A novel approach for DOA estimation in unknown correlated noise fields

The key of the subspace-based Direction Of Arrival （DOA） estimation lies in the estimation of signal subspace with high quality. In the case of uncorrelated signals while the signals are temporally correlated, a novel approach for the estimation of DOA in unknown correlated noise fields is proposed in this paper. The approach is based on the biorthogonality between a matrix and its Moore-Penrose pseudo inverse, and made no assumption on the spatial covariance matrix of the noise. The approach exploits the structural information of a set of spatio-temporal correlation matrices, and it can give a robust and precise estimation of signal subspace, so a precise estimation of DOA is obtained. Its performances are confirmed by computer simulation results.     

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.     

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     

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。      

Adaptive estimation in linear systems with unknown Markovian noise statistics

The asymptotic behavior of a Bayes optimal adaptive estimation scheme for a linear discrete-time dynamical system with unknown Markovian noise statistics is investigated. Noise influencing the state equation and the measurement equation is assumed to come from a group of Gaussian distributions having different means and covariances, with transitions from one noise source to another determined by a Markov transition matrix. The transition probability matrix is unknown and can take values only from a finite set. An example is simulated to illustrate the convergence.     

DOA estimation in unknown colored noise using covariance differencing and sparse signal recovery

A direction-of-arrival （DOA） estimation algorithm is presented based on covariance differencing and sparse signal recovery, in which the desired signal is embedded in noise with unknown covariance. The key point of the algorithm is to eliminate the noise component by forming the difference of original and transformed covariance matrix, as well as cast the DOA estimation considered as a sparse signal recovery problem. Concerning accuracy and complexity of estimation, the authors take a vectorization operation on difference matrix, and further enforce sparsity by reweighted l1-norm penalty. We utilize data-validation to select the regularization parameter properly. Meanwhile, a kind of symmetric grid division and refinement strategy is introduced to make the proposed algorithm effective and also to mitigate the effects of limiting estimates to a grid of spatial locations. Compared with the covariance-differencing-based multiple signal classification （MUSIC） method, the proposed is of salient features, including increased resolution, improved robustness to colored noise, distinguishing the false peaks easily, but with no requiring of prior knowledge of the number of sources.     

An auto-calibration method for spatially and temporally correlated noncircular sources in unknown noise fields

Angularly dependent gain and phase uncertainties are produced by the combined effects of multiple sensor errors. This paper proposes a direction-finding method for noncircular signals in the presence of angularly dependent gain/phase errors, which utilizes instrumental sensors to achieve auto-calibration and relies on an improved alternating projection procedure. By applying the principle of the extended 2-sided instrumental variable signal subspace fitting algorithm, the proposed method is effective for separating spatially and temporally correlated noncircular sources from the unknown colored (i.e., spatially correlated) noise. Considering that modeling errors of instrumental sensors are frequently encountered in practice, this paper also presents a theoretical derivation for the closed-form expression of the mean square error of the estimation under the influence of modeling errors of instrumental sensors in the first-order analysis. Finally, the results of two series of simulations are demonstrated. The first series of simulations verifies the effectiveness of the proposed auto-calibration method, and shows that noncircularity and temporal correlation of sources are informative for enhancing the calibration performance of our method. The results also prove that the proposed method performs better than the instrumental sensor method when applied to spatially and temporally correlated noncircular sources. Moreover, this performance advantage of our method is more prominent when signal-to-noise ratio is low, or in spatially correlated noise fields. The second series of simulations validates the theoretical prediction, and thus our statistical analysis has a high predictive value for calibration performance of the proposed method under the influence of modeling errors.     

Maximum likelihood DOA estimation and asymptotic Cramer-Rao boundsfor additive unknown colored noise

This paper is devoted to the maximum likelihood estimation of multiple sources in the presence of unknown noise. With the spatial noise covariance modeled as a function of certain unknown parameters, e.g., an autoregressive (AR) model, a direct and systematic way is developed to find the exact maximum likelihood (ML) estimates of all parameters associated with the direction finding problem, including the direction-of-arrival (DOA) angles Θ, the noise parameters α, the signal covariance Φ_s, and the noise power σ². We show that the estimates of the linear part of the parameter set Φ_s and σ² can be separated from the nonlinear parts Θ and α. Thus, the estimates of Φ_s and σ² become explicit functions of Θ and α. This results in a significant reduction in the dimensionality of the nonlinear optimization problem. Asymptotic analysis is performed on the estimates of Θ and α, and compact formulas are obtained for the Cramer-Rao bounds (CRB's). Finally, a Newton-type algorithm is designed to solve the nonlinear optimization problem, and simulations show that the asymptotic CRB agrees well with the results from Monte Carlo trials, even for small numbers of snapshots     

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     

Effect of partially polarized amplified spontaneous emission noise on Q-factor estimation using optical signal-to-noise ratio

We report on the effect of the partially polarized amplified spontaneous emission noise on the Q-factor estimation by using optical signal-to-noise ratio. The result shows that this effect is negligible even in a long-distance transmission system (number of spans: <40) as long as the polarization-dependent loss per span is smaller than 0.2 dB.     

Particle filter with ant colony optimization for frequency offset estimation in OFDM systems with unknown noise distribution

Orthogonal frequency division multiplexing (OFDM) is sensitive to carrier frequency offset (CFO) that causes inter-carrier interference (ICI). In this paper, a particle filter (PF) method augmented with ant colony optimization (ACO) is developed to estimate the CFO. The ACO for continuous domains is incorporated into PF to optimize the sampling process. Unlike the standard PF, resampling is not required in the method. Moreover, it does not require the noise distribution. Simulation results show that the proposed method is effective when estimating the CFO and can effectively combat the effect of ICI in OFDM systems.     

Asymptotically robust detection in partially contaminated noise

We consider the discrete-time detection of a known time-varying deterministic signal in white noise, where the univariate noise density is known perfectly only on an interval about the origin. We present a method to enhance the asymptotic performance of the detector by exploiting this knowledge, and at the same time preserve robustness properties of the detector to the remaining inexact knowledge of the univariate noise density. We then provide examples to show that improved performance is indeed obtained.This research was supported by the Air Force Office of Scientific Research under Grant AFOSR-82-0033.     

DOA estimation with a rotational uniform linear array (RULA) and unknown spatial noise covariance

Direction of Arrival (DOA) estimation is one of the major tasks in array signal processing. In this paper, a new DOA estimation method is proposed using a rotational uniform linear array (RULA) consisting of omnidirectional sensors. The main contribution of the proposed method is that the number of distinguishable signals is larger than the methods in the literature with a uniform linear array consisting of the same number of omnidirectional sensors. Moreover, the new method can effectively reduce unknown spatial noises using a generalized complement projection matrix under the RULA framework. Simulations are presented to illustrate the effectiveness of the proposed method and comparison with some existing DOA estimation methods is also made.     

Parameter estimation in chaotic noise

The problem of parameter estimation in chaotic noise is considered in this paper. Since a chaotic signal is inherently deterministic, a new complexity measure called the phase space volume (PSV) is introduced for estimation instead of using the conventional probabilistic measures. We show that the unknown parameters of a signal embedded in chaotic noise ran be obtained by minimizing the PSV (MPSV) of the output of an inverse filter of the received signal in a reconstructed phase space. Monte Carlo simulations are carried out to analyze the efficiency of the MPSV method for parameter estimation in chaotic noise. To illustrate the usefulness of the MPSV technique in solving real-life problems, the problem of sinusoidal frequency estimation in real radar clutter (unwanted radar backscatters) is considered. Modeling radar clutter as a chaotic process, we apply the MPSV technique to estimate the sinusoidal frequencies by estimating the coefficients of an autoregressive (AR) spectrum. The results show that the frequency estimates generated by the MPSV method are more accurate than those obtained by the standard least square (LS) technique     

Direction finding using noise covariance modeling

We consider the problem of direction finding in the presence of colored noise whose covariance matrix is unknown. We show that the ambient noise covariance matrix can be modeled by a sum of Hermitian matrices known up to a multiplicative scalar. Using this model, we estimate jointly the directions of arrival of the signals and the noise model parameters. We show that under certain conditions, it is possible to obtain unbiased and efficient estimates of the signal direction. The Cramer-Rao bound is used as the principal analysis tool. Computer simulations using the maximum likelihood estimator provide a validation of the analytical results     

Optimal, causal, simultaneous detection and estimation of random signal fields in a Gaussian noise field

Two methods are presented for simultaneously detecting a random signal field and estimating its Markovian parameters in the qq prurience of a white Gaussian noise field. The first method provides the Neyman-Pearson decision rule for determining the absence-or-presence of the signal and the maximum {em a posteriori} likelihood estimators of the parameters. The second provides the Bayes decision rule for the absence-or-presence of the signal and the minimum mean-square error estimators of the parameters. Both methods allow continual causal detection and estimation based on the continuous space-time data, and their detection and estimation statistics are given in terms of a single statistic, which is the solution of a certain stochastic integral equation. Furthermore, an approximate scheme is developed for recursively generating this statistic by using spatially and temporally sampled data.     

Minimax estimation of unknown deterministic signals in colorednoise

The estimation of a deterministic signal corrupted by random noise is considered. The strategy is to find a linear noncausal estimator which minimizes the maximum mean square error over an a priori set of signals. This signal set is specified in terms of frequency/energy constraints via the discrete Fourier transform. Exact filter expressions are given for the case of additive white noise. For the case of additive colored noise possessing a continuous power spectral density, a suboptimal filter is derived whose asymptotic performance is optimal. Asymptotic expressions for the minimax estimator error are developed for both cases. The minimax filter is applied to random data and is shown to solve asymptotically a certain worst-case Wiener filter problem     

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     

Parameter estimation for partially observed queues

We consider parameter estimation for a FIFO queue with deterministic service times and two independent arrival streams of “observed” and “unobserved” packets. The arrivals of unobserved packets are Poisson with an unknown rate λ while the arrivals of observed packets are arbitrary. Maximum likelihood estimation of λ is formulated based on the arrival times and waiting times of k observed packets. The likelihood function is derived in terms of the transition probabilities of the unfinished work process which are calculated recursively. Sufficient conditions for consistency, asymptotic normality, and asymptotic efficiency are given. The mean and variance of the MLE are measured in simulation experiments. Numerical results indicate that the MLE is consistent and asymptotically normal