Corresponding Cramer-Rao Lower Bound Research Articles

MLE-MPPL: A Maximum Likelihood Estimator for Multipolarimetric Phase Linking in MTInSAR

Multi-Temporal Synthetic Aperture Radar Interferometry (MTInSAR) is an efficient geodetic tool for earth surface displacement measurement, and the polarimetric capability of current and upcoming Synthetic Aperture Radar (SAR) satellites offers a new opportunity to further improve MTInSAR phase series estimation. However, none of existing estimators for multipolarimetric MTInSAR phase series of distributed scatters (DSs) is derived under the minimum root mean square error (RMSE) criterion. In this work, a maximum likelihood estimator for multipolarimetric phase linking (MLE-MPPL) is proposed and the corresponding Cramer-Rao lower bound (CRLB) is also derived by modeling the polarimetric interferometric coherence matrix as the Kronecker product of polarimetric coherence matrix and interferometric coherence matrix. In addition, a new metric called Pol-detR is proposed for performance evaluation of multipolarimetric MTInSAR phase series estimation in practical scenarios where the RMSE is not feasible any more. Experimental results based on both simulated and real data show that the proposed MLE-MPPL achieves the best estimation performance and is more robust against inter-channel interference than existing methods.

Unknown Transmit Power RSSD-Based Localization in a Gaussian Mixture Channel

Localization is a critical task in wireless sensor networks (WSNs). However, accurate localization is difficult when there is noise from external sources and faulty nodes, especially when the transmit power is unknown. Signal strength-based localization methods are popular due to their low complexity and simple implementation. In this paper, robust localization based on the received signal strength difference (RSSD) is considered with unknown transmit power and Gaussian mixture noise in the presence of faulty nodes. A robust fault-tolerant localization (RFLT) technique is proposed by reformulating the original localization problem using a generalized trust-region subproblem (GTRS) framework. The resulting problem is non-convex so it is divided into two subproblems using a regularization item (RI) with a block-update surrogate function (BUSF). The active set method (ASM) is used to obtain an initial solution. The corresponding Cramer–Rao lower bound (CRLB) for Gaussian mixture noise is derived as a benchmark. Simulation results are presented which show that the performance of the proposed method is superior to other state-of-the-art methods.

Directional source localization based on RSS-AOA combined measurements

Source localization plays an indispensable role in many applications. This paper addresses the directional source localization problem in a three-dimensional (3D) wireless sensor network using hybrid received-signal-strength (RSS) and angle-of-arrival (AOA) measurements. Both the position and transmission orientation of the source are to be estimated. In the considered positioning scenario, the angle and range measurements are respectively corresponding to the AOA model and RSS model that integrates the Gaussian-shaped radiation pattern. Given that the localization problem is non-convex and the unknown parameters therein are coupled together, this paper adopts the second-order cone relaxation and alternating optimization techniques in the proposed estimation algorithm. Moreover, to provide a performance benchmark for any localization method, the corresponding Cramer-Rao lower bounds (CRLB) of estimating the unknown position and transmission orientation of the source are derived. Numerical and simulation results demonstrate that the presented algorithm effectively resolves the problem, and its estimation performance is close to the CRLB for the localization with the hybrid measurements.

Gridless Sparse Direction Finding Method for Correlated Signals with Gain-Phase Errors

In practical array systems, the gain-phase errors among antennas degrade the performance of direction finding significantly. In this paper, a novel sparse system model for direction of arrival (DOA) estimation in the scenario with gain-phase errors is proposed by exploiting the signal sparsity in the spatial domain. In contrast to the existing sparse-based methods using the grids to construct the dictionary matrix, a novel gridless method based on atomic norm and convex optimization is proposed, where the gain-phase errors are described by a diagonal matrix. With the Schur complement, a semidefinite programming is formulated from the optimization problem, and can be solved efficiently. With the gain-phase errors, the corresponding Cram’er-Rao lower bound (CRLB) of direction finding is derived as an estimation benchmark. Simulation results show that the proposed method performs better than the state-of-the-art methods in the scenario with correlated signals and gain-phase errors.

RSS-Based Localization of Multiple Directional Sources With Unknown Transmit Powers and Orientations

Received-signal-strength (RSS)-based localization has received widespread attention recently. Due to the simple acquisition of the RSS measurements, the adequate inexpensive sensors in sensor networks are capable of providing the information needed for the positioning of multiple target sources. However, few studies have focused on the RSS-based localization of multiple directional sources that are common in reality. Based on the deduced parametric Optimal Maximum Likelihood (OML) solution, this paper proposes three new grid search-based algorithms, namely Alternating Projection (i.e., OMLAP) algorithm, Expectation-Maximization like (i.e., OMLEM) algorithm, and Particle Swarm Optimization (i.e., OMLPSO) algorithm. They can be utilized for estimating the transmit powers, locations, and orientations of multiple directional sources. Combining the interpolation process and proposed power threshold setting method, the search space is obviously reduced. Moreover, the corresponding Cramer-Rao lower bounds (CRLB) are also derived to characterize the estimation accuracy of the algorithms. Both the scenarios with different Signal-to-Noise Ratios (SNRs) and the scenarios with different sensor quantities are considered in the simulation, and the results demonstrate the effectiveness of the proposed algorithms and indicate that they are suitable for the parameter estimation of multiple directional sources.

Received-Signal-Strength-Based Localization in Wireless Sensor Networks

In this paper, an overview of recent developments in received-signal-strength (RSS)-based localization in wireless sensor networks is presented. Several important practical issues and their solutions are discussed. To save communication bandwidth and sensor energy, a maximum-likelihood estimator based on quantized data is presented along with its corresponding Cramer-Rao lower bound (CRLB) and optimal quantizer design schemes. For further system resource savings, an iterative sensor selection approach is presented to activate only the most informative sensors, by maximizing the mutual information or minimizing the posterior CRLB at each iteration. For a resource constrained WSN with imperfect wireless channels, channel-aware target localization is described, where the channel model is incorporated into the localization scheme itself, thereby improving performance without increasing communication overhead. Another practical issue involving the presence of malicious sensors called Byzantines is discussed and mitigation schemes are provided. A recent coding-theorybased approach which is both computationally inexpensive and robust to such malicious attacks is also discussed.

Algorithms and Performance Analysis for Stochastic Wiener System Identification

We analyze the statistical performance of identification of stochastic dynamical systems with non-linear measurement sensors. This includes stochastic Wiener systems, with linear dynamics, process noise and measured by a non-linear sensor with additive measurement noise. There are many possible system identification methods for such systems, including the maximum likelihood (ML) method and the prediction error method. The focus has mostly been on algorithms and implementation, and less is known about the statistical performance and the corresponding Cramer–Rao lower bound (CRLB) for identification of such non-linear systems. We derive expressions for the CRLB and the asymptotic normalized covariance matrix for certain Gaussian approximations of Wiener systems to show how a non-linear sensor affects the accuracy compared to a corresponding linear sensor. The key idea is to take second order statistics into account by using a common parametrization of the mean and the variance of the output process. This analysis also leads to an ML motivated identification method based on the conditional mean predictor and a Gaussian distribution approximation. The analysis is supported by numerical simulations.

Efficient Noise Variance Estimation Under Pilot Contamination for Massive MIMO Systems

Massive multiple input multiple output (MIMO) is expected to be one of the enabling technologies for fifth-generation cellular networks. One of the major challenges in massive MIMO systems is the accurate joint estimation of the channel and noise variance, which significantly affects the performance of wireless communications in practical scenarios. In this paper, we first derive a novel maximum likelihood estimator for the noise variance at the receiver of massive MIMO systems considering practical impairments such as pilot contamination. Then, this estimate is used to compute the minimum mean square error estimate of the channel. In order to measure the performance of the proposed noise variance estimator, we derive the corresponding Cramer–Rao lower bound (CRLB). Simulation results show that the estimator is efficient in certain scenarios, outperforming existing approaches in the literature. Furthermore, we develop the estimator and the CRLB for equal and different noise variance at the receive antennas. Although the proposed estimator is valid for all antenna array sizes, its use is particularly effective for massive MIMO systems.

Robust Localization with Crowd Sensors: A Data Cleansing Approach

In this paper, the source localization problem with crowd of anchor nodes is investigated, under the circumstances that abnormal data could be sporadically and randomly produced for the reason of either accidental equipment failures or random malicious behaviors. To cope with the problem that abnormal data brings, we formulate a generalized modeling of abnormal data in localization problem, which involves the impacts of both unexpected equipment failures and malicious data falsifications. The corresponding Cramer-Rao lower bound (CRLB) of the specific localization problem is derived. For the localization enhancement, we propose a data cleansing-based robust localization algorithm which exploits the low occupancy of channel band by sources and the sparsity of abnormal data. The data cleansing approach achieves both the new sensing data matrix that cleansed out abnormal data component and the estimated abnormal data matrix, which are respectively used for the correct source detection process and the position estimation process of the final source localization. The root mean squared error (RMSE) is derived to assess the performance of the proposed robust localization algorithm. Computer simulations show that the proposed data cleansing-based robust localization algorithm can effectively eliminate the impairment of the abnormal data and hence improve the localization performance evidently.

Optimal source localization problem based on TOA measurements

Determining an optimal emitting source location based on the time of arrival (TOA) measurements is one of the important problems in Wireless Sensor Networks (WSNs). The nonlinear least-squares (NLS) estimation technique is employed to obtain the location of an emitting source. This optimization problem has been formulated by the minimization of the sum of squared residuals between estimated and measured data as the objective function. This paper presents a hybridization of Genetic Algorithm (GA) for the determination of the global optimum solution with the local search Newton-Raphson (NR) method. The corresponding Cramer-Rao lower bound (CRLB) on the localization errors is derived, which gives a lower bound on the variance of any unbiased estimator. Simulation results under different signal-to-noise-ratio (SNR) conditions show that the proposed hybrid Genetic Algorithm-Newton-Raphson (GA-NR) improves the accuracy and efficiency of the optimal solution compared to the regular GA.

Method for estimating the central frequency of phase‐coded radar signals

The authors analyse radar carrier frequency estimation errors in digital electronic support measures receivers. They assume that the received signal is phase-coded, real-valued and embedded in white Gaussian noise. The corresponding Cramer–Rao lower bound (CRLB) for the frequency estimation is derived and formulated. The performance degradation can be significant when just a single traditional fast Fourier transform (FFT) algorithm used to estimate the central frequency of a phase-coded signal, where the spectrum is wide and distributes across many frequency bins. In this study, they assume that the FFT magnitude response of the received signal can be approximated by a raised Gaussian-shaped function. The corresponding Gaussian function fitting (GFF) estimator is proposed as a peak frequency estimation technique. Typical FFT-based frequency estimation process is simulated and the performance of the GFF estimator is compared with that of the most commonly used peak estimators: namely, with the maximum point (MAX) and the spline interpolation (INT). GFF estimator performance is also compared with a K-means-based peak estimator. With extensive set of simulations for different code lengths, they demonstrate that the proposed technique significantly reduces carrier (central) frequency estimation errors while illustrating CRLB within the figure.

Performance analysis of target parameters estimation using multiple widely separated antenna arrays

Target parameter estimation performance is investigated for a radar employing a set of widely separated transmitting and receiving antenna arrays. Cases with multiple extended targets are considered under two signal model assumptions: stochastic and deterministic. The general expressions for the corresponding Cramer-Rao lower bound (CRLB) and the asymptotic properties of the maximum-likelihood (ML) estimator are derived for a radar with $M_t$ arrays of $L_t$ transmitting elements and $M_r$ arrays of $L_r$ receiving elements for both types of signal models. It is shown that for an infinitely large product $M_tM_r$, and a finite $L_r$, the ML estimator is consistent and efficient under the stochastic model, while the deterministic model requires $M_tM_r$ to be finite and $L_r$ to be infinitely large in order to guarantee consistency and efficiency. Monte Carlo simulations further investigate the estimation performance of the proposed radar configuration in practical scenarios with finite $M_tM_r$ and $L_r$, and a fixed total number of available receiving antenna elements, $M_r L_r$. The numerical results demonstrate that grouping receiving elements into properly sized arrays reduces the mean squared error (MSE) and decreases the threshold SNR. In the numerical examples considered, the preferred configurations employ $M_t M_r > 1$. In fact, when $M_t M_r$ becomes too small, due to the loss of the geometric gain, the estimation performance becomes strongly dependent on the particular scenario and can degrade significantly, while the CRLB may become a poor prediction of the MSE even for high SNR. This suggests it may be advantageous to employ approaches where neither $M_tM_r$ nor $L_r$ are too small.

Convex Optimization Algorithms for Multiple Source Localization Based on Received Signal Strength Measurements

Decaying with the increasing of signal propagation distance, Received Signal Strength (RSS) is used in the wireless localization due to its low cost and easily implementation. When the transmit power is unavailable, two convex optimization algorithms including semi definite programming (SDP), second order cone and semi definite programming (SOC/SDP) are designed to estimate the source locations by relaxing the non-convex problem as convex optimization. The corresponding CramerRao lower bound (CRLB) of the problem is derived. The simulations demonstrate that the SOC/SDP algorithm provides the similar accuracy performance compared with the SDP algorithm. However the computational complexity of SOC/SDP is lower than that of the SDP due to the less variables and equality constraints. When perfect knowledge of the path loss exponent is available, the simulations also show that the accuracy performance of the proposed convex optimization algorithms degrades as the path loss exponent increases.

Clock Skew Estimation of Listening Nodes with Clock Correction upon Every Synchronization in Wireless Sensor Networks

Time synchronization is a significant component in Wireless Sensor Networks (WSNs) for maintaining synchrony among nodes. Most time synchronization protocols in WSNs utilize dedicated synchronization packets for clock accuracy optimization. However, in practical WSNs, the joint design of time synchronization and Media Access Control layer protocols should be considered. One approach is adding timestamps directly into data packets and the corresponding acknowledgements (ACKs). Therefore, communication and energy overhead could be saved greatly and time synchronization could be seamlessly integrated into networks. In this letter, we investigate the time synchronization scheme of listening nodes overhearing the neighboring two-way timing packet exchange based on periodical ACK mechanism and present an efficient clock skew estimation algorithm with clock correction upon every synchronization. The Maximum Likelihood Estimator (MLE) of clock skew for Gaussian random packet delay is derived, and the corresponding Cramer-Rao Lower Bound (CRLB) is obtained. In addition, the MLE shows that the clock skew could be estimated without any prior knowledge of the fixed packet delay and the time of adjustment. Simulation results verify that the MLE is efficient.

基于外辐射源的约束总体最小二乘定位算法及其理论性能分析

A passive location and tracking system that locates “silent” targets using an external illuminator is of great importance in a variety of applications. Based on the spatial-, time-, and frequency-domain measures of this system, a constrained total least squares (CTLS) localization theory and a method that takes into account observer location and velocity errors are proposed. First, four kinds of joint location mechanisms are summarized, and the algebraic processing method that transforms nonlinear measurement equations into pseudo-linear equalities was considered. Then, the CTLS optimization model was constructed for locating the target, and the Newton iteration algorithm to solve the optimization problem was derived. Subsequently, an analytical performance analysis was performed for the CTLS location algorithm including observer location and velocity errors; thus, a performance analysis frame was built. In addition, the CTLS location variance was mathematically proved to be able to reach the corresponding Cramer-Rao lower bound (CRLB). Finally, some numerical experiments were devised to corroborate the advantage of the presented algorithm and the effectiveness of the theoretical analysis.

Parameter Estimation From Quantized Observations in Multiplicative Noise Environments

The problem of distributed parameter estimation from binary quantized observations is studied when the unquantized observations are corrupted by combined multiplicative and additive Gaussian noise. These results are applicable to sensor networks where the sensors observe a parameter in combined additive and nonadditive noise and to a case where dispersed receivers are employed with analog communication over fading channels where the receivers employ binary quantization before noise-free digital communications to a fusion center. We first discuss the case in which all the quantizers use an identical threshold. The parameter identifiability condition is given, and, surprisingly, it is shown that unless the common threshold is chosen properly and the parameter lies in an open interval, the parameter will not generally be identifiable, in contrast to the additive noise case. The best achievable mean square error (MSE) performance is characterized by deriving the corresponding Cramer-Rao Lower Bound (CRLB). A closed-form expression describing the corresponding maximum likelihood (ML) estimator is presented. The stability of the performance of the ML estimators is improved when a nonidentical threshold strategy is utilized to estimate the unknown parameter. The thresholds are designed by maximizing the minimum asymptotic relative efficiency (ARE) between quantized and unquantized ML estimators. Although the ML estimation problem is nonconvex, it is shown that one can relax the optimization to make it convex. The solution to the relaxed problem is used as an initial solution in a gradient algorithm to solve the original problem. Next, the case where both the variances of the additive noise and multiplicative noise are unknown is studied. The corresponding CRLB is obtained, and the ML estimation problem is transformed to a convex optimization, which can be solved efficiently. Finally, numerical simulations are performed to substantiate the theoretical analysis.

The constrained-total-least-squares localization algorithm and performance analysis based on an external illuminator

A passive location and tracking system that locates silent targets using an external illuminator is of great importance in a variety of applications. Based on the spatial-, time-, and frequency-domain measures of this system, a constrained total least squares(CTLS) localization theory and a method that takes into account observer location and velocity errors are proposed. First, four kinds of joint location mechanisms are summarized, and the algebraic processing method that transforms nonlinear measurement equations into pseudo-linear equalities was considered. Then, the CTLS optimization model was constructed for locating the target, and the Newton iteration algorithm to solve the optimization problem was derived. Subsequently, an analytical performance analysis was performed for the CTLS location algorithm including observer location and velocity errors;thus, a performance analysis frame was built. In addition, the CTLS location variance was mathematically proved to be able to reach the corresponding Cram′er-Rao lower bound(CRLB). Finally, some numerical experiments were devised to corroborate the advantage of the presented algorithm and the effectiveness of the theoretical analysis.

Multiple 3D Target Tracking in Binary Wireless Sensor Network

In this paper, 3D target tracking in wireless sensor network (WSN) is considered. Unlike other published work, this paper addresses the problem of estimating locations of different targets in 3D environment. Each sensor makes T observations to get local decisions. These decisions are modulated using on-off keying (OOK) and sent to a central node called fusion center. The fusion center detects all decisions sent from all sensors to process them in order to estimate the targets locations using maximum likelihood estimator. This approach generalizes the existing work in the literature about a single target localization if we considered that the number of targets equal one. The corresponding Cramer-Rao lower bound (CRLB) has been also derived. Furthermore, simulation results are provided and discussed. 

Maximum Likelihood SNR Estimation of Linearly-Modulated Signals Over Time-Varying Flat-Fading SIMO Channels

In this paper, we tackle for the first time the problem of maximum likelihood (ML) estimation of the signal-to-noise ratio (SNR) parameter over time-varying single-input multiple-output (SIMO) channels. Both the data-aided (DA) and the non-data-aided (NDA) schemes are investigated. Unlike classical techniques where the channel is assumed to be slowly time-varying and, therefore, considered as constant over the entire observation period, we address the more challenging problem of instantaneous (i.e., short-term or local) SNR estimation over fast time-varying channels. The channel variations are tracked locally using a polynomial-in-time expansion. First, we derive in closed-form expressions the DA ML estimator and its bias. The latter is subsequently subtracted in order to obtain a new unbiased DA estimator whose variance and the corresponding Cram\'er-Rao lower bound (CRLB) are also derived in closed form. Due to the extreme nonlinearity of the log-likelihood function (LLF) in the NDA case, we resort to the expectation-maximization (EM) technique to iteratively obtain the exact NDA ML SNR estimates within very few iterations. Most remarkably, the new EM-based NDA estimator is applicable to any linearly-modulated signal and provides sufficiently accurate soft estimates (i.e., soft detection) for each of the unknown transmitted symbols. Therefore, hard detection can be easily embedded in the iteration loop in order to improve its performance at low to moderate SNR levels. We show by extensive computer simulations that the new estimators are able to accurately estimate the instantaneous per-antenna SNRs as they coincide with the DA CRLB over a wide range of practical SNRs.

Adaptive quantization for distributed estimation in cluster-based wireless sensor networks

In this paper, the problem of parameter estimation in cluster-based wireless sensor networks is studied. Particularly, we focus on how to choose a suitable threshold in the one-bit adaptive quantization scheme. An adaptive quantization scheme for parameter estimation in cluster-free sensor networks is extended to this scenario. Intra-cluster and inter-cluster maximum likelihood estimators (MLEs) as well as the corresponding Crame´r–Rao lower bounds (CRLBs) are derived. Due to the energy constraint of sensors, the performance–energy tradeoff of parameter estimation is also investigated. Simulation results show that parameter estimation in cluster-based sensor networks with adaptive quantization can be more energy-efficient than that in cluster-free sensor networks, while achieving close performance as the number of sensors increases.