The Exact Cramer-Rao Bound for Gaussian Autoregressive Processes

An explicit expression is derived for the Cramer-Rao bound(CRB)on unbiased estimates of the parameters of autoregressiveprocesses, given a finite number of measurements. The expressionconverges to the well-known asymptotic form of the CRB when thenumber of measurements tends to infinity. The behavior of thebound is illustrated by some numerical examples.     

Performance of diversely polarized antenna arrays for correlatedsignals

The performance of direction finding systems in a correlated signal environment utilizing diversely polarized antenna arrays is investigated. The Cramer-Rao bound (CRB) is used to evaluate the accuracy of the estimated directions of arrival (DOAs). Compact closed form formulas are presented for the CRB corresponding to the joint estimation of the DOAs, signal covariance matrix, signal polarization parameters, and noise variance. The CRB is evaluated numerically for selected examples, to provide insights into the potential improvement in direction-finding accuracy due to polarization diversity     

Posterior Cramer-Rao bounds for multi-target tracking

This study is concerned with multi-target tracking (MTT). The Cramer-Rao lower bound (CRB) is the basic tool for investigating estimation performance. Though basically defined for estimation of deterministic parameters, it has been extended to stochastic ones in a Bayesian setting. In the target tracking area, we have thus to deal with the estimation of the whole trajectory, itself described by a Markovian model. This leads up to the recursive formulation of the posterior CRB (PCRB). The aim of the work presented here is to extend this calculation of the PCRB to MTT under various assumptions.     

DOA and steering vector estimation using a partially calibratedarray

We consider the problem of estimating directions of arrival (DOAs) using an array of sensors, where some of the sensors are perfectly calibrated, while others are uncalibrated. We identify a cost function whose minimizer is a statistically consistent and efficient estimator of the unknown parameters-the DOAs and the gains and phases of the uncalibrated sensors. Next we present an iterative algorithm for finding the minimum of that cost function The proposed algorithm is guaranteed to converge. The performance of the estimation algorithm is compared with the Cramer Rao bound (CRB). The derivation of the bound is also included. It is shown that DOA accuracy can be improved by adding uncalibrated sensors to a precisely calibrated array. Moreover, the number of sources that can be resolved may be larger than the number that can be resolved by the calibrated portion of the array     

Range, radial velocity, and acceleration MLE using radar LFM pulsetrain

An efficient implementation of the maximum likelihood estimator (MLE) is presented for the estimation of target range, radial velocity, and acceleration when the radar waveform consists of a wideband linear frequency modulated (LFM) pulse train. Analytic properties of the associated wideband ambiguity function are derived; in particular the ambiguity function, with acceleration set to zero, is derived in closed form. Convexity and symmetry properties of the ambiguity function over range, velocity, and acceleration are presented; these are useful for determining region and speed of convergence for recursive algorithms used to compute the MLE. In addition, the Cramer-Rao bound (CRB) is computed in closed form which shows that the velocity bound is decoupled from the corresponding bounds in range and acceleration. A fast MLE is then proposed which uses the Hough transform (HT) to initialize the MLE algorithm. Monte Carlo simulations show that the MLE attains the CRB for low to moderate signal-to-noise depending on the a priori estimates of range, velocity, and acceleration     

DOA estimation using one-bit quantized measurements

The effects 1-bit quantization of the input samples has on the direction-of-arrival (DOA) estimation accuracy are considered. The signal model assumes a single stochastic Gaussian point source that is embedded in white Gaussian noise (WGN). The inherent limitations governed by the extreme clipping of the input data are analyzed using the Cramer-Rao bound (CRB) that is derived for a two-sensor array. In addition, several estimators for the I-bit estimation are discussed. Numerical and analytical analyses of the estimation error reveal weak dependency on signal-to-noise ratio (SNR) with singular behavior of the estimation error in certain DOA angles.     

DOA estimation under unknown mutual coupling and multipath

We propose a new method for direction of arrival (DOA) estimation in the presence of multipath propagation and mutual coupling for a frequency hopping (FH) system. With the use of pilot symbols and assuming perfect time-frequency synchronization for a linear array, we take mutual coupling and multipath propagation into account, and derive a maximum likelihood (ML) estimator for both the mutual coupling matrix and DOA estimation. We then formulate an iterative alternating minimization (AM) algorithm for finding the mutual coupling and DOA parameters in an alternate manner. Simulation results illustrating the performance of the algorithm and comparison with the Cramer-Rao bound (CRB) are also presented.     

一种针对恒模信号的运动单站直接定位算法

相比于常规的"测向+位置估计"两步定位模式,以Weiss等提出的目标直接位置确定(DPD)算法具有估计精度高、分辨能力强和无需数据关联等诸多优点。基于该类定位算法的基本理念,提出了一种利用单个运动天线阵列对恒模(即相位调制)信号的DPD算法。首先,依据最大似然(ML)准则以及恒模信号的恒包络特征,建立了相应的直接定位优化模型;接着,根据优化函数的代数特征提出了一种有效的多参量交替迭代算法,用以获得ML估计器的最优数值解;此外,推导了针对恒模信源的位置直接估计方差的克拉美罗界(CRB),从而为新算法的定位精度提供定量的理论下界。仿真实验表明:相比于已有的基于单个运动天线阵列的直接定位算法以及传统的两步定位算法,通过利用恒模信号的恒包络特征可以明显提高目标直接定位的估计精度。     

基于声波传播时间估计的气流速度测量方法

 为实现对亚声速和超声速气流速度的统一测量,提出了一种基于声传感器的新型测量方法。首先,根据声波在亚声速和超声速气流中的传播特性,利用特定的测量装置建立了声波传播时间与气流速度之间的数学模型,从而将气流速度的测量问题转化为声波传播时间的测量问题。然后,在此基础上,利用计时法和最大似然估计(MLE)方法来估计声波传播时间;其中,计时法在实时性上优势明显,而MLE方法则在可靠性上优于前者。最后,分别从阵元位置扰动性、计时误差和克拉美-罗界(CRB)3个方面对所提算法的性能进行了分析与仿真验证。结果表明,该算法能够实现对亚声速和超声速气流速度的精确测量。     

基于四线性分解的双基地MIMO雷达的角度和多普勒频率联合估计

研究双基地多输入多输出(Multiple-input Multiple-output, MIMO)雷达中的角度和多普勒频率联合估计问题,提出了一种基于四线性分解(Quadrilinear Decomposition)的离开角(Direction of Departure, DOD)、波达角(Direction of Arrival, DOA)和多普勒频率的联合估计算法。通过对接收端匹配滤波器的输出进行延迟操作,得到符合四线性模型的数据,根据四线性交替最小二乘(Quadrilinear Alternating Least Squares, QALS)进行迭代,得到方向矩阵和多普勒频率矩阵的估计,进而得到角度和频率的估计。该算法无需谱峰搜索,无需知道反射系数,可实现角度和频率的自动配对,且能用于非均匀阵,该算法的角度估计性能优于多维ESPRIT方法和三线性交替最小二乘(Trilinear Alternating Least Squares, TALS)方法。论文分析了所提算法复杂度,并推导了克拉美-罗界(Cramer-Rao bound, CRB)。仿真结果验证了该算法的有效性。     

CRB: Sinusoid-Sources' Estimation using Collocated Dipoles/Loops

This work derives new asymptotic Cram?r-Rao lower bounds (CRB) for the estimation of multiple pure-tone incident signals' azimuth-elevation arrival-angles, polarization parameters, frequencies, amplitudes, and temporal phases—based on data collected by spatially collocated but orthogonally oriented dipoles and/or loops. The incident sources are pure-tones at distinct, deterministic but unknown frequencies, in contrast to the case of all incident sources at one common known frequency, as has been investigated in the existing research literature on the CRB for diversely-polarized direction-finding. The derived CRBs are closed-form expressions, explicitly in terms of the signal parameters. The new CRBs presented here reveal how a constituent dipole and/or loop's presence and orientation may affect estimation precision, thereby offering guidelines to the system engineer on what dipole(s) and/or loop(s) to include or to omit in constructing the electromagnetic vector-sensor.     

A two-step weighted least-squares method for joint estimation of source and sensor locations: A general framework

It is well known that the Two-step Weighted Least-Squares(TWLS) is a widely used method for source localization and sensor position refinement. For this reason, we propose a unified framework of the TWLS method for joint estimation of multiple disjoint sources and sensor locations in this paper. Unlike some existing works, the presented method is based on more general measurement model, and therefore it can be applied to many different localization scenarios.Besides, it does not have the initialization and local convergence problem. The closed-form expression for the covariance matrix of the proposed TWLS estimator is also derived by exploiting the first-order perturbation analysis. Moreover, the estimation accuracy of the TWLS method is shown analytically to achieve the Cramér-Rao Bound(CRB) before the threshold effect takes place. The theoretical analysis is also performed in a common mathematical framework, rather than aiming at some specific signal metrics. Finally, two numerical experiments are performed to support the theoretical development in this paper.     

Correlation between horizontal and vertical monopulse measurements

Many radar systems use the monopulse ratio to extract angle of arrival (AOA) measurements in both azimuth and elevation angles. The accuracies of each such measurement are reasonably well known: each measurement is, conditioned on the sum-signal return, Gaussian-distributed with calculable bias (relative to the true AOA), and variance. However, we note that the two monopulse ratios are functions of basic radar measurements that are not entirely independent, specifically in that the sum signal is common to both. The effect of this is that the monopulse ratios are dependent, and a simple explicit expression is given for their correlation; this is of considerable interest when the measurements are supplied to a tracking algorithm that requires a measurement covariance matrix. The system performance improvement when this is taken into account is quantified: while it makes little difference for a tracking radar with small pointing errors, there are more substantial gains when a target is allowed to stray within the beam, as with a rotating (track-while-scan) radar or when a single radar dwell interrogates two or more targets at different ranges. But in any case, the correct covariance expression is so simple that there is little reason not to use it. We additionally derive the Cramer-Rao lower bound (CRLB) on joint azimuth/elevation angle estimation and discover that it differs only slightly from the covariance matrix corresponding to the individual monopulse ratios. Hence, using the individual monopulse ratios and their simple joint accuracy expression is an adequate and quick approximation of the optimal maximum likelihood procedure for single resolved targets.     

Detecting small moving objects using temporal hypothesis testing

This paper addresses the problem of detecting small, moving, low amplitude objects in image sequences that also contain moving nuisance objects and background noise. We formulate this problem in the context of a hypothesis testing procedure on individual pixel temporal profiles, leading to a computationally efficient statistical test. The technique assumes we have reasonable deterministic and statistical models for the temporal behavior of the background noise, target, and clutter, on a single pixel basis. Based on these models we develop a generalized likelihood ratio test (GLRT) and perfect measurement performance analysis, and present the resulting decision rule. We also propose a parameter estimation technique and compare its performance to the Cramer Rao bound (CRB). We demonstrate the effectiveness of the technique by applying the resulting algorithm to real world infrared (IR) image sequences containing targets of opportunity. The approach could also be applicable to other image sequence processing scenarios, using acquisition systems besides IR imaging, such as detection of small moving objects or structures in a biomedical or biological imaging scenario, or the detection of satellites, meteors or other celestial bodies in night sky imagery acquired using a telescope     

一种基于多普勒频率的恒模信号直接定位方法

相比于传统的差分多普勒(DD)两步定位方法,以Amar和Weiss提出的基于多普勒频率的单步直接定位方法在低信噪比和小样本条件下具有更高的定位精度。在该类新型定位体制的基础上,提出了一种基于多普勒频率的恒模信号直接定位方法。首先,依据最大似然(ML)准则以及恒模信号的恒包络特征,建立相应的直接定位优化模型。然后,根据目标函数的代数特征将全部未知参量分成两组,并提出一种有效的多参量交替迭代算法,用以获得该优化问题的最优数值解。新算法包含了针对这两组未知参量的Newton型迭代公式,用以避免网格搜索,并能实现多维参数的"解耦合"估计。最后,推导出针对恒模信号的目标位置直接估计方差的克拉美罗界(CRB)。数值实验验证了新方法的优越性。     

Optimal Cooperative Sensing using a Team of UAVs

The authors investigates the joint optimal estimation of both the position and velocity of a ground moving target (GMT) using pulse Doppler radars on-board unmanned aerial vehicles (UAVs). The problem of cooperative estimation using a UAV team and the optimization of the team's configuration to achieve optimal GMT position and velocity estimates are addressed. Based on the Cramer-Rao bound, the minimum achievable error variance of the GMT position and velocity estimates is derived. The expression of the minimum achievable estimation error variance for unbiased estimation provided by the Cramer-Rao bound is minimized yielding the optimal configuration of the UAV team. Our solution is complete in that it addresses various GMT tracking scenarios and an arbitrary number of UAVs. Optimal sensor geometries for typical applications are illustrated     

直达与非直达环境中的多目标解耦直接定位方法

针对直达（LOS）与非直达（NLOS）环境中的定位问题,提出了一种波形已知条件下的单阵地多目标直接定位（DPD）算法。该算法针对发射时间已知和未知两种情况,利用多径信号到达角度与时延关于障碍物（或反射体）、观测站与目标位置参数的数学关系,建立了三维目标位置的最大似然（ML）函数,无需估计测量参数,避免了传统两步定位方法所需的非直达径识别与数据关联。为了克服多目标定位中的高维非线性优化问题,该算法利用独立波形信息将多目标定位解耦为对各个目标单独求解。通过对目标函数有效近似,算法在发射时间已知和未知两种情况下均仅需三维网格搜索,比相应的两步定位方法具有更低的计算量。此外,基于多径定位场景,推导了发射时间已知和未知两种情况下的位置估计克拉美罗界（CRB）。仿真结果表明：算法的定位性能能够逼近相应的克拉美罗界,比传统两步定位方法和子空间直接定位算法具有更高的定位精度。     

Particle filters for tracking with out-of-sequence measurements

An extension is presented to the particle filtering toolbox that enables nonlinear/non-Gaussian filtering to be performed in the presence of out-of-sequence measurements (OOSMs) with arbitrary lag, without the need to adopt linearising approximations in the filter and without the degradation of performance that would occur if the OOSMs were simply discarded. An estimate of the performance of the OOSM particle filter (OOSM-PF) is obtained for bearings-only tracking scenarios with a single target and a small number of sensors. These performance estimates are then compared with the posterior Cramer-Rao lower bound (CRLB) for the state estimate rms error and similar performance estimates obtained from the oosm extended Kalman filter (OOSM-EKF) algorithms recently introduced in the literature. For a mildly nonlinear bearings-only tracking problem the OOSM-PF and OOSM-EKF are shown to achieve broadly similar performance.     

Approximate Maximum Likelihood Algorithm for Moving Source Localization Using TDOA and FDOA Measurements

 A closed-form approximate maximum likelihood (AML) algorithm for estimating the position and velocity of a moving source is proposed by utilizing the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements of a signal received at a number of receivers. The maximum likelihood (ML) technique is a powerful tool to solve this problem. But a direct approach that uses the ML estimator to solve the localization problem is exhaustive search in the solution space, and it is very computationally expensive, and prohibits real-time processing. On the basis of ML function, a closed-form approximate solution to the ML equations can be obtained, which can allow real-time implementation as well as global convergence. Simulation results show that the proposed estimator achieves better performance than the two-step weighted least squares (WLS) approach, which makes it possible to attain the Cram閞-Rao lower bound (CRLB) at a sufficiently high noise level before the threshold effect occurs.     

A Linear-correction Least-squares Approach for Geolocation Using FDOA Measurements Only

  A linear-correction least-squares(LCLS) estimation procedure is proposed for geolocation using frequency difference of arrival (FDOA) measurements only. We first analyze the measurements of FDOA, and further derive the Cram閞-Rao lower bound (CRLB) of geolocation using FDOA measurements. For the localization model is a nonlinear least squares(LS) estimator with a nonlinear constrained, a linearizing method is used to convert the model to a linear least squares estimator with a nonlinear constrained. The Gauss-Newton iteration method is developed to conquer the source localization problem. From the analysis of solving Lagrange multiplier, the algorithm is a generalization of linear-correction least squares estimation procedure under the condition of geolocation using FDOA measurements only. The algorithm is compared with common least squares estimation. Comparisons of their estimation accuracy and the CRLB are made, and the proposed method attains the CRLB. Simulation results are included to corroborate the theoretical development.