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.     

Comparative Analysis of ML-PSO DOA Estimation with Conventional Techniques in Varied Multipath Channel Environment

In the field of array signal processing, direction of arrival (DOA) estimation is a prime area of research. DOA estimation and adaptive beamforming (ABF) are the main issues in smart antennas, which dynamically find the direction of desired and interfering users and finds the optimum weights for beamforming. There are numerous spectral and eigen structure algorithms for estimating the direction of narrow band sources. However, in a complex multipath channel environment, received signals from different directions are fully or partially correlated, which prevents the applications of high resolution techniques to estimate the direction of incoming signals. Maximum likelihood (ML) is an efficient technique for DOA estimation in a low signal to noise ratio (SNR) and coherent channel environment. In this paper, we use particle swarm optimization (PSO) for estimating ML solution by optimizing complex non linear multimodal function over a high dimensional space in linear arrays. PSO-ML estimator has been compared with conventional DOA estimation techniques in uncorrelated, partially correlated and coherent channel environment. The performance of PSO-ML estimator and conventional algorithms are analyzed in varying partially correlated channel environment. The simulation results demonstrate that PSO based estimator gives superior statistical performance.     

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     

Maximum-likelihood source localization and unknown sensor locationestimation for wideband signals in the near-field

In this paper, we derive the maximum-likelihood (ML) location estimator for wideband sources in the near field of the sensor array. The ML estimator is optimized in a single step, as opposed to other estimators that are optimized separately in relative time-delay and source location estimations. For the multisource case, we propose and demonstrate an efficient alternating projection procedure based on sequential iterative search on single-source parameters. The proposed algorithm is shown to yield superior performance over other suboptimal techniques, including the wideband MUSIC and the two-step least-squares methods, and is efficient with respect to the derived Cramer-Rao bound (CRB). From the CRB analysis, we find that better source location estimates can be obtained for high-frequency signals than low-frequency signals. In addition, large range estimation error results when the source signal is unknown, but such unknown parameter does not have much impact on angle estimation. In some applications, the locations of some sensors may be unknown and must be estimated. The proposed method is extended to estimate the range from a source to an unknown sensor location. After a number of source-location frames, the location of the uncalibrated sensor can be determined based on a least-squares unknown sensor location estimator     

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.      

Accurate direction‐of‐arrival estimation of multiple sources using a genetic approach

In this paper, we present an accurate direction‐of‐arrival (DOA) estimation method, which is based on the maximum likelihood (ML) principle and implemented using a modified and refined genetic algorithm (GA). With the newly introduced features—intelligent initialization and the emperor‐selective (EMS) mating scheme, carefully selected crossover and mutation operators and fine‐tuned parameters such as the population size, the probability of crossover and mutation etc., the GA‐ML estimator achieves fast global convergence. A GA operator and parameter standard is suggested for this application, which is independent of the source and array configurations except the number of sources. Simulation results demonstrate that in general scenarios, the proposed estimator is the most efficient in computation and its statistical performance is the best among all popular ML‐based DOA estimation methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Maximum likelihood array processing for stochastic coherent sources

Maximum likelihood (ML) estimation in array signal processing for the stochastic noncoherent signal case is well documented in the literature. We focus on the equally relevant case of stochastic coherent signals. Explicit large-sample realizations are derived for the ML estimates of the noise power and the (singular) signal covariance matrix. The asymptotic properties of the estimates are examined, and some numerical examples are provided. In addition, we show the surprising fact that the ML estimates of the signal parameters obtained by ignoring the information that the sources are coherent coincide in large samples with the ML estimates obtained by exploiting the coherent source information. Thus, the ML signal parameter estimator derived for the noncoherent case (or its large-sample realizations) asymptotically achieves the lowest possible estimation error variance (corresponding to the coherent Cramer-Rao bound)     

Application of natural computing algorithms to maximum likelihood estimation of direction of arrival

This work presents a study of the performance of populational meta-heuristics belonging to the field of natural computing when applied to the problem of direction of arrival (DOA) estimation, as well as an overview of the literature about the use of such techniques in this problem. These heuristics offer a promising alternative to the conventional approaches in DOA estimation, as they search for the global optima of the maximum likelihood (ML) function in a framework characterized by an elegant balance between global exploration and local improvement, which are interesting features in the context of multimodal optimization, to which the ML-DOA estimation problem belongs. Thus, we shall analyze whether these algorithms are capable of implementing the ML estimator, i.e., finding the global optima of the ML function. In this work, we selected three representative natural computing algorithms to perform DOA estimation: differential evolution, clonal selection algorithm, and the particle swarm. Simulation results involving different scenarios confirm that these methods can reach the performance of the ML estimator, regardless of the number of sources and/or their nature. Moreover, the number of points evaluated by such methods is quite inferior to that associated with a grid search, which gives support to their application.     

Direction of Arrival Estimation in the Presence of Noise Coupling in Antenna Arrays

The direction of arrival (DOA) estimation problem in the presence of signal and noise coupling in antenna arrays is addressed. In many applications, such as smart antenna, radar and navigation systems, the noise coupling between different antenna array elements is often neglected in the antenna modeling and thus, may significantly degrade the system performance. Utilizing the exact noise covariance matrix enables to achieve high-performance source localization by taking into account the colored properties of the array noise. The noise covariance matrix of the antenna array consists of both the external noise sources from sky, ground and interference, and the internal noise sources from amplifiers and loads. Computation of the internal noise covariance matrix is implemented using the theory of noisy linear networks combined with the method of moments (MoM). Based on this noise statistical analysis, a new four-port antenna element consisting of two orthogonal loops is proposed with enhanced source localization performance. The maximum likelihood (ML) estimator and the Cramer-Rao lower bound (CRLB) for DOA estimation in the presence of noise coupling is derived. Simulation results show that the noise coupling in antenna arrays may substantially alter the source localization performance. The performance of a mismatched ML estimator based on a model which ignores the noise coupling shows significant performance degradation due to noise coupling. These results demonstrate the importance of the noise coupling modeling in the DOA estimation algorithms.     

共形阵列非圆信号波达方向估计算法

受共形载体变曲率结构的影响,各天线单元指向不尽相同,使得共形天线阵列呈现极化多样性。因此,共形天线阵列的建模过程中需考虑不同阵元的极化响应特性。基于柱面共形天线阵列的快拍数据模型,利用非圆信号的特性对阵列输出进行扩展,基于秩亏理论和子空间原理实现信号波达方向（DOA）估计,所提方法估计精度高,不需要参数配对。存在相干信源时,提出对扩展后的虚拟阵列进行划分,对划分出的子阵进行虚拟的空间平滑,实现解相干的预处理操作。仿真结果表明该方法能有效应用于柱面共形阵列非圆信号DOA估计,并提高了空间分辨率。      

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     

Computationally Efficient Maximum Likelihood Approach to DOA Estimation of a Scattered Source

The problem of Direction-Of-Arrival (DOA)estimation in the presence of local scatterers using a uniform linear array(ULA) of sensors is addressed. We consider two models depending on whether theform of the azimuthal power distribution is explicitly known or not. For bothmodels, the block-diagonal structure of the associated Fisher InformationMatrix (FIM) is exploited to decouple the estimation of the DOA from that ofthe other model parameters. An asymptotically efficient Maximum Likelihood(ML)DOA estimator is derived which entails solving a 1-D minimization problemonly.Furthermore, the 1-D criterion can be expressed as a simple Fourier Transform.A numerical comparison with the Cramér-Rao Bound (CRB) illustrates thefactthat our computationally very simple DOA estimators are statisticallyefficientfor a wide range of scenarios.     

基于高阶累积量的近场通信波达方向估计算法

通过对近场通信波达方向准确估计,提高目标信源的定位能力.传统方法中对近场源通信信源的波达方向估计采用多普勒估计方法,由于近场通信的空间信源为窄带信号,多普勒估计会导致DOA估计频谱失真.提出一种基于高阶累积量的近场通信波达方向估计算法.采用均匀间隔线列阵构建近场通信的信号模型,进行近场源目标特征构建,提取近场源通信信号的斜度和峰度等特征,采用高阶累积量特征提取方法,分别求得对应近场通信信源的方位角、频率和距离三维参数,使得每个信源的参数自动配对,提高了近场通信DOA波达方向估计的效率和精度,实现近场源通信信号的波达方向估计算法改进.仿真实验结果表明,采用该方法进行近场方法波达方向估计的精度较高,对信源方位的定位准确,性能优越于传统方法,在近场通信中具有较好的应用价值.     

A Refined Genetic Algorithm for Accurate and Reliable DOA Estimation with a Sensor Array

Maximum likelihood (ML) direction-of-arrival (DOA) estimation algorithm is a nearly optimal technique. In this paper, we present a modified and refined genetic algorithm (GA) to find the exact solutions to the complex, multi-modal, multivariate and highly nonlinear likelihood function. With the newly introduced features such as intelligent initialization and the emperor-selective mating scheme, carefully selected crossover and mutation operators, and fine-tuned parameters such as the population size, the probability of crossover and mutation, the GA-ML estimator achieves fast global convergence. The GA-ML estimator has been compared with various DOA estimation methods in a variety of scenarios, and the simulation results demonstrate that in most scenarios the proposed GA-ML estimator is the fastest and its performance is the best among popular ML-based DOA estimation methods.     

Joint signal detection and parameter estimation in multiusercommunications

The problem of simultaneously detecting the information bits and estimating signal amplitudes and phases in a K-user asynchronous direct-sequence spread-spectrum multiple-access communication system is addressed. The joint maximum-likelihood (ML) estimator has a computational complexity that is exponential in the total number of bits transmitted and thus does not represent a practical solution to the problem. An estimator that combines a suboptimum tree-search algorithm with a recursive least-squares estimator of complex signal amplitude is considered. The complexity of this estimator is O(K²) computations per decoded bit, and its performance is very close to that of the joint ML receiver. This receiver has the advantage that the transmitted signal powers and phases are extracted from the received signal in an adaptive fashion without using a test sequence     

Maximum-likelihood direction-of-arrival estimation in the presenceof unknown nonuniform noise

We consider the problem of estimating directions of arrival (DOAs) of multiple sources observed on the background of nonuniform white noise with an arbitrary diagonal covariance matrix. A new deterministic maximum likelihood (ML) DOA estimator is derived. Its implementation is based on an iterative procedure which includes a stepwise concentration of the log-likelihood (LL) function with respect to the signal and noise nuisance parameters and requires only a few iterations to converge. New closed-form expressions for the deterministic and stochastic direction estimation Cramer-Rao bounds (CRBs) are derived for the considered nonuniform model. Our expressions can be viewed as an extension of the well-known results by Stoica and Nehorai (1989, 1990) and Weiss and Friedlander (1993) to a more general noise model than the commonly used uniform one. In addition, these expressions extend the results obtained by Matveyev et al. (see Circuits, Syst., Signal Process., vol.18, p.479-87, 1999) to the multiple source case. Comparisons with the above-mentioned earlier results help to discover several interesting properties of DOA estimation in the nonuniform noise case. To compare the estimation performance of the proposed ML technique with the results of our CRB analysis and with the performance of conventional “uniform” ML, simulation results are presented. Additionally, we test our technique using experimental seismic array data. Our simulations and experimental results both validate essential performance improvements achieved by means of the approach proposed     

Exact maximum likelihood time delay estimation for shortobservation intervals

An exact solution is presented to the problem of maximum likelihood time delay estimation for a Gaussian source signal observed at two different locations in the presence of additive, spatially uncorrelated Gaussian white noise. The solution is valid for arbitrarily small observation intervals; that is, the assumption T≫τ _c, |d| made in the derivation of the conventional asymptotic maximum likelihood (AML) time delay estimator (where τ _c is the correlation time of the various random processes involved and d is the differential time delay) is relaxed. The resulting exact maximum likelihood (EML) instrumentation is shown to consist of a finite-time delay-and-sum beamformer, followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one-dimensional integral equation with nonconstant weight. The solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities. Finally, the performance of the EML and AML estimators is compared by means of computer simulations     

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     

Robust maximum likelihood bearing estimation in contaminatedGaussian noise

A robust maximum likelihood (ML) direction-of-arrival (DOA) estimation method that is insensitive to outliers and distributional uncertainties in Gaussian noise is presented. The algorithm has been shown to perform much better than the Gaussian ML algorithm when the underlying noise distribution deviates even slightly from Gaussian while still performing almost as well in pure Gaussian noise. As with the Gaussian ML estimation, it is still capable of handling correlated signals as well as single snapshot cases. Performance of the algorithm is analyzed using the unique resolution test procedure which determines whether a DOA estimation algorithm, at a given confidence level, can resolve two dominant sources with very close DOAs     

协方差矩阵稀疏表示在网格失配波达方向估计中的应用

该文利用了入射信号在空域的稀疏性,将波达方向（DOA）估计问题描述为在网格划分的空间协方差矩阵稀疏表示模型,并将其松弛为一个凸问题,从而提出了一种网格匹配下的交替迭代方法（AIEGM）。传统的基于稀疏重构的波达方向估计算法由于其模型的局限性,一旦入射角不在预先设定的离散化网格上,就会造成估计性能的急剧恶化。针对这个问题,该算法可以在离散化网格比较粗糙的前提下,通过交替迭代的方法求解一系列基追踪去噪（BPDN）问题,对于不在网格上的真实角度估计值进行修正,从而达到更精确的波达方向估计。仿真结果证明了AIEGM算法的有效性。