Unknown Noise Covariance Matrix Research Articles

A Tunable Adaptive Detector for Distributed Targets When Signal Mismatch Occurs

Aiming at the problem of detecting a distributed target when signal mismatch occurs, this paper proposes a tunable detector parameterized by an adjustable parameter. By adjusting the parameter, the tunable detector can achieve robust or selective detection of mismatched signals. Moreover, the proposed tunable detector, with a proper tunable parameter, can provide higher detection probability compared with existing detectors in the case of no signal mismatch. In addition, the proposed tunable detector possesses the constant false alarm rate property with the unknown noise covariance matrix.

Detector Design and Performance Analysis for Target Detection in Subspace Interference

It is often difficult to obtain sufficient training data for adaptive signal detection, which is required to calculate the unknown noise covariance matrix. Additionally, interference is frequently present, which complicates the detecting issue. We provide a two-step method, termed interference cancellation before detection (ICBD), to address the issue of signal detection in the unknown Gaussian noise and subspace interference. The first involves projecting the test and training data to the interference-orthogonal subspace in order to suppress the interference. Utilizing traditional adaptive detector design ideas is the next stage. Due to the smaller dimension of the projected data, the ICBD-based detectors can function with little training data. The ICBD has two additional benefits over traditional detectors. Lower computational burden and proper operation with interference being in the training data are two additional benefits of ICBD-based detectors over conventional ones. We also give the statistical properties of the ICBD-based detectors and demonstrate their equivalence with the traditional ones in the special case of a large amount of training data containing no interference.

GLRT-based detectors for FDA-MIMO radar with training data

In this paper, a frequency diverse array multi-input multi-output (FDA-MIMO) radar is used for moving target detection in unknown Gaussian noise. To solve the detection problem, we adopt the generalized likelihood ratio test (GLRT) and use the training data to estimate the unknown noise covariance matrix. We derive the GLRT both for the case of unknown target velocity and the case of known target velocity. We also derive the analytical expressions for the probabilities of detection (PDs) and the probabilities of false alarm (PFAs), which indicate that the proposed detectors have constant false alarm rate (CFAR) properties. Theoretical and simulation results show that the proposed detectors can provide higher PDs than the existing detectors which do not utilize training data.

Simple Direction-of-Arrival Estimation under Nonuniform Noise Scenarios

The uniform white noise assumption is widely applied for most of the existing direction-of-arrival (DOA) estimation methods, which however may degrade the estimation performance. In practical scenarios, the nonuniform white noise model is more adequate. In this paper, we propose a simple DOA estimation method in the presence of spatially nonuniform white noise with unknown noise covariance matrix. The estimate of the unknown noise covariance matrix is firstly obtained by utilizing the shift property of identical subarrays. Then, the received signal covariance matrix is prewhitened by the estimated noise covariance, which can overcome the biased estimate of signal covariance. Finally, the performance improvement is verified by simulations in terms of resolution ability and estimation accuracy.

Persymmetric Adaptive Array Detection of Spread Spectrum Signals

The spread spectrum signal detection problem is examined in colored noise with an unknown covariance matrix. When the receiver is equipped with a symmetrically spaced linear array, persymmetry exists in the received data. We exploit the persymmetric structures to design adaptive detectors according to the principles of generalized likelihood ratio test (GLRT), Wald test, and Rao test. It turns out that the proposed GLRT has the same form as the proposed Wald test, and the Rao test does not exist. We prove that the proposed detector exhibits a constant false alarm rate against the unknown noise covariance matrix. Numerical examples demonstrate that the proposed detector has better performance than its non-persymmetric counterpart.

Recursive Extended Instrumental Variable Based LCMV Beamformers for Planar Radial Coprime Arrays Under Spatially Colored Noise

This article proposes a new recursive linearly constrained minimum variance (LCMV) beamformer based on the extended instrumental variable (EIV) method for planar radial coprime arrays (PRCAs) under spatially colored noise. The proposed recursive LCMV beamformer is able to deal with multiple constraints with high precision and low complexity and can be applicable to various array geometrical configurations. Taking advantage of the EIV vector, the proposed beamformer can effectively combat the additive color noise with unknown noise covariance matrix. We develop our recursive LCMV beamformer based on the square-root (SR) EIV algorithm due to its improved numerical stability than the conventional EIV-based algorithms. Furthermore, we studied a class of planar arrays called PRCAs, which consists of a set of linear coprime arrays arranged radially at various azimuth angles. The coprime array property is utilized to enlarge the array aperture leading to higher resolution and stronger interference rejection and it offers additional flexibility in the tradeoffs between array complexity and performance. Simulation results demonstrate that the proposed recursive SREIV-based LCMV beamformer outperforms the conventional QR decomposition based LCMV beamformers in the resolution and suppression of interferences under various scenarios. The PRCAs tested outperform the uniform rectangular arrays with the same number of elements. Moreover, better performance can be achieved with more linear subarrays at the expense of increased complexity.

Pinsker estimators for local helioseismology: inversion of travel times for mass-conserving flows

A major goal of helioseismology is the three-dimensional reconstruction of the three velocity components of convective flows in the solar interior from sets of wave travel-time measurements. For small amplitude flows, the forward problem is described in good approximation by a large system of convolution equations. The input observations are highly noisy random vectors with a known dense covariance matrix. This leads to a large statistical linear inverse problem. Whereas for deterministic linear inverse problems several computationally efficient minimax optimal regularization methods exist, only one minimax-optimal linear estimator exists for statistical linear inverse problems: the Pinsker estimator. However, it is often computationally inefficient because it requires a singular value decomposition of the forward operator or it is not applicable because of an unknown noise covariance matrix, so it is rarely used for real-world problems. These limitations do not apply in helioseismology. We present a simplified proof of the optimality properties of the Pinsker estimator and show that it yields significantly better reconstructions than traditional inversion methods used in helioseismology, i.e. regularized least squares (Tikhonov regularization) and SOLA (approximate inverse) methods. Moreover, we discuss the incorporation of the mass conservation constraint in the Pinsker scheme using staggered grids. With this improvement we can reconstruct not only horizontal, but also vertical velocity components that are much smaller in amplitude.

Kalman Filter With Recursive Covariance Estimation—Sequentially Estimating Process Noise Covariance

The Kalman filter has been found to be useful in vast areas. However, it is well known that the successful use of the standard Kalman filter is greatly restricted by the strict requirements on a priori information of the model structure and statistics information of the process, and measurement noises. Generally speaking, the covariance matrix of process noise is harder to be determined than that of the measurement noise by routine experiments, since the statistical property of process noise cannot be obtained directly by collecting a large number of sensor data due to the intrinsic coupling of process noise and system dynamics. Considering such background of wide applications, this paper introduces one algorithm, recursive covariance estimation (RCE) algorithm, to estimate the unknown covariance matrix of noise from a sample of signals corrupted with the noise. Based on this idea, for a class of discrete-time linear-time-invariant systems where the covariance matrix of process noise is completely unknown, a new Kalman filtering algorithm named, Kalman filter with RCE, is presented to resolve this challenging problem of state estimation without the statistical information of process noise, and the rigorous stability analysis is given to show that this algorithm is optimal in the sense that the covariance matrix and state estimations are asymptotically consistent with the ideal Kalman filter when the exact covariance matrix of process noise is completely known a priori. Extensive simulation studies have also verified the theoretical results and the effectiveness of the proposed algorithm.

Recursive Generalized Total Least Squares with Noise Covariance Estimation

We propose a recursive generalized total least-squares (RGTLS) estimator that is used in parallel with a noise covariance estimator (NCE) to solve the errors-in-variables problem for multi-input-single-output linear systems with unknown noise covariance matrix. Simulation experiments show that the suggested RGTLS with NCE procedure outperforms the common recursive least squares (RLS) and recursive total instrumental variables (RTIV) estimators when all measured inputs and the measured output are noisy. Moreover, when all measured inputs are noise-free, RGTLS with NCE performs similarly to RLS, which in this special case is the optimal estimator, and again RTIV was inferior compared with the RGTLS and NCE procedure.

Performance Enhancement of Subspace Detection With a Diversely Polarized Antenna

Based on a diversely polarized antenna, we address the problems of adaptive detection and performance enhancement in partially homogeneous environments where the test and training data samples share the same noise covariance matrix up to an unknown scaling factor. The matched subspace detector and adaptive subspace detector are employed to handle the detection problem for the cases of known and unknown noise covariance matrix structure, respectively. The performance of the two detectors is evaluated in terms of their probabilities of false alarm and detection. In particular, a waveform design algorithm to enhance the detection performance of the two detectors is proposed. A numerical analysis is presented to demonstrate the potential performance improvement obtained with this algorithm.

Diffuse Noise Suppression Using Crystal-Shaped Microphone Arrays

This paper describes novel methods for diffuse noise suppression using crystal-shaped microphone arrays. The two-stage processing of the observed signals by the Minimum Variance Distortionless Response (MVDR) beamformer and the subsequent Wiener post-filter is effective for diffuse noise suppression and gives the linear minimum mean square error (LMMSE) estimator of the target signal. It is essential in this framework to accurately estimate the short-time power spectrum and the steering vectors of the target signal from the noisy observations. Our methods diagonalize the spatial noise covariance matrix and utilizes the denoised off-diagonal entries of the spatial covariance matrix to accurately estimate the short-time power spectrum and the steering vectors of the target signal. We employ crystal arrays, certain classes of crystal-shaped array geometries, which make it possible to diagonalize the unknown noise covariance matrix by a constant unitary matrix regardless of its value as long as noise meets an isotropy condition. It is shown through experiments with simulated and real environmental noise that the proposed methods outperform previous methods substantially for real world noise and in the presence of reverberation.

Generalized Correlation Decomposition-Based Blind Channel Estimation in DS-CDMA Systems With Unknown Wide-Sense Stationary Noise

A new blind subspace-based channel estimation technique is proposed for direct-sequence code-division multiple access (DS-CDMA) systems operating in the presence of unknown wide-sense stationary noise. Unlike most of the blind techniques developed for unknown correlated noise environments so far, the proposed algorithm is applicable to an arbitrary symbol constellation and does not require any auxiliary antennas at the receiver or any prior knowledge of the interfering user spreading codes. Our approach exploits the centro-Hermitian property of the unknown noise covariance matrix and makes use of the generalized correlation decomposition (GCD) to obtain an accurate estimate of the noise subspace and, consequently, of the user-of- interest channel vector. We also obtain optimal values of the GCD weighting matrices which maximally preserve the orthogonality of the estimated noise subspace to the actual signal subspace in the high signal-to-noise ratio (SNR) regime. It is shown that such an optimal choice of the weighting matrices transforms GCD to an extended form of the conventional canonical correlation decomposition (CCD). Simulation results further demonstrate that if such an extended CCD-based approach is used to estimate the user-of-interest channel vector, then the estimation performance can be substantially improved as compared to earlier SVD-based blind channel estimation techniques.

Adaptive Detection With Bounded Steering Vectors Mismatch Angle

We address the problem of detecting a signal of interest (SOI), using multiple observations in the primary data, in a background of noise with unknown covariance matrix. We consider a situation where the signal's signature is not known perfectly, but its angle with a nominal and known signature is bounded. Furthermore, we consider a possible scaling inhomogeneity between the primary and the secondary noise covariance matrix. First, assuming that the noise covariance matrix is known, we derive the generalized-likelihood ratio test (GLRT), which involves solving a semidefinite programming problem. Next, we substitute the unknown noise covariance matrix for its estimate obtained from secondary data, to yield the final detector. The latter is compared with a detector that assumes a known signal's signature

On posterior distributions for signals in Gaussian noise with unknown covariance matrix

A Bayesian approach to estimate parameters of signals embedded in complex Gaussian noise with unknown color is presented. The study specifically focuses on a Bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems. By integrating out uncertainties regarding the noise color, an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited. Several noninformative priors for the covariance matrix, such as the reference prior, the Jeffreys prior, and modifications to this, are considered. Some of the priors result in analytical solutions, whereas others demand numerical approximations. In the linear signal model, connections are made between the standard Adaptive Maximum Likelihood (AML) estimate and a Bayesian solution using the Jeffreys prior. With adjustments to the Jeffreys prior, correspondence to the regularized solution is also established. This in turn enables a formal treatment of the regularization parameter. Simulations indicate that significant improvements, compared to the AML estimator, can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior. The simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged.

A modified likelihood function approach to DOA estimation in the presence of unknown spatially correlated Gaussian noise using a uniform linear 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.

Adaptive control scheme using real time tuning of the parameter estimator

The paper presents an adaptive controller for discrete-time systems, with parameters modelled as a Gauss–Markov process with an unknown noise covariance matrix. The cost function adopted in the optimisation of the control performance is the sum of the output variances up to M steps ahead in time. Optimal predictors are used to estimate the future outputs y(k + i), i = 1, 2, ..., M, that are needed in the solution of the optimisation problem that yields the value of the control signal at a given time k. The estimates for the system parameters are obtained using a Kalman filter, together with an algorithm to tune the covariance matrix in real time. The adaptation mechanism reduces the risk of divergence of the Kalman filter, as shown by simulation results that illustrate the actual performance of the new controller under uncertainty in the noise covariance matrix.

Estimation of the number of signals in the presence of unknown correlated sensor noise

A method for estimating the number of transmitted signals in the presence of spatially correlated sensor noise is proposed. The procedure is developed under the assumption that the unknown noise covariance matrix is a band matrix. In practice, it is quite robust with respect to this finite correlation length assumption. In sensor array processing, this amounts to assuming that the noise field is locally correlated spatially. Since spatial stationarity of the noise is not necessary, it also applies, for instance, to white noise with different power along the array. Simulations indicate that the asymptotic analysis holds for quite small sample sizes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Estimation of the directions of arrival of signals in unknown correlated 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. >

A new CFAR detection test for radar

In a well-known paper [2], Reed, Mallett, and Brennan (RMB ) discuss an adaptive procedure for the detection of a signal of known form in the presence of noise (or interference) which is assumed to be Gaussian, but whose covariance matrix is totally unknown. Two sets of input data are distinguished and are called for convenience the primary and the secondary input data. The primary data allow for the possibility of signal presence, while the secondary inputs are assumed to contain only noise, independent of and statistically identical to the noise components of the primary data. In the RMB procedure, the secondary inputs are used to form an estimate of the noise covariance, from which a weight vector for the detection of the known signal is determined. This weight vector is applied then to the primary data in the form of a standard colored-noise matched filter. The implication is that the output of this filter is compared with a threshold in order to achieve signal detection. However, no special rule is given for the determination of the threshold needed to control the probability of a false alarm ( PFA ) . In [l] Kelly used the generalized likelihood ratio to derive a hypothesis test for the above problem. This test exhibits the desirable property that its PFA is independent of the covariance matrix (level and structure) of the actual noise encountered; i.e., it is a CFAR (constant false alarm rate) test. However, the derivation of Kelly’s test statistic is complicated. A new related test is developed in this paper. This test is based on the optimal decision rule for a known signal template and a known noise covariance matrix. Then this classical test variable is normalized to have a unit variance. In the usual case of an unknown noise covariance matrix, the maximum likelihood estimate