE channel estimation and PDP estimation can be achieved by the iterative joint estimation procedure. Analysis and simulation results show that in different communication scenarios, the PDP estimation error of the proposed method can approach the Cramér-Rao lower bound (CRLB) after a finite number of iterations. Moreover, the mean square error of channel estimation is close to the performance of accurate PDP-assisted LMMSE.

A Robust Track Error Estimation Method for Airborne SAR Based on Accuracy Analysis Model

With the development of miniaturization technology, Synthetic Aperture Radar (SAR) can be equipped on small carriers such as small Unmanned Aerial Vehicles (UAVs). In order to lower the cost, the accuracy of navigation equipment carried by UAV SAR is usually limited, so it is challenging to meet the requirements of SAR imaging and locating accuracy. Therefore, accurately estimating SAR tracks becomes a crucial issue. So, for the motion error estimation model widely used in current literature, this paper derives the accuracy limits of the model for the first time. The derived Cramer–Rao Lower Bound (CRLB) specifies the factors affecting the estimation accuracy, which provides new insights into the estimation model. The in-depth analysis of how the factors affect CRLB can guide the setting of the parameters while using the estimation method. Moreover, based on the accuracy analysis model, this paper improves the WTLS-based autofocus method (WTA) by selecting the appropriate estimation kernel step. The proposed method can suppress noise more effectively and further ensure estimation accuracy compared to WTA. Airborne SAR data experiments in the high-resolution condition obtain trajectory estimation results of 0.02 m.

An Approximate MSE Expression for Maximum Likelihood and Other Implicitly Defined Estimators of Non-Random Parameters

An approximate mean square error (MSE) expression for the performance analysis of implicitly defined estimators of non-random parameters is proposed. An implicitly defined estimator (IDE) declares the minimizer/maximizer of a selected cost/reward function as the parameter estimate. The maximum likelihood (ML) and the least squares estimators are among the well known examples of this class. In this paper, an exact MSE expression for implicitly defined estimators with a symmetric and unimodal objective function is given. It is shown that the expression reduces to the Cramer-Rao lower bound (CRLB) and misspecified CRLB in the large sample size regime for ML and misspecified ML estimation, respectively. The expression is shown to yield the Ziv-Zakai bound (without the valley filling function) for the maximum a posteriori (MAP) estimator when it is used in a Bayesian setting, that is, when an a-priori distribution is assigned to the unknown parameter. In addition, extension of the suggested expression to the case of nuisance parameters is studied and some approximations are given to ease the computations for this case. Numerical results indicate that the suggested MSE expression not only predicts the estimator performance in the asymptotic region; but it is also applicable for the threshold region analysis, even for IDEs whose objective functions do not satisfy the symmetry and unimodality assumptions. Advantages of the suggested MSE expression are its conceptual simplicity and its relatively straightforward numerical calculation due to the reduction of the estimation problem to a binary hypothesis testing problem, similar to the usage of Ziv-Zakai bounds in random parameter estimation problems.

Optimizing Current Profiles for Efficient Online Estimation of Battery Equivalent Circuit Model Parameters Based on Cramer–Rao Lower Bound

Battery management systems (BMS) are important for ensuring the safety, efficiency and reliability of a battery pack. Estimating the internal equivalent circuit model (ECM) parameters of a battery, such as the internal open circuit voltage, battery resistance and relaxation parameters, is a crucial requirement in BMSs. Numerous approaches to estimating ECM parameters have been reported in the literature. However, existing approaches consider ECM identification as a joint estimation problem that estimates the state of charge together with the ECM parameters. In this paper, an approach is presented to decouple the problem into ECM identification alone. Using the proposed approach, the internal open circuit voltage and the ECM parameters can be estimated without requiring the knowledge of the state of charge of the battery. The proposed approach is applied to estimate the open circuit voltage and internal resistance of a battery.

Data Fusion Methods for Indoor Positioning Systems Based on Channel State Information Fingerprinting.

Indoor signals are susceptible to NLOS propagation effects, multipath effects, and a dynamic environment, posing more challenges than outdoor signals despite decades of advancements in location services. In modern Wi-Fi networks that support both MIMO and OFDM techniques, Channel State Information (CSI) is now used as an enhanced wireless channel metric replacing the Wi-Fi received signal strength (RSS) fingerprinting method. The indoor multipath effects, however, make it less robust and stable. This study proposes a positive knowledge transfer-based heterogeneous data fusion method for representing the different scenarios of temporal variations in CSI-based fingerprint measurements generated in a complex indoor environment targeting indoor parking lots, while reducing the training calibration overhead. Extensive experiments were performed with real-world scenarios of the indoor parking phenomenon. Results revealed that the proposed algorithm proved to be an efficient algorithm with consistent positioning accuracy across all potential variations. In addition to improving indoor parking location accuracy, the proposed algorithm provides computationally robust and efficient location estimates in dynamic environments. A Cramer-Rao lower bound (CRLB) analysis was also used to estimate the lower bound of the parking lot location error variance under various temporal variation scenarios. Based on analytical derivations, we prove that the lower bound of the variance of the location estimator depends on the (i) angle of the base stations, (ii) number of base stations, (iii) distance between the target and the base station, djr (iv) correlation of the measurements, ρrjai and (v) signal propagation parameters σC and γ.

Data-Aided SNR Estimation for Bandlimited Optical Intensity Channels.

Not only for radio frequency but also for optical communication systems, knowledge of the signal-to-noise ratio (SNR) is essential, e.g., for an adaptive network, where modulation schemes and/or error correction methods should be selected according to the varying channel states. In the current paper, this topic is discussed for a bandlimited optical intensity link under the assumption that the data symbols are known to the receiver unit in form of pilot sequences. This requires a unipolar signal design regarding the symbol constellation, but also a non-negative pulse shape satisfying the Nyquist criterion is necessary. Focusing on this kind of scenario, the modified Cramer-Rao lower bound is derived, representing the theoretical limit of the error performance of the data-aided SNR estimator developed in this context. Furthermore, we derive and analyze a maximum likelihood algorithm for SNR estimation, which turns out to be particularly simple for specific values of the excess bandwidth, among them the most attractive case of minimum bandwidth occupation. Numerical results confirming the analytical work conclude the paper.

Improved Target Localization in Multiwaveform Multiband Hybrid Multistatic Radar Networks

This study proposes an algorithm to improve the target localization performance. This is implemented in a multiwaveform multiband hybrid (passive and active) multistatic radar network scenario that utilizes broadcasting signals for radar sensing in addition to the radar waveforms. Multiwaveform multiband radar receivers can exploit the broadcast signals transmitted by noncooperative transmitters, such as communication or broadcasting systems, for target sensing in addition to radar waveform. Hence, multiple measurements of the targets can be acquired and fused to improve target detection and parameter estimation. Because of utilizing various waveforms, each transmitter–receiver (Tx–Rx) pair has a different range and velocity estimation accuracy, which is also affected by the bistatic geometry of the bistatic pairs. Taking this into account, this study proposes a target localization algorithm based on bistatic Cramér–Rao lower bounds (CRLBs) for multistatic multiband radar networks. It is shown that modeling the entire network, evaluating the bistatic range CRLB of each bistatic pair in advance, and utilizing this information while estimating the target location significantly improve the localization accuracy. Moreover, the proposed algorithm also includes a target height estimation correction stage to achieve better 3-D localization accuracy.

Novel cooperative localization method of over-the-horizon shortwave emitters based on direction-of-arrival and time-difference-of-arrival measurements

The location of long-distance shortwave radiation sources is practically significant in national defense security. Shortwave multistation-positioning systems mainly include direction-of-arrival (DOA) and time-difference-of-arrival (TDOA) intersection locations. These two systems have their advantages and disadvantages in terms of performance. To combine the advantages of these two positioning technologies, this study focuses on the cooperative positioning problem based on DOA and TDOA measurements. First, the two-dimensional DOA and TDOA observation models are established for the over-the-horizon propagation scenario. Subsequently, the Cramér-Rao lower bound (CRLB) of positioning accuracy is derived in the presence of measurement errors in virtual ionosphere height. Aiming at the strong nonlinear characteristics of the observation equations, a new DOA/TDOA cooperative localization method is proposed. Finally, the asymptotic efficiency of the new estimator is proved using the first-order error analysis theory. The new method comprises two stages. In the first stage, the shortwave DOA observation equation is transformed into a pseudo-linear observation equation by introducing an auxiliary variable, as well as solving a quadratic equation with one unknown. Next, an optimization model with two quadratic equality constraints is constructed, and an iterative optimization algorithm based on matrix $~\bf{QR}$ decomposition is proposed to obtain an intermediate estimate of the emitter position. The second stage transforms the shortwave TDOA observation equation into a pseudo-linear observation equation by combining the intermediate estimates given in the first phase. Also, an iterative-constrained weighted-least-squares estimator based on the matrix decomposition is proposed again to localize the emitter. Simulation results show that the new cooperative localization method has good global convergence, the localization performance can reach the CRLB, and the obtained cooperative gain is quite high.

Robust Vehicle Positioning Based on Multi-Epoch and Multi-Antenna TOAs in Harsh Environments

For radio-based time-of-arrival (TOA) positioning systems applied in harsh environments, obstacles in the surroundings and on the vehicle itself will block the signals from the anchors, reduce the number of available TOA measurements and thus degrade the localization performance. Conventional multi-antenna positioning technique requires a good initialization to avoid local minima, and suffers from location ambiguity due to insufficient number of TOA measurements and/or poor geometry of anchors at a single epoch. In this paper, taking advantage of the multi-epoch and multi-antenna (MEMA) TOA measurements bridged by inter-epoch constraints to utilize more information and improve the geometry of visible anchors, we propose a new positioning method, namely MEMA-TOA method. A new initialization method based on semidefinite programming (SDP), namely MEMA-SDP, is first designed to address the initialization problem of the MEMA-TOA method. Then, an iterative refinement step is developed to obtain the optimal positioning result based on the MEMA-SDP initialization. We derive the Cramér-Rao lower bound (CRLB) to analyze the accuracy of the new MEMA-TOA method theoretically, and show its superior positioning performance over the conventional single-epoch and multi-antenna (SEMA) localization method. Simulation results in harsh environments demonstrate that i) the new MEMA-SDP provides an initial estimation that is close to the real location, and empirically guarantees the global optimality of the final refined positioning solution, and ii) compared with the conventional SEMA method, the new MEMA-TOA method has higher positioning accuracy without location ambiguity, consistent with the theoretical analysis.

Reweighted Robust Particle Filtering Approach for Target Tracking in Automotive Radar Application

In view of the decline of filtering accuracy caused by measured outliers in target tracking application, a novel reweighted robust particle filter is proposed to acquire accurate state estimates in an automotive radar system. To infer the importance of each entry in the multidimensional contaminated measurement vector, we employ a weight vector, which follows a Gamma distribution, to model the measured noise and carry out accurate state estimates. Additionally, the particle filter method is employed to perform approximate posterior inference of state estimates in the nonlinear model. The Cramer–Rao lower bound is provided for the performance evaluation of the proposed method. Both simulation and experimental results demonstrate the superiorities of the proposed algorithm over other robust solutions.

The Identification of Mean Quantum Potential with Fisher Information Leads to a Strong Uncertainty Relation

The Cramer-Rao bound, satisfied by classical Fisher information, a key quantity in information theory, has been shown in different contexts to give rise to the Heisenberg uncertainty principle of quantum mechanics. In this paper, we show that the identification of the mean quantum potential, an important notion in Bohmian mechanics, with the Fisher information, leads, through the Cramer-Rao bound, to an uncertainty principle which is stronger, in general, than both Heisenberg and Robertson-Schrodinger uncertainty relations, allowing to experimentally test the validity of such an identification.

Gaussian processes based extended target tracking in polar coordinates with input uncertainty

Gaussian processes (GP) based extended target tracking (ETT) technique has attracted the attention of scientists since it can accurately estimate the kinematic states as well as the contour states of irregular-shape targets. Most traditional GP methods consider the ETT problem with a linear measurement model in Cartesian coordinates and ignore the input uncertainty of GP for simplicity. In many applications, however, measurements are generated with high-resolution sensors in polar coordinates. More importantly, the ignorance of input uncertainty in GP will exacerbate the tracking performance in these cases. In order to track an irregular-shape extended target in polar coordinates with input uncertainty, an improved GP-based probabilistic data association (IGP-PDA) algorithm is developed that includes the following enhancements: Firstly, a nonlinear measurement model is used and the unbiased converted measurement technique is invoked for the ETT problem. Secondly, the analytical form of statistical properties of the input uncertainty due to measurements noise and predicted error is derived. Thirdly, an IGP-PDA taking into account measurement origin uncertainty as well as input uncertainty is proposed, and three sub-optimal implementations are provided. Last, the posterior Cramer-Rao lower bound of ETT with input uncertainty is derived. Simulation results verify the effectiveness of the proposed method.

H∞ filter-based dynamic joint carrier frequency offset and linear phase noise tracking for CO-OFDM systems.

In this paper, a robust, dynamic and joint carrier frequency offset (CFO) and linear phase noise (LPN) tracking algorithm is proposed by utilizing the H∞ filter for CO-OFDM systems. The dynamic tracking is implemented by Bayesian filtering which time-recursively updates the posterior state estimates based on the prior state information and calculated gain. To improve the robustness of the dynamic tracking process against the uncertainty of the noise terms, we adopt the H∞ filter, which designs a min-max optimization problem with a performance bound-defined constraint. The H∞ filter obtains the state estimates by limiting the worst-case estimation error, which guarantees its robustness to the system uncertain noise terms and external disturbances. After the joint estimation of CFO and LPN, we then propose a decision-feedback algorithm to achieve accurate signal detection. The accuracy and robustness of the proposed algorithm are verified in an 88.9 Gb/s 16-QAM CO-OFDM system, by comparing with the conventional extended Kalman filter (EKF) and Gaussian particle filter (GPF). Simulation results show that the MSEs of CFO and LPN by H∞ filter can reach the Cramér-Rao Lower Bounds (CRLBs) as well as the Bayesian CRLB (BCRLB), and possesses excellent noise and chromatic dispersion (CD) tolerance.

Robust blind separation of smooth graph signals using minimization of graph regularized mutual information

The smoothness of the graph signals on predefined/constructed graphs appears in many natural applications of processing unstructured (i.e., graph-based) data. In the case of latent sources being smooth graph signals, blind source separation (BSS) quality can be significantly improved by exploiting graph signal smoothness along with the classic measures of statistical independence. In this paper, we propose a BSS method benefiting from the minimization of mutual information as a well-known independence criterion and also graph signal smoothness term of the estimated latent sources, and show that its performance is superior and fairly robust to the state-of-the-art classic and Graph Signal Processing (GSP)-based methods, even in the case of the smooth graph signal sources being heavily corrupted. In addition to the comprehensive complexity and convergence analysis, the asymptotic performance of the proposed method is comprehensively compared with that of the Cramér-Rao Bound (CRB) to illustrate the asymptotic efficiency of the proposed method. Besides, the comprehensive real-world analysis of separating mixed images and speech signals illustrates the applicability and efficiency of the proposed method in real-world scenarios.

3-D Sound Source Localization With a Ternary Microphone Array Based on TDOA-ILD Algorithm

Sound source localization is essential for machine-based environmental perception when visual methods are not applicable. Current array-based localization methods often require many microphones and are heavy in computing. This article proposes an efficient algorithm for estimating the location of a single source in 3-D based on the time difference of arrival and interaural level difference (TDOA-ILD) with an array of only three microphones. The algorithm can compute the source 3-D coordinates with a closed-form analytical solution derived from geometrical modeling and identify the actual source from the cone of the confusion area through clustering methods. This article comprehensively evaluates the algorithm’s behavior for various array sizes and source locations in a wide range of signal-to-noise ratios (SNRs) with simulative models to prove that the algorithm achieves estimation errors close to its Cramer–Rao lower bound (CRLB). Experimental evaluation demonstrates the method’s localization feasibility in outdoor open environments with off-the-shelf sensing equipment. The proposed scheme is suitable for real-time 3-D sound source localization with cost-effective hardware.

Performance Analysis of ML-based Low Complex CFO Estimation for MIMO-OFDMA Uplink Systems

Carrier frequency offset (CFO) estimation in multiuser multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) system is investigated in this study. MIMO-OFDM is very sensitive to CFOs due to oscillator frequency mismatch and/or Doppler shift. Inaccurate CFO estimation results in intercarrier interference (ICI) through the loss of orthogonality among subcarriers. In this paper, the performance of the ML and APFE algorithm is analyzed for CFO estimation. ML becomes extremely complex due to the multidimensional exhaustive search issue, which is the basic concern in ML estimation. However, making use of the iterative low complex APFE method, here this multidimensional search is replaced with a sequence of mono-dimensional searches. This results in an estimation algorithm of reasonable complexity which is suitable for practical applications. In addition, ML accuracy is compared with the Cramer-Rao bound (CRB).

Proximity Detection During Epidemics: Direct UWB TOA Versus Machine Learning Based RSSI.

In this paper, we compare the direct TOA-based UWB technology with the RSSI-based BLE technology using machine learning algorithms for proximity detection during epidemics in terms of complexity of implementation, availability in existing smart phones, and precision of the results. We establish the theoretical limits on the precision and confidence of proximity estimation for both technologies using the Cramer Rao Lower Bound (CRLB) and validate the theoretical foundations using empirical data gathered in diverse practical operating scenarios. We perform our empirical experiments at eight distances in three flat environments and one non-flat environment encompassing both Line of Sight (LOS) and Obstructed-LOS (OLOS) situations. We also analyze the effects of various postures (eight angles) of the person carrying the sensor, and four on-body locations of the sensor. To estimate the range with BLE RSSI, we use 14 features for training the Gradient Boosted Machines (GBM) learning algorithm and we compare the precision of results with those obtained from memoryless UWB TOA ranging algorithm. We show that the memoryless UWB TOA algorithm achieves 93.60% confidence, slightly outperforming the 92.85% confidence of the BLE RSSI with more complex GBM machine learning (ML) algorithm and the need for substantial training. The training process for the RSSI-based BLE social distance measurements involved 3000 measurements to create a training dataset for each scenario and post-processing of data to extract 14 features of RSSI, and the ML classification algorithm consumed 200 s of computational time. The memoryless UWB ranging algorithm achieves more robust results without any need for training in less than 0.5 s of computation time.Graphical