Complex amplitude estimation in the known steering matrix and generalized waveform case

We consider a special growth-curve (SGC) model with a known steering matrix and generalized waveform in the presence of unknown interference and noise. Several estimators of the complex amplitude based on this model are derived, including the methods of approximate maximum likelihood (AML), minimum variance distortionless response (MVDR), and amplitude and phase estimation (APES). We analyze the statistical properties of these estimators and show that in the presence of temporally white but spatially correlated noise and interference, AML is asymptotically statistically efficient for a large snapshot number while MVDR and APES are asymptotically equivalent but not statistically efficient. Via several numerical examples, we also show that when the noise and interference are both spatially and temporally correlated, the APES estimator can achieve better estimation accuracy and exhibit greater robustness than the other methods.     

Space-time fading channel estimation and symbol detection inunknown spatially correlated noise

We present maximum likelihood (ML) methods for space-time fading channel estimation with an antenna array in spatially correlated noise having unknown covariance; the results are applied to symbol detection. The received signal is modeled as a linear combination of multipath-delayed and Doppler-shifted copies of the transmitted waveform. We consider structured and unstructured array response models and derive the Cramer-Rao bound (CRB) for the unknown directions of arrival, time delays, and Doppler shifts. We also develop methods for spatial and temporal interference suppression. Finally, we propose coherent matched-filter and concentrated-likelihood receivers that account for the spatial noise covariance and analyze their performance     

Maximum likelihood estimation, analysis, and applications ofexponential polynomial signals

We model complex signals by approximating the phase and the logarithm of the time-varying amplitude of the signal as a finite order polynomial. We refer to a signal that has this form as an exponential polynomial signal (EPS). We derive an iterative maximum-likelihood (ML) estimation algorithm to estimate the unknown parameters of the EPS model. The initialization of the ML algorithm can be performed by using the result of a related paper. A statistical analysis of the ML algorithm is performed using a finite-order Taylor expansion of the mean squared error (MSE) of the estimate about the variance of the additive noise. This perturbation analysis gives a method of predicting the MSE of the estimate for any choice of the signal parameters. The MSE from the perturbation analysis is compared with the MSE from a Monte Carlo simulation and the Cramer-Rao Bound (CRB). The CRB for this model is also derived     

Robust adaptive array for wireless communications

In the application of a receiver antenna array to wireless communications, a known signal preamble is used for estimating the propagation vector at the beginning of each data frame. The estimated propagation vector is then used in linear combining of array inputs for interference suppression and demodulation of a desired user's information data stream. Since the training preamble is usually very short, conventional training methods, which estimate the propagation vector based solely on the training preamble, may incur large estimation errors. In many wireless channels, the ambient noise is known to be decidedly non-Gaussian, due to impulsive phenomena. The conventional training methods may suffer further from such impulsive noise. Moreover, performance of linear combining techniques can degrade substantially in the presence of impulsive noise. We first propose a new technique for propagation vector estimation which exploits the whole frame of the received signal. It is shown that as the length of the signal frame tends to infinity, in the absence of noise, this method can recover the propagation vector of the desired user exactly, given a small number of training symbols for that user. We then develop robust techniques for propagation vector estimation and array combining in the presence of impulsive noise. These techniques are nonlinear in nature and are based on the M-estimation method. It is seen that the proposed robust methods offer performance improvement over linear techniques in non-Gaussian noise, with little attendant increase in computational complexity. Finally, we address the extension of the proposed techniques to dispersive channels with intersymbol interference     

On robust Capon beamforming and diagonal loading

The Capon (1969) beamformer has better resolution and much better interference rejection capability than the standard (data-independent) beamformer, provided that the array steering vector corresponding to the signal of interest (SOI) is accurately known. However, whenever the knowledge of the SOI steering vector is imprecise (as is often the case in practice), the performance of the Capon beamformer may become worse than that of the standard beamformer. Diagonal loading (including its extended versions) has been a popular approach to improve the robustness of the Capon beamformer. We show that a natural extension of the Capon beamformer to the case of uncertain steering vectors also belongs to the class of diagonal loading approaches, but the amount of diagonal loading can be precisely calculated based on the uncertainty set of the steering vector. The proposed robust Capon beamformer can be efficiently computed at a comparable cost with that of the standard Capon beamformer. Its excellent performance for SOI power estimation is demonstrated via a number of numerical examples.     

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.     

Direction-of-arrival estimation based on spatial–temporal statistics without knowing the source number

Direction-of-arrival (DOA) estimation is a central problem in array processing and has a variety of applications. In this paper, a new algorithm for finding DOAs of multiple temporally correlated signals is devised. The proposed approach is based on the joint diagonalization structure of a set of spatio-temporal correlation matrices. Unlike the subspace-based DOA estimators, it is not necessary to estimate the noise or signal subspace explicitly. Moreover, the proposed method can provide the spatial spectrum and estimate the DOAs even when the number of sources is not known a priori. Interestingly, it is revealed that the well-known MUSIC method is a special case of our algorithm. Simulation results validate that the developed approach is superior to conventional DOA estimators in terms of resolution capability, estimation accuracy, and robustness against array model errors.     

Capon multiuser receiver for CDMA systems with space-time coding

We present in this paper a linear blind multiuser receiver, referred to as the Capon receiver, for code-division multiple-access (CDMA) systems utilizing multiple transmit antennas and space-time (ST) block coding. The Capon receiver is designed by exploiting signal structures imposed by both spreading and ST coding. We highlight the unique ST coding induced structure, which is shown to be critical in establishing several analytical results, including self-interference (i.e., spatially mixed signals of the same user) cancellation, receiver output signal-to-interference-and-noise ratio (SINR), and blind channel estimation of the Capon receiver. To resolve the scalar ambiguity intrinsic to all blind schemes, we propose a semi-blind implementation of the Capon receiver, which capitalizes on periodically inserted pilots and the interference suppression ability of the Capon filters, for (slowly) time-varying channels. Numerical examples are presented to compare the Capon receiver with several other training-assisted and (semi-)blind receivers and to illustrate the performance gain of ST-coded CDMA systems over those without ST coding     

四元数域干扰加噪声协方差矩阵重构鲁棒自适应波束形成

基于电磁矢量传感器阵列的四元数Capon波束形成器较传统的复数域Capon波束形成器有更好的性能。但是该方法在存在指向误差和极化失配的情况下性能急剧下降,甚至会出现信号相消现象。本文将协方差矩阵重构方法推广于四元数Capon波束形成中,通过利用Q-Capon的极化-角度谱估计得到干扰和噪声的功率来对干扰加噪声协方差矩阵进行重构,避免了对角加载方法中对对角加载因子的求解,而且能够有效克服指向误差与极化失配带来的性能下降。计算机仿真表明,该方法相较于其他四元数域的方法有着更好的性能。      

基于贪婪算法的高分辨信号源DOA估计

确定辐射源的来波方向(DOA)是阵列信号处理的重要研究内容,已经广泛应用于雷达、声纳和无线通信等领域。本文研究了远场窄带信号源的DOA高分辨估计问题。利用信号来波方向在空域具有稀疏性的特点,建立了远场窄带信号源的稀疏表示模型。根据协方差矩阵的特征值分解和贪婪匹配追踪算法原理提出了一种基于特征值分解的多重正交匹配追踪算法（EIG MOMP）。首先,利用特征值分解对阵列接收数据进行降维处理。这一降维操作使得问题转化为了一个具有多重观测向量（MMV）的欠定方程求解问题。接着利用MOMP算法对降维后的数据进行处理,最终得到信号的DOA估计值。该算法实现了在低信噪比下远场窄带信号源的高分辨DOA估计,并具有较低的运算复杂度。将本文提出的算法与传统的Capon算法、多重信号分类算法（MUSIC）以及正交匹配追踪算法（OMP）进行了对比。结果证明,该算法在低信噪比下能取得较好的DOA估计效果,可以针对任意的相干信号源,并且具有高分辨率的优点。      

A modified likelihood function approach to DOA estimation in thepresence of unknown spatially correlated Gaussian noise using a uniformlinear array

The problem of modified ML estimation of DOAs of multiple source signals incident on a uniform linear array (ULA) in the presence of unknown spatially correlated Gaussian noise is addressed here. Unlike previous work, the proposed method does not impose any structural constraints or parameterization of the signal and noise covariances. It is shown that the characterization suggested here provides a very convenient framework for obtaining an intuitively appealing estimate of the unknown noise covariance matrix via a suitable projection of the observed covariance matrix onto a subspace that is orthogonal complement of the so-called signal subspace. This leads to a formulation of an expression for a so-called modified likelihood function, which can be maximized to obtain the unknown DOAs. For the case of an arbitrary array geometry, this function has explicit dependence on the unknown noise covariance matrix. This explicit dependence can be avoided for the special case of a uniform linear array by using a simple polynomial characterization of the latter. A simple approximate version of this function is then developed that can be maximized via the-well-known IQML algorithm or its variants. An exact estimate based on the maximization of the modified likelihood function is obtained by using nonlinear optimization techniques where the approximate estimates are used for initialization. The proposed estimator is shown to outperform the MAP estimator of Reilly et al. (1992). Extensive simulations have been carried out to show the validity of the proposed algorithm and to compare it with some previous solutions     

Relationships between adaptive minimum variance beamforming andoptimal source localization

For many years, the popular minimum variance (MV) adaptive beamformer has been well known for not having been derived as a maximum likelihood (ML) estimator. This paper demonstrates that by use of a judicious decomposition of the signal and noise, the log-likelihood function of source location is, in fact, directly proportional to the adaptive MV beamformer output power. In the proposed model, the measurement consists of an unknown temporal signal whose spatial wavefront is known as a function of its unknown location, which is embedded in complex Gaussian noise with unknown but positive definite covariance. Further, in cases where the available observation time is insufficient, a constrained ML estimator is derived here that is closely related to MV beamforming with a diagonally loaded data covariance matrix estimate. The performance of the constrained ML estimator compares favorably with robust MV techniques, giving slightly better root-mean-square error (RMSE) angle-of-arrival estimation of a plane-wave signal in interference. More importantly, however, the fact that such optimal ML techniques are closely related to conventional robust MV methods, such as diagonal loading, lends theoretical justification to the use of these practical approaches     

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.     

Radio-astronomical imaging in the presence of strong radiointerference

Radio-astronomical observations are increasingly contaminated by interference, and suppression techniques become essential. A powerful candidate for interference mitigation is adaptive spatial filtering. We study the effect of spatial filtering techniques on radio-astronomical imaging. Current deconvolution procedures, such as CLEAN, are shown to be unsuitable for spatially filtered data, and the necessary corrections are derived. To that end, we reformulate the imaging (deconvolution/calibration) process as a sequential estimation of the locations of astronomical sources. This not only leads to an extended CLEAN algorithm, but also the formulation allows the insertion of other array signal processing techniques for direction finding and gives estimates of the expected image quality and the amount of interference suppression that can be achieved. Finally, a maximum-likelihood (ML) procedure for the imaging is derived, and an approximate ML image formation technique is proposed to overcome the computational burden involved. Some of the effects of the new algorithms are shown in simulated images     

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     

Doubly constrained robust Capon beamformer

The standard Capon beamformer (SCB) is known to have better resolution and much better interference rejection capability than the standard data-independent beamformer when the array steering vector is accurately known. However, the major problem of the SCB is that it lacks robustness in the presence of array steering vector errors. In this paper, we will first provide a complete analysis of a norm constrained Capon beamforming (NCCB) approach, which uses a norm constraint on the weight vector to improve the robustness against array steering vector errors and noise. Our analysis of NCCB is thorough and sheds more light on the choice of the norm constraint than what was commonly known. We also provide a natural extension of the SCB, which has been obtained via covariance matrix fitting, to the case of uncertain steering vectors by enforcing a double constraint on the array steering vector, viz. a constant norm constraint and a spherical uncertainty set constraint, which we refer to as the doubly constrained robust Capon beamformer (DCRCB). NCCB and DCRCB can both be efficiently computed at a comparable cost with that of the SCB. Performance comparisons of NCCB, DCRCB, and several other adaptive beamformers via a number of numerical examples are also presented.     

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     

冲激噪声背景下基于幅度预处理的测向新方法

冲激噪声环境下的测向算法大多基于分数低阶统计量提出的,其计算复杂度相对传统的二阶矩测向算法大大增加.通过对冲激噪声幅值特征进行分析,分别提出基于阵列接收数据幅度均值和中值进行幅度预处理的测向新方法.两种方法都是利用了冲激噪声分布的统计特性,首先根据阵列接收数据估计出幅值门限,然后对阵列接收数据进行幅度预处理,可以有效抑制噪声的冲激特性,从而可以利用传统的二阶矩类测向算法进行波达方向估计.理论分析和实验仿真结果表明,两种方法均处理简便,计算复杂度低,无需估计先验参数;基于中值的幅度预处理方法更是具有估计精度优良,对低信噪比时的测向性能有较大改善,且适用于强冲激噪声环境等优点.     

基于稳健Capon波束形成技术的矢量相关测向方法

为了弥补阵列天线导向矢量失配对测向性能的影响,提出基于稳健Capon波束形成技术的矢量相关测向方法。与直接使用相位差的常规相关干涉测向技术不同,该方法首先利用稳健Capon波束形成技术估计目标信号的真实导向矢量;然后通过导向矢量的相关拟合确定目标信号方向。通过仿真分析,得出了以测向为衡量标准时不确定集约束参数的选择原则。仿真结果表明该方法能够弥补阵列流型失配的影响、准确测量目标信号方向。     

DOA Estimation and Detection in Colored Noise Using Additional Noise-Only Data

In a typical array processing scenario, noise acting on the array can not be assumed spatially white. It is in many cases necessary to use quiet periods, when only noise is received, to estimate the noise covariance. If estimation of the signal parameters and noise covariance is performed jointly, performance can be improved. This is especially true when stationarity considerations limit the amount of available valid noise-only data. An asymptotically valid approximative maximum likelihood method (AML) for the estimation problem is derived in this work. The resulting criterion is, when concentrated with respect to the signal parameters, relatively simple. In numerical experiments, AML shows promising small-sample performance compared to earlier methods. The criterion function is also well suited for numerical optimization. The new criterion function allows for the development of a novel, MODE-like, non-iterative estimation procedure if the array belongs to the important class of uniform linear arrays. The resulting procedure retains the asymptotic properties of maximum likelihood, and numerical simulations indicate superior threshold performance when compared to an optimally weighted subspace fitting (WSF) formulation of MODE. For the detection problem, no method has been presented that takes the unknown noise covariance into account. Here, a well known detection scheme for WSF is extended to work in this scenario as well. The derivations of this scheme further stress the importance of using the correct weighting in WSF when the noise covariance is unknown. It is also shown that the minimum value of the criterion function associated with AML can be used for the detection purpose. Numerical experiments indicate very promising performance for the AML-detection scheme.