Toeplitz Approximation Method Research Articles

An Improved Toeplitz Approximation Method for Coherent DOA Estimation in Impulsive Noise Environments.

Direction of arrival (DOA) estimation is an important research topic in array signal processing and widely applied in practical engineering. However, when signal sources are highly correlated or coherent, conventional subspace-based DOA estimation algorithms will perform poorly due to the rank deficiency in the received data covariance matrix. Moreover, conventional DOA estimation algorithms are usually developed under Gaussian-distributed background noise, which will deteriorate significantly in impulsive noise environments. In this paper, a novel method is presented to estimate the DOA of coherent signals in impulsive noise environments. A novel correntropy-based generalized covariance (CEGC) operator is defined and proof of boundedness is given to ensure the effectiveness of the proposed method in impulsive noise environments. Furthermore, an improved Toeplitz approximation method combined CEGC operator is proposed to estimate the DOA of coherent sources. Compared to other existing algorithms, the proposed method can avoid array aperture loss and perform more effectively, even in cases of intense impulsive noise and low snapshot numbers. Finally, comprehensive Monte-Carlo simulations are performed to verify the superiority of the proposed method under various impulsive noise conditions.

A fourth-order cumulant orthonormal propagator rooting method based on Toeplitz approximation

A novel Toeplitz fourth-order cumulant (FOC) orthonormal propagator rooting method (TFOC ‐ OPRM) of direction-of-arrival (DOA) estimation for uniform linear array (ULA) is proposed in this paper. Specifically, the modified (i.e., reduced-dimension) FOC (MFOC) matrix is achieved at first via removing the redundant information encompassed in the primary FOC matrix; then, the TFOC matrix which possesses Toeplitz structure can be recovered by utilizing the Toeplitz approximation method. To reduce the computational complexity, an effective method based on the polynomial rooting technology is adopted. Finally, the DOAs of incident signals can be estimated by exploiting orthonormal propagator rooting method. The theoretical analysis coupled with simulation results show that the proposed resultant algorithm can reduce the computational complexity significantly, as well as improve the estimation performance in both spatially white noise environment and spatially color noise environment.

TAM and MUSIC Approach for Impact-Source Localization under Deformation Conditions.

As an impact-source-localization technique, Lamb waves are commonly used to detect low-velocity impact in composite structures. However, the performance of Lamb waves is susceptible under deformation conditions. In this paper, a novel approach combined the Toeplitz approximation method (TAM) and multiple-signal classification (MUSIC) (TAM-MUSIC) to improve impact-source-localization (angle and distance in polar coordinates) accuracy under deformation conditions. The method divided a two-dimensional search of direction and distance into two one-dimensional searches. The impact direction was calculated by the TAM, which was introduced into the steering vector of MUSIC to estimate the distance by scanning the whole monitoring area. An epoxy laminate plate experiment showed that the phase and amplitude of uniform linear-array signals had different average plate curvature that led to poor impact-source-localization accuracy using the MUSIC method. TAM provided high-resolution direction-finding capability, suitable for the pretreatment of Lamb waves. Results showed that the present method, with a small amount of computation and low memory requirement, had higher location-estimation accuracy than that of traditional methods under deformation conditions.

Sum and difference coarray based MIMO radar array optimization with its application for DOA estimation

Based on the sum and difference coarrays, multiple-input multiple-output (MIMO) radar with minimum redundancy (MR) concept, referred to as MR MIMO, can considerably increase the spatial degrees of freedom (DOFs). However, traditional MR MIMO needs computational search to determine the position of each element. In this paper, a modified MR monostatic MIMO configuration is proposed, referred to as MMRM MIMO. In the proposed system, the MMRM MIMO radar is consisted of several levels of uniform linear array, which brings the advantage that the position of each element can be determined without computational search. Furthermore, it offers more than $$N^{2}$$ DOFs for an N-elemental array. In order to utilize the extended DOFs of MMRM MIMO radar for direction-of-arrival (DOA) estimation, an average Toeplitz approximation method (TAM) is employed, which achieves robust performance even under low signal-to-noise ratio, few snapshots and array error. Numerous simulation results are provided to demonstrate the effectiveness of the proposed method for DOA estimation.

Model-based processing of a holographic moiré

A state space model for the determination of dual phase distributions in a holographic moiré in the presence of nonsinusoidal waveforms, random noise, and miscalibration of the piezoelectric (PZT) devices is proposed. The extraction of these phase terms requires incorporating two PZTs into the moiré setup. A Toeplitz approximation method (TAM) is applied for phase determination, and modification to the Toeplitz covariance matrix formed from the phase-shifted moiré fringes by application of a denoising step in the state-feedback matrix is proposed. This step ensures that the phase terms can even be estimated at a signal-to-noise ratio much lower than that of the original TAM or by our previously suggested polynomial based method.

Genetic Algorithm Based Mixed Spectrum Estimation for Discrimination of Sinusoids and ARMA Noise

This paper is concerned with a method for estimating a mixed spectrum which is composed of multiple sinusoids and an ARMA noise with continuous spectrum. In order to avoid simultaneous minimization of a prediction error criterion with respect to all unknown model parameters, we give an efficient iterative algorithm for estimating the frequencies of the sinusoids and other parameters separately. By adopting the genetic algorithm to choose initial values of the ARMA parameters in the iterative estimation, we can attain a globally optimal estimates of unknown parameters. For executing the proposed hybrid genetic algorithm, a modification of the Toeplitz approximation method is given to estimate the frequencies in the presence of the MA noise, by using a shifted correlation matrix of the filtered observed data. The effectiveness of the proposed algorithm is validated in numerical simulations using the bench mark test data.

Unitary Hessenberg and state-space model based methods for the harmonic retrieval problem

The connections between two families of methods for the harmonic retrieval problem are explored: the recently derived unitary Hessenberg methods and the state-space model based methods. It is shown that in a noise-free situation, the unitary Hessenberg methods can be understood as one particular case of a very popular high resolution algorithm, the Toeplitz approximation method (TAM) of S.Y. Kung.

Two-dimensional angle-of-arrival estimation for uniform planar arrays with sensor position errors

The one-dimensional bearing estimation problem of linearly periodic arrays with sensor position errors has been tackled by the Toeplitz approximation method (TAM), iterative TAM, modified TAM, and iterative MTAM without calibrating the sensor positions. This paper extends these methods to the two-dimensional situation using a uniform planar array with sensor position errors to estimate the 2-D angle-of-arrivals (AOAs), azimuth and elevation angles of the emitting sources. Based on the block Toeplitz property and eigenstructure of the ideal covariance matrix observed by the unperturbed array, we extend the methods to alleviate the effect caused by the random perturbations in sensor position. The Music algorithm incorporating a 2-D AOA searching is applied for bearing estimation. Further, the 1-D processing approach presented earlier for solving the 2-D AOA estimation problem can also be applied to reduce the computational burden of 2-D AOA searching.

Bearing estimation without calibration for randomly perturbed arrays

The bearing estimation problem of linearly periodic arrays with the presence of sensor position errors is discussed. Conventional approaches for solving this problem generally consist of two steps: calibrating sensor locations and then performing bearing estimation based on the calibrated sensor locations. Approaches to carrying out bearing estimation based on the Toeplitz and eigenstructure reconstruction of the covariance matrix without the need for calibration are proposed. The Toeplitz approximation method (TAM) and a modification of it (MTAM) are used to reconstruct a matrix with Toeplitz structure. To further enhance the capabilities of the TAM and MTAM, an iterative algorithm incorporating with the TAM (ITAM) and the MTAM (IMTAM) is proposed to iteratively reconstruct both the Toeplitz and the desired eigenstructure from the observed covariance matrix. Computer simulations show that the MTAM is more effective than the TAM. Moreover, the iterative methods are superior to the noniterative ones at the price of computations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Performance comparison of subspace rotation and MUSIC methods for direction estimation

The statistical performance of subspace rotation (SR) methods (such as the Toeplitz approximation method and a variant of ESPRIT) for direction estimation using arrays composed of matched sensor doublets is studied. The distributional properties of these methods are established, and a compact explicit formula for the covariance matrix of their estimation error is provided. Next, using this formula and a similar formula for MUSIC covariance matrix, it is shown that the SR methods are statistically less efficient than MUSIC, at least for a sufficiently large number of snapshots. The difference in statistical performance between the commonly used SR method and MUSIC may be substantial if the number of sensors in the array is large. An optimally weighted SR method which may approach the MUSIC level of statistical performance for one direction parameter (specified by the user) is introduced. >

Sensitivity considerations in state-space model-based harmonic retrieval methods

State-space models are considered for estimating the frequencies of multiple sinusoids, and robust coordinate systems for frequency estimation are identified. It is shown that the ideal parameter set from a parameter sensitivity point of view involves estimating a unitary state transition matrix and then computing its eigenvalues to obtain an estimate of the frequencies. Procedures to estimate such matrices from covariance and time-series data are examined. It is shown that two state-space methods, the Toeplitz approximation method and direct data approximation, estimate robust state transition matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Block Toeplitz Approximation

In this paper the two-dimensional harmonic retrieval problem is addressed by extending a high resolution one-dimensional technique, the Iterated Toeplitz Approximation Method (ITAM), to the two-dimensional problem. The two-dimensional version of the algorithm (ITAM2-D) attempts to improve the frequency estimates by modeling the covariance matrix as a low rank Toeplitz-block Toeplitz matrix and is derived from the one-dimensional algorithm in a straightforward manner. It is shown that ITAM2-D may be used effectively as a preprocessor for orther algorithms such as the all-pole modeling technique of Jackson and Chien. Computer simulations are presented to examine the performance of the ITAM2-D algorithm.

State-space and singular-value decomposition-based approximation methods for the harmonic retrieval problem

We present new high-resolution methods for the problem of retrieving sinusoidal processes from noisy measurements. The approach taken is by use of the so-called principal-components method, which is a singular-value-decomposition-based approximate modeling method. The low-rank property and the algebraic structure of both the data matrix and the covariance matrix (under noise-free conditions) form the basis of exact modeling methods. In a noisy environment, however, the rank property is often perturbed, and singular-value decomposition is used to obtain a low-rank approximant in factored form. The underlying algebraic structure of these factors leads naturally to least-squares estimates of the state-space parameters of the sinusoidal process. This forms the basis of the Toeplitz approximation method, which offers a robust Pisarenko-like spectral estimate from the covariance sequence. Furthermore, the principle of Pisarenko’s method is extended to harmonic retrieval directly from time-series data, which leads to a direct-data approximation method. Our simulation results indicate that favorable resolution capability (compared with existing methods) can be achieved by the above methods. The application of these principles to two-dimensional signals is also discussed.

New Unbiased Methods for Narrowband Spectral Estimation

The problem of unbiased spectral estimation of narrowband signals in additive noise, colored or white, is addressed in this paper. Both recursive and non-recursive methods are presented. First, a recursive method, the adaptive notch filter working directly on the data is developed. Certain features of its realization i.e. cascade, modified gradient etc. are highlighted. Next, a non-recursive method, the Toeplitz approximation method working on second order statistics i.e. autocovariance is proposed. A approximation procedure based on singular value decomposition is discussed.