Discrete fast algorithms for two-dimensional linear prediction on apolar raster

New generalized split Levinson and Schur algorithms for the two-dimensional linear least squares prediction problem on a polar raster are derived. The algorithms compute the prediction filter for estimating a random field at the edge of a disk from noisy observations inside the disk. The covariance function of the random field is assumed to have a Toeplitz-plus-Hankel structure for both its radial part and its transverse (angular) part. This assumption is valid for some types of random fields, such as isotropic random fields. The algorithms generalize the split Levinson and Schur algorithms in two ways: (1) to two dimensions; and (2) to Toeplitz-plus-Hankel covariances     

A resampling method for estimating the signal subspace of spatio-temporal EEG/MEG data

Source localization using spatio-temporal electroencephalography (EEG) and magnetoencephalography (MEG) data is usually performed by means of signal subspace methods. The first step of these methods is the estimation of a set of vectors that spans a subspace containing as well as possible the signal of interest. This estimation is usually performed by means of a singular value decomposition (SVD) of the data matrix: The rank of the signal subspace (denoted by r) is estimated from a plot in which the singular values are plotted against their rank order, and the signal subspace itself is estimated by the first r singular vectors. The main problem with this method is that it is strongly affected by spatial covariance in the noise. Therefore, two methods are proposed that are much less affected by this spatial covariance, and old and a new method. The old method involves prewhitening of the data matrix, making use of an estimate of the spatial noise covariance matrix. The new method is based on the matrix product of two average data matrices, resulting from a random partition of a set of stochastically independent replications of the spatio-temporal data matrix. The estimated signal subspace is obtained by first filtering out the asymmetric and negative definite components of this matrix product and then retaining the eigenvectors that correspond to the r largest eigenvalues of this filtered data matrix. The main advantages of the partition-based eigen decomposition over prewhited SVD is that 1) it does not require an estimate of the spatial noise covariance matrix and 2b) that it allows one to make use of a resampling distribution (the so-called partitioning distribution) as a natural quantification of the uncertainty in the estimated rank. The performance of three methods (SVD with and without prewhitening, and the partition-based method) is compared in a simulation study. From this study, it could be concluded that prewhited SVD and the partition-based eigen decomposition perform equally well when the amplitude time series are constant, but that the partition-based method performs better when the amplitude time series are variable.     

Noise covariance incorporated MEG-MUSIC algorithm: a method formultiple-dipole estimation tolerant of the influence of background brainactivity

This paper proposes a method of localizing multiple current dipoles from spatio-temporal biomagnetic data. The method is based on the multiple signal classification (MUSIC) algorithm and is tolerant of the influence of background brain activity. In this method, the noise covariance matrix is estimated using a portion of the data that contains noise, but does not contain any signal information. Then, a modified noise subspace projector is formed using the generalized eigenvectors of the noise and measured-data covariance matrices. The MUSIC localizer is calculated using this noise subspace projector and the noise covariance matrix. The results from a computer simulation have verified the effectiveness of the method. The method was then applied to source estimation for auditory-evoked fields elicited by syllable speech sounds. The results strongly suggest the method's effectiveness in removing the influence of background activity     

Interference mitigation in STAP using the two-dimensional Wold decomposition model

We propose a novel parametric approach for modeling, estimation, and detection in space-time adaptive processing (STAP) radar systems. The proposed parametric interference mitigation procedures can be applied even when information in only a single range gate is available, thus achieving high performance gain when the data in the different range gates cannot be assumed stationary. The model is based on the Wold-like decomposition of two-dimensional (2D) random fields. It is first shown that the same parametric model that results from the 2D Wold-like orthogonal decomposition naturally arises as the physical model in the problem of space-time processing of airborne radar data. We exploit this correspondence to derive computationally efficient fully adaptive and partially adaptive detection algorithms. Having estimated the models of the noise and interference components of the field, the estimated parameters are substituted into the parametric expression of the interference-plus-noise covariance matrix. Hence, an estimate of the fully adaptive weight vector is obtained, and a corresponding test is derived. Moreover, we prove that it is sufficient to estimate only the spectral support parameters of each interference component in order to obtain a projection matrix onto the subspace orthogonal to the interference subspace. The resulting partially adaptive detector is simple to implement, as only a very small number of unknown parameters need to be estimated, rather than the field covariance matrix. The performance of the proposed methods is illustrated using numerical examples.     

MIMO阵列恒定包络波形设计

提出了MIMO阵列系统利用白高斯随机过程设计恒定包络波形算法,实现给定发射波形的协方差矩阵.该算法首先对白高斯随机过程去白化得到高斯随机变量,建立恒定包络波形相关矩阵与高斯随机变量相关矩阵之间的关系,再采用无记忆性非线性投影函数把高斯随机变量投影到恒定包络随机变量,从而解出恒定包络波形.与现有的阵列系统波形设计算法相比...     

Estimation of structured covariance matrices

Covariance matrices from stationary time series are Toeplitz. Multichannel and multidimensional processes have covariance matrices of block Toeplitz form. In these cases and many other situations, one knows that the actual covariance matrix belongs to a particular subclass of covariance matrices. This paper discusses a method for estimating a covariance matrix of specified structure from vector samples of the random process. The theoretical foundation of the method is to assume that the random process is zero-mean multivariate Gaussian, and to find the maximum-likelihood covariance matrix that has the specified structure. An existence proof is given and the solution is interpreted in terms of a minimum-entropy principle. The necessary gradient conditions that must be satisfied by the maximum-likelihood solution are derived and unique and nonunique analytic solutions for some simple problems are presented. A major contribution of this paper is an iterative algorithm that solves the necessary gradient equations for moderate-sized problems with reasonable computational ease. Theoretical convergence properties of the basic algorithm are investigated and robust modifications discussed. In doing maximum-entropy spectral analysis of a sine wave in white noise from a single vector sample, this new estimation procedure causes no splitting of the spectral line in contrast to the Burg technique.     

Estimation of the directions of arrival of signals in unknowncorrelated noise. I. The MAP approach and its implementation

The authors propose a method of direction of arrival (DOA) estimation of signals in the presence of noise whose covariance matrix is unknown and arbitrary, other than being positive definite. They examine the projection of the data onto the noise subspace. The conditional probability density function (PDF) of the projected data given the signal parameters and the unknown projected noise covariance matrix is first formed. The a posteriori PDF of the signal parameters alone is then obtained by assigning a noninformative a priori PDF to the unknown noise covariance matrix and integrating out this quantity. A simple criterion for the maximum a posteriori (MAP) estimate of the DOAs of the signals is established. Some properties of this criterion are discussed, and an efficient numerical algorithm for the implementation of this criterion is developed. The advantage of this method is that the noise covariance matrix does not have to be known, nor must it be estimated     

Semiblind Channel Estimation for MIMO Single Carrier With Frequency-Domain Equalization Systems

We propose a semiblind channel-estimation method for multiple-input multiple-output (MIMO) single carrier with frequency-domain equalization systems. By taking advantage of periodic precoding and the block circulant channel model after cyclic prefix removal, we obtain the channel-product matrices by solving a series of decoupled linear systems, which is gained from the covariance matrix of the received data. Then, the channel-impulse-response matrix is obtained by computing the positive eigenvalues and eigenvectors of a Hermitian matrix formed from the channel-product matrices. We also propose an optimal design of the precoding sequence, which minimizes the noise effect and numerical error in covariance matrix estimation, and discuss the impact of the optimal sequence on channel equalization. With the proposed framework, the method is shown to be robust with respect to channel-order overestimation, and the identifiability condition is simply that the channel-impulse-response matrix has full column rank. Due to the identifiability condition, the method is applicable to MIMO channels with more transmitters or more receivers. Simulations are used to demonstrate the performance of the proposed method.      

一种改进的快速高分辨DOA估计方法

快速子空间类高分辨DOA估计方法(Propagator Method,PM)算法是从阵列接收数据的协方差矩阵中提取噪声子空间的基,此方法是利用被噪声污染的数据来实现对噪声子空间的逼近。利用阵列分割的消噪预处理原理对PM算法进行改进,改进后的算法是在剔除噪声之后的数据协方差矩阵中提取噪声子空间的基。仿真试验表明改进的算法会带来阵列孔径损失,但是在大快拍数的条件下DOA估计的分辨力有明显的提高而算法的计算量没有增加。     

Subspace-based error and erasure correction with DFT codes for wireless channels

This paper deals with the decoding of lowpass discrete Fourier transform (DFT) codes in the presence of both errors and erasures. We propose a subspace-based approach for the error localization that is similar to the subspace approaches followed in the array signal processing for direction-of-arrival (DOA) estimation. The basic idea is to divide a vector space into two orthogonal subspaces of which one is spanned by the error locator vectors. The locations of the errors are estimated from the spanning eigenvectors of the complement subspace. However, unlike the subspace approach in DOA estimation, which is similar to estimating the subspaces from the syndrome covariance matrix after a projection, in the proposed approach, the subspaces are estimated from the modified syndrome covariance matrix after a whitening transform. Simulation results with a Gauss-Markov source reveal that the proposed algorithm is more efficient than the coding theoretic approach on impulsive channels as well as the subspace approach with projection on lossy channels.     

一种新颖的DOA估计算法

子空间类波达方向(Direction Of Arrival, DOA)估计算法的关键在于得到高质量的信号子空间估计。该文利用矩阵伪逆的双正交性,针对源信号不相关而其本身是色信号的情况,给出了一种新颖的DOA估计算法,它不需要知道噪声统计特性。该算法利用一组空时相关矩阵的结构化信息,能稳健而精确地估计出信号子空间,从而得到DOA的精确估计。仿真实验证实了所给算法的有效性。     

Covariance Matrix Estimation With Heterogeneous Samples

We consider the problem of estimating the covariance matrix M_p of an observation vector, using heterogeneous training samples, i.e., samples whose covariance matrices are not exactly M_p. More precisely, we assume that the training samples can be clustered into K groups, each one containing L_k, snapshots sharing the same covariance matrix M_k. Furthermore, a Bayesian approach is proposed in which the matrices M_k. are assumed to be random with some prior distribution. We consider two different assumptions for M_p. In a fully Bayesian framework, M_p is assumed to be random with a given prior distribution. Under this assumption, we derive the minimum mean-square error (MMSE) estimator of M_p which is implemented using a Gibbs-sampling strategy. Moreover, a simpler scheme based on a weighted sample covariance matrix (SCM) is also considered. The weights minimizing the mean square error (MSE) of the estimated covariance matrix are derived. Furthermore, we consider estimators based on colored or diagonal loading of the weighted SCM, and we determine theoretically the optimal level of loading. Finally, in order to relax the a priori assumptions about the covariance matrix M_p, the second part of the paper assumes that this matrix is deterministic and derives its maximum-likelihood estimator. Numerical simulations are presented to illustrate the performance of the different estimation schemes.     

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.     

Simple polynomial detectors for CDMA downlink transmissions on frequency-selective channels

In code-division multiple-access (CDMA) transmissions, computing the multiuser minimum mean-squared error (MMSE) detector coefficients requires the inversion of the covariance matrix associated to the received vector signal, an operation usually difficult to implement when the spreading factor and the number of users are large. It is therefore interesting to approximate the inverse by a matrix polynomial. In this correspondence, means for computing the polynomial coefficients are proposed in the context of CDMA downlink transmissions on frequency-selective channels, the users having possibly different powers. Derivations are made in the asymptotic regime where the spreading factor and the number of users grow toward infinity at the same rate. Results pertaining to the mathematics of large random matrices, and in particular to free probability theory, are used. Spreading matrices are modeled as isometric random matrices (spreading vectors orthonormality is a natural assumption in downlink) and also as random matrices with independent and identically distributed (i.i.d.) elements.     

Fast estimation of continuous Karhunen-Loeve eigenfunctions usingwavelets

This paper develops a new wavelet method for the fast estimation of continuous Karhunen-Loeve eigenfunctions. The method of snapshots is modified by projecting the ensemble functions onto orthogonal or biorthogonal interpolating function spaces. Under well-behaved piecewise smooth polynomial ensemble functions, the size of the covariance matrix produced is greatly reduced, without sacrificing much accuracy. Moreover, the covariance matrix C˜ may be easily decomposed such that C˜ = A^T A, and thus, the more stable singular value decomposition (SVD) algorithm may be applied. An interpolating scheme that reduces the computation of projecting the ensemble functions onto the biorthogonal subspace to a single sample is also developed. Furthermore, by projecting the ensemble functions onto wavelet spaces, the covariance matrix may be sparsified by a multiresolution decomposition. Error bounds for the eigenvalues between the sparsified and nonsparsified covariance matrix are also derived     

Two-Dimensional DOA Estimation by?Cross-Correlation Matrix Stacking

In this paper, we present a novel scheme to improve the two-dimensional (2-D) direction-of-arrival (DOA) estimation performance for narrowband signals impinging on two orthogonal uniform linear arrays (ULAs). The proposed scheme exploits the cross-correlation matrix information between subarray data to construct a stacking matrix and derive an expanded signal subspace representation through the singular value decomposition (SVD). This method enables the alleviation of the effects of additive noise. In particular, 2-D DOA estimation can be achieved by computing two rotation matrices with the same set of eigenvectors obtained by partitioning the expanded signal subspace. The pair matching procedure for elevation and azimuth angles is implemented by permutation test. Simulation results demonstrate that the proposed method performs better than the existing techniques in DOA estimation as well as the detection of successful pair matching.     

Tracking of signal subspace projectors

A new subspace tracking approach that directly operates on the projection matrix onto the signal subspace instead of tracking its unitary eigenbasis is proposed. Therefore, the difficulties arising from introducing a calculus on the manifold of projection matrices are overcome by a proper parameterization of the projectors. Simulation results are presented from the field of angular frequency retrieval in array signal processing     

A novel wideband DOA estimator based on Khatri-Rao subspace approach

A novel DOA estimation method for uncorrelated wideband sources named focusing Khatri-Rao subspace method (FKR) is proposed based on coherent signal-subspace method (CSM) and Khatri-Rao (KR) subspace. Compared with the conventional CSM that simply averages the covariance matrices of different frequency bins after focusing, FKR transforms the covariance matrices into a higher dimensional matrix through KR product. This method has three major advantages: (1) it achieves a higher resolution than CSM, (2) the root mean square error of DOA estimation from FKR is smaller than that of CSM when the initial angles are inaccurate and (3) it performs well even when the number of sensors is reduced to about half of the sources. The performance of the FKR method is demonstrated and analyzed through the computer simulations.     

Generalized Consistent Estimation on Low-Rank Krylov Subspaces of Arbitrarily High Dimension

The problem of Krylov subspace estimation based on the sample covariance matrix is addressed. The focus is on signal processing applications where the Krylov subspace is defined by the unknown second-order statistics of the observed samples and the signature vector associated with the desired parameter. In particular, the consistency of traditionally optimal sample estimators is revised and analytically characterized under a practically more relevant asymptotic regime, according to which not only the number of samples but also the observation dimension grow without bound at the same rate. Furthermore, an improved construction of a class of Krylov subspace estimators is proposed based on the generalized consistent estimation of a set of vector-valued power functions of the observation covariance matrix. To that effect, an extension of some known results from random matrix theory on the estimation of certain spectral functions of the covariance matrix to the convergence of not only the covariance eigenspectrum but also the associated eigensubspaces is provided. A new family of estimators is derived that generalizes conventional implementations by proving to be consistent for observations of arbitrarily high dimension. The proposed estimators are shown to outperform traditional constructions via the numerical evaluation of the solution to two fundamental problems in sensor array signal processing, namely the problem of estimating the power of an intended source and the estimation of the principal eigenspace and dominant eigenmodes of a structured covariance matrix.     

Semiblind Channel Estimation for Multiuser MIMO-CDMA Systems with Orthogonal Space-Time Block Codes

In this paper, we concern the channel estimation for a wireless communication system in which the techniques of multiple-input multiple-output, code division multiple access (CDMA) and orthogonal space-time block codes (OSTBCs) are integrated together for the purpose of achieving high data rate. We show that a composite channel information (CCI) vector can be formed, which contains the effects of channel state information, spreading coding and OSTBCs. From the standpoint of the MUltiple SIgnal Classification method, such CCI vector must lie in the signal subspace spanned by the dominant eigenvectors of the received data covariance matrix. Also, this CCI vector is located in another subspace which is associated with the CDMA and OSTBC codes and can be computed off-line. Using the vector space projections method, this CCI vector can be viewed as the intersection of these two subspaces and thus can be computed by alternative projections. In order to reduce the computation complexity, we propose an equivalent but computationally effective single-step solution in which the channel estimation amounts to searching for the principal eigenvector of a certain matrix with moderate size. Additionally, only one training block is required to overcome the problem of sign ambiguity. Numerical results demonstrate that, in addition to improving the bandwidth efficiency, the proposed method offers better performance in terms of channel estimation accuracy and bit-error-rate as compared with the standard nonblind least-squares channel estimation approach.