Computationally efficient angle estimation for signals with knownwaveforms

This paper presents a large sample decoupled maximum likelihood (DEML) angle estimator for uncorrelated narrowband plane waves with known waveforms and unknown amplitudes arriving at a sensor array in the presence of unknown and arbitrary spatially colored noise. The DEML estimator decouples the multidimensional problem of the exact ML estimator to a set of 1-D problems and, hence, is computationally efficient. We shall derive the asymptotic statistical performance of the DEML estimator and compare the performance with its Cramer-Rao bound (CRB), i.e., the best possible performance for the class of asymptotically unbiased estimators. We will show that the DEML estimator is asymptotically statistically efficient for uncorrelated signals with known waveforms. We will also show that for moderately correlated signals with known waveforms, the DEML estimator is no longer a large sample maximum likelihood (ML) estimator, but the DEML estimator may still be used for angle estimation, and the performance degradation relative to the CRB is small. We shall show that the DEML estimator can also be used to estimate the arrival angles of desired signals with known waveforms in the presence of interfering or jamming signals by modeling the interfering or jamming signals as random processes with an unknown spatial covariance matrix. Finally, several numerical examples showing the performance of the DEML estimator are presented in this paper     

非均匀噪声环境下宽带近场源定位的两种改进方法

研究了宽带近场信号源基于最大似然方法和相关信号子空间方法在非均匀噪声下的被动定位算法，并进行了比较。这两种算法均可在传感器任意分布的情况下有效地进行信号源定位。最大似然法采用了迭代的方法来估计噪声的协方差矩。而信号子空间法给出了聚焦阵构造的新方法。仿真试验证明了方法的有效性和稳健性。     

Maximum likelihood multiple-source localization using acoustic energy measurements with wireless sensor networks

A maximum likelihood (ML) acoustic source location estimation method is presented for the application in a wireless ad hoc sensor network. This method uses acoustic signal energy measurements taken at individual sensors of an ad hoc wireless sensor network to estimate the locations of multiple acoustic sources. Compared to the existing acoustic energy based source localization methods, this proposed ML method delivers more accurate results and offers the enhanced capability of multiple source localization. A multiresolution search algorithm and an expectation-maximization (EM) like iterative algorithm are proposed to expedite the computation of source locations. The Crame/spl acute/r-Rao Bound (CRB) of the ML source location estimate has been derived. The CRB is used to analyze the impacts of sensor placement to the accuracy of location estimates for single target scenario. Extensive simulations have been conducted. It is observed that the proposed ML method consistently outperforms existing acoustic energy based source localization methods. An example applying this method to track military vehicles using real world experiment data also demonstrates the performance advantage of this proposed method over a previously proposed acoustic energy source localization method.     

Maximum likelihood direction-of-arrival estimation in unknown noise fields using sparse sensor arrays

We address the problem of maximum likelihood (ML) direction-of-arrival (DOA) estimation in unknown spatially correlated noise fields using sparse sensor arrays composed of multiple widely separated subarrays. In such arrays, intersubarray spacings are substantially larger than the signal wavelength, and therefore, sensor noises can be assumed to be uncorrelated between different subarrays. This leads to a block-diagonal structure of the noise covariance matrix which enables a substantial reduction of the number of nuisance noise parameters and ensures the identifiability of the underlying DOA estimation problem. A new deterministic ML DOA estimator is derived for this class of sparse sensor arrays. The proposed approach concentrates the ML estimation problem with respect to all nuisance parameters. In contrast to the analytic concentration used in conventional ML techniques, the implementation of the proposed estimator is based on an iterative procedure, which includes a stepwise concentration of the log-likelihood (LL) function. The proposed algorithm is shown to have a straightforward extension to the case of uncalibrated arrays with unknown sensor gains and phases. It is free of any further structural constraints or parametric model restrictions that are usually imposed on the noise covariance matrix and received signals in most existing ML-based approaches to DOA estimation in spatially correlated noise.     

Maximum likelihood array processing in spatially correlated noisefields using parameterized signals

This paper deals with the problem of estimating signal parameters using an array of sensors. This problem is of interest in a variety of applications, such as radar and sonar source localization. A vast number of estimation techniques have been proposed in the literature during the past two decades. Most of these can deliver consistent estimates only if the covariance matrix of the background noise is known. In many applications, the aforementioned assumption is unrealistic. Recently, a number of contributions have addressed the problem of signal parameter estimation in unknown noise environments based on various assumptions on the noise. Herein, a different approach is taken. We assume instead that the signals are partially known. The received signals are modeled as linear combinations of certain known basis functions. The exact maximum likelihood (ML) estimator for the problem at hand is derived, as well as computationally more attractive approximation. The Cramer-Rao lower bound (CRB) on the estimation error variance is also derived and found to coincide with the CRB, assuming an arbitrary deterministic model and known noise covariance     

Estimating parameters of multiple wideband polynomial-phase sourcesin sensor arrays

We consider the problem of estimating the parameters of multiple wideband polynomial-phase signal (PPS) sources in sensor arrays. A new maximum likelihood (ML) direction-of-arrival (DOA) estimator is introduced, and the exact Cramer-Rao bound (CRB) is derived for the general case of multiple constant-amplitude polynomial-phase sources. Since the proposed exact ML estimator is computationally intensive, an approximate solution is proposed, originating from the analysis of the log-likelihood (LL) function in the single chirp signal case. As a result, a new form of spatio-temporal matched filter (referred to as the chirp beamformer) is derived, which is applicable to "well-separated" sources that have distinct time-frequency or/and spatial signatures. This beamforming approach requires solving a three-dimensional (3-D) optimization problem and, therefore, enjoys essentially simpler implementation than that entailed by the exact ML. Simulation results are presented, illustrating the performance of the estimators and validating our theoretical CRB analysis     

Acoustic vector-sensor processing in the presence of a reflectingboundary

We consider the passive direction-of-arrival (DOA) estimation problem using arrays of acoustic vector sensors located in a fluid at or near a reflecting boundary. We formulate a general measurement model applicable to any planar surface, derive an expression for the Cramer-Rao bound (CRB) on the azimuth and elevation of a single source, and obtain a bound on the mean-square angular error (MSAE). We then examine two applications of great practical interest: hull-mounted and seabed arrays. For the former, we use three models for the hull: an ideal rigid surface for high frequency, an ideal pressure-release surface for low frequency, and a more complex, realistic layered model. For the seabed scenario, we model the ocean floor as an absorptive liquid layer. For each application, we use the CRB, MSAE bound, and beam patterns to quantify the advantages of using velocity and/or vector sensors instead of pressure sensors. For the hull-mounted application, we show that normal component velocity sensors overcome the well-known, low-frequency problem of small pressure signals without the need for an undesirable “stand-off” distance. For the seabed scenario, we also derive a fast wideband estimator of the source location using a single vector sensor     

基于正交训练序列的MIMO系统联合最大似然时频同步和信道估计

该文推导了多输入多输出(MIMO)系统中的符号定时、频偏和信道参数的联合最大似然(ML)估计。针对联合ML估计没有闭合的表达式、数值计算复杂度高的问题,该文提出了一种基于重复结构的正交训练序列的简化估计算法。该估计算法形式简单、复杂度低,且仍为最大似然估计。最后仿真分析了最大似然参数估计的均方误差与接收信噪比和天线数目的关系,并与Cramer-Rao界作了比较,表明了该算法的有效性。     

Source localization with distributed sensor arrays and partial spatial coherence

Multiple sensor arrays provide the means for highly accurate localization of the (x,y) position of a source. In some applications, such as microphone arrays receiving aeroacoustic signals from ground vehicles, random fluctuations in the air lead to frequency-selective coherence losses in the signals that arrive at widely separated sensors. We present performance analysis for localization of a wideband source using multiple, distributed sensor arrays. The wavefronts are modeled with perfect spatial coherence over individual arrays and frequency-selective coherence between distinct arrays, and the sensor signals are modeled as wideband, Gaussian random processes. Analysis of the Cramer-Rao bound (CRB) on source localization accuracy reveals that a distributed processing scheme involving bearing estimation at the individual arrays and time-delay estimation (TDE) between sensors on different arrays performs nearly as well as the optimum scheme while requiring less communication bandwidth with a central processing node. We develop Ziv-Zakai bounds for TDE with partially coherent signals in order to study the achievability of the CRB. This analysis shows that a threshold value of coherence is required in order to achieve accurate time-delay estimates, and the threshold coherence value depends on the source signal bandwidth, the additive noise level, and the observation time. Results are included based on processing measured aeroacoustic data from ground vehicles to illustrate the frequency-dependent signal coherence and the TDE performance.     

A Method for the Direction-of-Arrival Estimation of Incoherently Distributed Sources

We consider the problem of estimating the nominal direction of arrival (DOA) of an incoherently distributed source. This problem is encountered due to the presence of local scatterers in the vicinity of a transmitter or due to signals propagating through a random inhomogeneous medium. Since the spatial covariance matrix has full rank for an incoherently distributed source, the performance of most high-resolution DOA estimation algorithms conceived under coherently distributed sources, as well as point source models, degrades when scattering is present. In addition, several DOA estimation techniques devised under a distributed source model require a high-dimensional nonlinear optimization problem. In this paper, we propose a novel method based on the conventional beamforming approach, which estimates the nominal DOA from a spatial maximum peak of the output power. The proposed method is computationally more attractive than the maximum likelihood (ML) estimator, although the performance degrades in comparison with the ML estimator, whose asymptotic performance is equivalent to the Cramer–Rao bound (CRB). We derive and compare the asymptotic performances of the proposed method and the redundancy-averaged covariance matching (RACM) method in the single-source case. The simulation results illustrate that the asymptotic performance of the proposed method is better than that of the RACM method.      

Maximum likelihood estimation and Cramer-Rao bounds for directionof arrival parameters of a large sensor array

A maximum likelihood (ML) method is developed for estimation of direction of arrival (DOA) and associated parameters of narrowband signals based on the Taylor's series expansion of the inverse of the data covariance matrix R for large M, M specifying number of sensors in the array. The stochastic ML criterion function can thus be simplified resulting in a computationally efficient algorithm for DOA estimation. The more important result is the derivation of asymptotic (large M) expressions for the Cramer-Rao lower bound (CRB) on the covariance matrix of all unknown DOA angles for the general D source case. The derived bound is expressed explicitly as a function of snapshots, signal-to-noise ratio (SNR), sensors, separation, and correlation between signal sources. Using the condition of positive definiteness of the Fisher information matrix a resolution criterion is proposed which gives a tight lower limit on the minimum resolvable angle     

Source localization in a waveguide using polynomial rooting

Source localization in acoustic waveguides involves a multidimensional search procedure. We propose a new algorithm in which the search in the depth direction is replaced by polynomial rooting. Using the proposed algorithm, range and depth estimation by a vertical array requires a 1-D search procedure. For a 3-D localization problem (i.e., range, depth, and direction-of-arrival (DOA) estimation), the algorithm involves a 2-D search procedure. Consequently, the proposed algorithm requires significantly less computation than other methods that are based on a brute-force search procedure over the source location parameters. In order to evaluate the performance of the algorithm, an error analysis is carried out, and Monte-Carlo simulations are performed. The results are compared with the Cramer-Rao bound (CRB) and to the maximum likelihood (ML) simulation performance. The algorithm is shown to be efficient, while being computationally simpler than the ML or the Bartlett processors. The disadvantage of the algorithm is that its SNR threshold occurs in lower SNR than in the ML algorithm     

同步时钟偏差存在下的时差定位性能分析及改进的定位方法

同步时钟偏差会显著增加时差（TDOA）定位误差,该文针对这一问题进行了理论性能分析,并提出了改进方法.首先,分析了时钟偏差存在下参数估计方差的克拉美罗界（CRB）,给出了关于目标位置估计方差更为闭式的CRB表达式,随后基于最大似然（ML）估计准则和泰勒级数（TS）定位方法,定量推导了时钟偏差对于TDOA定位精度的影响.接着,提出了可抑制时钟偏差的降维TS定位方法,并且给出了时钟偏差的ML闭式解.最后,数值实验验证了文中理论分析的有效性,并且新方法可以有效抑制同步时钟偏差的影响.     

Optimal and suboptimal broad-band source location estimation

Two maximum-likelihood (ML) estimators are considered for direction-of-arrival (DOA) estimation of broadband sources with unknown spectral parameters. One is based on the assumption that the sources radiate stochastic-Gaussian signals and therefore is called the stochastic-Gaussian ML (SGML) estimator; the other, using estimates of the actual signals (not their assumed distribution), is called the conditional ML (CML) estimator. Neither is efficient if the source spectral parameters are completely arbitrary and unknown, but the problem can be avoided for a version of the SGML estimation if the signal and noise spectra are known to satisfy certain smoothness conditions. While this version of the SGML is formally superior to the CML, it is demonstrated that the performance difference is small with underconditions not infrequently encountered in practice. When these are satisfied, the computationally simpler CML can be used without significant loss. The required conditions become more stringent as the source separation decreases or correlation between sources increases. A closed-form analytic expression is obtained for the small-error variance of the CML estimator of the DOA of the nth source in the presence of N-1 other sources     

被动虚拟阵列的CRB界及最大似然估计方法研究

针对利用机载运动平台对窄带微波信号进行侦测的背景,研究了被动虚拟阵列（PASA）对窄带微波信号的参数估计性能。在考虑方向角、频率和幅度均为未知参数的条件下,推导了方向角估计的克拉美劳界（CRB）的表达式,同时给出了PASA合成孔径长度的选取方案。另外,本文给出了PASA对方位角估计的最大似然（ML）估计算法。研究表明,随着合成孔径长度和信噪比的增加,ML估计误差可以很快地收敛于CRB,但存在阈值效应。计算机仿真结果验证了本文研究结果的正确性。     

基于特征分析和二次规划的窄带信号超分辨率频率估计

本文研究了窄带信号的超分辨率频率估计.在单次实验数据和信号包络函数形式未知的条件下,通过推广应用于分布源DOA估计的特征分析方法,并利用最优化理论中的二次规划算法,提出了一种窄带信号的超分辨率频率估计方法,本文称之为EQP方法.通过仿真分析,将该方法和其它方法进行了比较,同时还将估计结果的均方误差(MSE)的数值仿真结果和克拉美-罗界(CRB)作了比较,都表明本文提出的方法有效实现了窄带信号的超分辨率频率估计.     

Oblique projection for source estimation in a competitive environment: Algorithm and statistical analysis

In this paper, we study the problem where the aim is to estimate the source (complex amplitude) parameter of a single signal contaminated by a structured interference (constituted by the other signals) and by a background Gaussian noise. To solve this problem, we propose an estimator based on a partially estimated oblique projection. We derive closed-form expressions of the variance of this estimator and of the Cramér–Rao bound (CRB) associated with the considered model. In particular, we show that the proposed estimator is (i) asymptotically (for large number of sensors) efficient in the sense that its variance meets the CRB for a single signal in noise and (ii) for a small of moderate number of sensors, the variance remains close to the CRB without structured interference for well separated bearings.     

Maximum-likelihood bearing estimation with partly calibrated arraysin spatially correlated noise fields

The problem of using a partly calibrated array for maximum likelihood (ML) bearing estimation of possibly coherent signals buried in unknown correlated noise fields is shown to admit a neat solution under fairly general conditions. More exactly, this paper assumes that the array contains some calibrated sensors, whose number is only required to be larger than the number of signals impinging on the array, and also that the noise in the calibrated sensors is uncorrelated with the noise in the other sensors. These two noise vectors, however, may have arbitrary spatial autocovariance matrices. Under these assumptions the many nuisance parameters (viz., the elements of the signal and noise covariance matrices and the transfer and location characteristics of the uncalibrated sensors) can be eliminated from the likelihood function, leaving a significantly simplified concentrated likelihood whose maximum yields the ML bearing estimates. The ML estimator introduced in this paper, and referred to as MLE, is shown to be asymptotically equivalent to a recently proposed subspace-based bearing estimator called UNCLE and rederived herein by a much simpler approach than in the original work. A statistical analysis derives the asymptotic distribution of the MLE and UNCLE estimates, and proves that they are asymptotically equivalent and statistically efficient. In a simulation study, the MLE and UNCLE methods are found to possess very similar finite-sample properties as well. As UNCLE is computationally more efficient, it may be the preferred technique in a given application     

Linear Regression With Gaussian Model Uncertainty: Algorithms and Bounds

In this paper, we consider the problem of estimating an unknown deterministic parameter vector in a linear regression model with random Gaussian uncertainty in the mixing matrix. We prove that the maximum-likelihood (ML) estimator is a (de)regularized least squares estimator and develop three alternative approaches for finding the regularization parameter that maximizes the likelihood. We analyze the performance using the Cramer-Rao bound (CRB) on the mean squared error, and show that the degradation in performance due the uncertainty is not as severe as may be expected. Next, we address the problem again assuming that the variances of the noise and the elements in the model matrix are unknown and derive the associated CRB and ML estimator. We compare our methods to known results on linear regression in the error in variables (EIV) model. We discuss the similarity between these two competing approaches, and provide a thorough comparison that sheds light on their theoretical and practical differences.     

Maximum Likelihood Localization of a Diffusive Point Source Using Binary Observations

In this paper, we investigate the problem of localization of a diffusive point source of gas based on binary observations provided by a distributed chemical sensor network. We motivate the use of the maximum likelihood (ML) estimator for this scenario by proving that it is consistent and asymptotically efficient, when the density of the sensors becomes infinite. We utilize two different estimation approaches, ML estimation based on all the observations (i.e., batch processing) and approximate ML estimation using only new observations and the previous estimate (i.e., real time processing). The performance of these estimators is compared with theoretical bounds and is shown to achieve excellent performance, even with a finite number of sensors