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.     

Gaussian Cramer-Rao bound for direction estimation of noncircular signals in unknown noise fields

This paper focuses on the stochastic Cramer-Rao bound (CRB) on direction of arrival (DOA) estimation accuracy for noncircular Gaussian sources in the general case of an arbitrary unknown Gaussian noise field parameterized by a vector of unknowns. Explicit closed-form expressions of the stochastic CRB for DOA parameters alone are obtained directly from the Slepian-Bangs formula for general noncircular complex Gaussian distributions. As a special case, the CRB under the nonuniform white noise assumption is derived. Our expressions can be viewed as extensions of the well-known results by Stoica and Nehorai, Ottersten et al., Weiss and Friedlander, Pesavento and Gershman, and Gershman et al. Some properties of these CRBs are proved and finally, these bounds are numerically compared with the conventional CRBs under the circular complex Gaussian distribution for different unknown noise field models.     

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.     

Pseudo-randomly generated estimator banks: a new tool for improvingthe threshold performance of direction finding

A new powerful tool for improving the threshold performance of direction finding is considered. The main idea of our approach is to reduce the number of outliers in the DOA estimates using a previously proposed joint estimation strategy (JES). For this purpose, multiple different DOA estimators are calculated in a parallel manner for the same batch of data (i.e. for a single data record). Employing these estimators simultaneously, the JES improves the threshold performance because it removes outliers and exploits only “successful” estimators that are sorted out using a hypothesis testing procedure. We consider an efficient modification of the JES with application to the pseudo-randomly generated eigenstructure estimator banks based on secondand higher order statistics. Weighted MUSIC estimators based on the covariance and contracted quadricovariance matrices are chosen as appropriate underlying techniques for the second- and fourth-order estimator banks, respectively. Computer simulations with uncorrelated sources verify the dramatic improvements of threshold performance as compared with the conventional second- and fourth-order MUSIC algorithms. Simulations also show that in the second-order case, the threshold performance of our technique is close to that of the WSF method and stochastic/deterministic ML methods, which are known today as the most powerful (in the sense of estimation performance) and, at the same time, as the most computationally expensive DOA estimation techniques. The computational cost of our algorithm is much lower than that of the WSF and ML techniques because no multidimensional optimization is required     

非均匀噪声背景下混合信号DOA估计算法

针对非均匀噪声背景下非相关信源与相干信源并存时波达方向(DOA)估计问题，提出了基于迭代最小二乘和空间差分平滑的混合信号DOA估计算法。首先，该算法利用迭代最小二乘方法得到噪声协方差矩阵估计，然后对数据协方差矩阵进行“去噪”处理，利用子空间旋转不变技术实现非相关信源DOA估计；其次，基于空间差分法消除非相关信号并构造新矩阵进行前后向空间平滑，利用求根MUSIC算法估计相干信源DOA。相比于传统算法，该算法能估计更多的信源数，在低信噪比情况下DOA估计性能更优越。仿真实验结果验证了该算法的有效性。      

非均匀噪声条件下的互质阵列欠定DOA估计方法

针对基于互质阵列的欠定DOA估计方法在非均匀噪声条件下性能下降的问题,该文提出一种基于协方差矩阵重构和矩阵填充的鲁棒DOA估计方法。首先,将接收数据协方差矩阵分解,得到包含非均匀噪声项的对角阵;然后,选取对角线元素中的最小值,替换其余对角线元素,进而得到重构后的数据协方差矩阵;最后,对重构后的协方差矩阵进行扩展和矩阵填充,结合子空间方法进行DOA估计。理论分析和仿真结果表明,相对于现有方法,该文方法有效地抑制了非均匀噪声的影响,有更好的DOA估计性能。     

CRLB for DOA Estimation in Gaussian and Non-Gaussian Mixed Environments

The determination of Cramer-Rao lower bound (CRLB) as an optimality criterion for the problem of Direction-of-arrival (DOA) estimation is a very important issue. Several CRLBs on DOA estimation have been derived for Gaussian noise. However, a practical channel is affected by not only Gaussian background noise but also non-Gaussian noise such as impulsive interference. This paper derives the deterministic CRLB for Gaussian and non-Gaussian mixed environments. Since non-parametric kernel method is used to build the probability density function (PDF) of non-Gaussian noise, the CRLB derived is suitable for various noise distributions with or without symmetric PDF. The relationship between the CRLB for Gaussian noise and the proposed CRLB is also investigated. Theoretical analysis shows that the proposed CRLB provides a unified representation for both the cases of Gaussian and mixed environments. Computer simulations are included to verify the derived CRLB in different noise environments.     

Direction of Arrival Estimation using EM-ESPRIT with Nonuniform Arrays

This paper deals with the problem of the Direction Of Arrival (DOA) estimation with nonuniform linear arrays. The proposed method is based on the Expectation Maximization method where ESPRIT is used in the maximization step. The key idea is to iteratively interpolate the data to a virtual uniform linear array in order to apply ESPRIT to estimate the DOA. The iterative approach allows one to improve the interpolation using the previously estimated DOA. One of this method’s novelties lies in its capacity of dealing with any nonuniform array geometry. This technique manifests significant performance and computational advantages over previous algorithms such as Spectral MUSIC, EM-IQML and the method based on manifold separation technique. EM-ESPRIT is shown to be more robust to additive noise. Furthermore, EM-ESPRIT fully exploits the advantages of using a nonuniform array over a uniform array: simulations show that for the same aperture and with a smaller number of sensors, the nonuniform array presents almost identical performance as the equivalent uniform array.     

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.     

Direction finding in partly calibrated sensor arrays composed of multiple subarrays

We consider the direction-finding problem in partly calibrated arrays composed of several calibrated and identically oriented (but possibly nonidentical) subarrays that are displaced by unknown (and possibly time-varying) vector translations. A new search-free eigenstructure-based direction-finding approach is proposed for such class of sensor arrays. It is referred to as the rank-reduction (RARE) estimator and enjoys simple implementation that entails computing the eigendecomposition of the sample array covariance matrix and polynomial rooting. Closed-form expressions for the deterministic Cramer-Rao bounds (CRBs) on direction-of-arrival (DOA) estimation for the considered class of sensor arrays are derived. Comparison of these expressions with simulation results show that the finite-sample performance of RARE algorithms in both time-invariant and time-varying array cases is close to the corresponding bounds. Moreover, comparisons of the derived CRBs with the well-known bounds for the fully calibrated time-invariant array case help to discover several interesting properties of DOA estimation in partly calibrated and time-varying arrays.     

Direction of Arrival Estimation of Multiple UWB Signals

This paper deals with the problem of estimation of direction of arrivals (DOA) of a multiple ultra-wideband (UWB) pulse postion modulation signals incident on a smart antenna in the presence of white Gaussian noise. We transform the received signal into frequency domain in order to split the array output into multiple frequency channels. Corresponding frequency channels data of the array is arranged into a model similar to narrowband DOA estimation. Iterative quadratic maximum likelihood algorithm is applied to yield DOA estimates. These separate estimates at different frequencies are combined into a single estimate of DOA for each source in an appropriate manner. The performance of the proposed method is studied via extensive computer simulations. It is seen that the technique can successfully resolve the DOA of the closely-spaced UWB signals.     

浅海非高斯噪声下基于变分贝叶斯推断的波达角估计

传统基于高斯统计特性的波达角(DOA)估计方法在高斯背景噪声中可以获得较好的估计性能,然而受脉冲噪声影响的浅海环境噪声不再服从高斯分布,若直接利用传统波达角估计方法会引入较大误差。为提升非高斯噪声环境下的波达角估计性能,该文提出一种浅海非高斯噪声下的基于变分贝叶斯推断的波达角估计方法。首先利用信号与脉冲噪声的稀疏性构建多测量向量稀疏信号恢复(SSR)模型;其次,考虑信号的共稀疏特性与脉冲噪声的独立稀疏性,构建层次化贝叶斯估计框架;然后利用变分贝叶斯推断估计信号与噪声的后验概率估计。稀疏信号模型中考虑离网格误差,利用根稀疏贝叶斯估计实现离网格误差修正,解决离网格误差引起的基失配问题;最后通过迭代更新获得较为精确的波达角估计,同时消除脉冲噪声的影响。仿真结果表明:所提方法在非高斯噪声环境下具有较好的波达角估计性能,同时对于脉冲噪声具有较强的抗干扰特性。     

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。     

Direction finding for wide-band signals using an interpolated array

The authors derive a new direction-finding algorithm for multiple wideband signals received by an arbitrary array and analyze its performance. Using an interpolation technique, they generate a set of virtual arrays, each for a different frequency band, having the same array manifold. The convergence matrices of these arrays are added to produce a composite covariance matrix. Direction-of-arrival (DOA) estimates are obtained by eigendecomposition of this composite covariance matrix using the narrowband MUSIC algorithm or its variants. Closed-form expressions for the asymptotic covariance matrix of the DOA estimation errors are derived using a perturbation analysis, evaluated for specific cases, and compared with the Cramer-Rao lower bound. Special attention is given to correlated and coherent signals. The formulas for the error covariance are quite general and can be modified to provide results for other wideband DOA estimation algorithms     

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.     

有限采样影响下秩减估计器的波达方向估计性能分析

首先从理论上分析有限采样影响下秩减估计器的波达方向估计性能,然后基于信号(或噪声)子空间的正交投影矩阵扰动定理,分别推导秩减估计器方位估计偏差的一阶和二阶闭式表达式,在此基础上给出其方位估计均方误差、偏置以及测向成功概率的理论计算公式,最后针对若干重要的秩减估计器给出数值实验,实验结果验证了所提理论推导的有效性。     

High-resolution direction finding: the missing data case

This paper considers the problem of estimating the direction-of-arrival (DOA) of one or more signals using an array of sensors, where some of the sensors fail to work before the measurement is completed. Methods for estimating the array output covariance matrix are discussed. In particular, the maximum-likelihood (ML) estimate of this covariance matrix and its asymptotic accuracy are derived and discussed. Different covariance matrix estimates are used for DOA estimation together with the MUSIC algorithm and with a covariance matching technique. In contrast to MUSIC, the covariance matching technique can utilize information on the estimation accuracy of the array covariance matrix, and it is demonstrated that this yields a significant performance gain     

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     

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.     

Nested planar array: configuration design,optimal array and DOA estimation

Nested array enables to enhance localisation resolution and achieve under-determined direction of arrival (DOA) estimation. In this paper, we improve the traditional nested planar array to achieve more degrees of freedom (DOFs) and better angle estimation performance. The closed-form expressions for sensor positions of the improved array are given and the optimal array configuration for largest available DOFs is derived. Meanwhile, a computationally efficient DOA estimation algorithm is proposed. Specifically, we utilise two dimensional Discrete Fourier Transform (2D DFT) method to obtain the coarse DOA estimates; Subsequently, we achieve the fine DOA estimates by 2D spatial smoothing multiple signals classification (SS-MUSIC) algorithm. The proposed algorithm enjoys the same estimation accuracy as SS-MUSIC algorithm but with lower complexity because the coarse DOA estimates enable to shrink the range of spectral search. In addition, estimation of the number of signals is not required by 2D DFT method. Extensive simulation results testify the effectiveness of the proposed algorithm.