Noise Subspace Research Articles

Robust adaptive beamforming for cylindrical uniform conformal arrays based on low-rank covariance matrix reconstruction

Recently, conformal arrays have attracted considerable interest because such arrays can provide reduced radar cross-section and increased angle coverage. In this article, we devise a robust adaptive beamforming (RAB) approach using cylindrical uniform conformal array (CUCA). Firstly, we derive the minimum variance distortionless response (MVDR) beamformer for the CUCA by utilizing the noise subspace of interference covariance matrix (ICM) and steering vector (SV) of the signal-of-interest (SOI). Subsequently, the ICM is reconstructed by estimating the noise-free covariance matrix of the CUCA outputs and the interference projection matrix. Specifically, the noise-free covariance matrix can be regarded as multiple low-rank covariance matrices, and each low-rank matrix is reconstructed by formulating a nuclear norm minimization (NNM) problem. With the reconstructed covariance matrix, the 2-D DOAs of sources are determined by employing 2-D MUSIC spectrum to form the interference projection matrix. In addition, the SOI SV is estimated by solving a quadratically constrained quadratic programming (QCQP) problem. Numerical results demonstrate that the proposed approach is obviously superior to the existing RAB techniques.

High-resolution ultrasound imaging based on spatio-temporal MUSIC

Abstract Time-reversal MUltiple SIgnal Classification (TR-MUSIC) is known for its super-resolution capability in locating point scatterers. However, a prerequisite for the application of the TR-MUSIC method is a thorough understanding of the properties of the medium and its background Green’s function. In general, it is difficult to obtain an accurate model of the Green’s function in advance for non-uniform or complex media, necessitating the use of approximate Green’s functions. When TR-MUSIC relies on these approximations, the impact on lateral resolution may be minimal, but the axial resolution is significantly reduced. In addition, we have proposed the Range-MUSIC (R-MUSIC) method, which exploits the linear relationship between the rotation of the echo phases and the positions of the scatterers. This approach uses subbands of different frequencies, differentiating multiple point scatterers by the speed of phase rotation associated with each frequency. By analyzing the profile in the time domain of echoes within the noise subspace, R-MUSIC achieves super-resolution in the axial direction without improving lateral resolution. Therefore, our study combines R-MUSIC with TR-MUSIC to improve axial resolution while maintaining lateral resolution. Through simulation and experimental evidence, we have shown that this combined approach yields better imaging performance than either method used independently.

Source number estimation based on sequential ratio for the invariant subspace of matrices

In this paper, we propose a method to estimate the number of sources based on the sequential ratio for the invariant subspace of matrices that can provide accurate results in scenarios involving colored noise and a small number of snapshots. We first form a pair of projection matrices on each eigenvector of the sample covariance matrices by using the invariant subspace of matrices. We then use the generalized cosine-squared method to evaluate the differences of M pairs of projection matrices, according to the gap between the projection matrices corresponding to the eigenvectors of the signals and noise. This yields a distinct transition from the signal subspace to the noise subspace. Based on this, we develop a sequential ratio-based method to precisely determine the number of sources by identifying the position at which the ratio of differences significantly decreases. The proposed method can identify the index value at which a significant drop occurs in the number of sources according to the transition from the signal subspace to the noise subspace. The simulation results showed that the proposed method is superior to the current methods, and excels particularly in conditions involving colored noise.

Robust adaptive beamforming for cylindrical uniform conformal array

As we all know, conformal arrays can provide smaller radar cross-section (RCS) and greater angle coverage than conventional linear arrays and planar arrays. In this paper, we develop a robust adaptive beamforming (RAB) approach using the cylindrical uniform conformal array. Firstly, we reconstruct the interference-plus-noise covariance matrix (INCM) by estimating the steering vectors (SVs) and powers of all interferences. Specifically, we estimate the interference SVs by exploiting the MUSIC spectrum and corrected by solving a minimization problem based on the orthogonality between the interferences SVs and noise subspace, and the interferences powers are further estimated by utilizing the non-correlation between different signal SVs. With the reconstructed INCM, the SV of the signal-of-interest is estimated by solving a quadratically constrained quadratic programming problem. Numerical results demonstrate that the proposed approach is obviously superior to the existing RAB techniques.

Deblending and interpolation of subsampled blended seismic data based on damped randomized singular spectrum analysis

AbstractWhen compared to traditional seismic data acquisition, irregular blended acquisition significantly promotes the acquisition efficiency. Yet, the blending noise of subsampled blended data introduces new obstacles for the subsequent processing of seismic data. Due to the predictability of linear events in the frequency–space domain, the constructed Hankel matrices exhibit low‐rank characteristics. However, the blending noise of subsampled blended data increases the rank, so deblending and interpolation can be implemented via rank‐reduction algorithms such as the singular spectrum analysis. The significant computing cost of the singular value decomposition, however, makes the traditional singular spectrum analysis inefficient. An alternative algorithm, known as the randomized singular spectrum analysis, employs the randomized singular value decomposition instead of the traditional singular value decomposition for rank‐reduction. The randomized singular spectrum analysis significantly enhances the efficiency of the decomposition process, particularly when dealing with large Hankel matrices. There still remains some random noise when using the singular spectrum analysis or randomized singular spectrum analysis for subsampled blended data, because the noise subspace and signal subspace are coupled together. Thus, we incorporate a damping operator into the randomized singular value decomposition and propose a novel damped randomized singular spectrum analysis method. The damped randomized singular spectrum analysis combines the advantages of the randomized singular value decomposition and the damping operator to enhance the computational efficiency and suppress the remaining noise. Moreover, an iterative projected gradient descent strategy is introduced to achieve deblended and interpolated seismic data for subsequent processing. Examples from synthetic data and field data are used to demonstrate the effectiveness and superiority of the proposed damped randomized singular spectrum analysis method, which enhances the accuracy and efficiency during simultaneous deblending and interpolation.

DIRECTION OF ARRIVAL USING PCA NEURALNETWORKS

This paper adapted the neural network for the estimating of the direction of arrival (DOA). It usesan unsupervised adaptive neural network with APEX algorithm to extract the principal componentsthat in turn, are used by Capon method to estimate the DOA, where by the PCA neural network wetake signal subspace only and use it in Capon (i.e. we will ignore the noise subspace, and take thesignal subspace only)

Direction of Arrival Estimation Method Based on Eigenvalues and Eigenvectors for Coherent Signals in Impulsive Noise

In this paper, a Toeplitz construction method based on eigenvalues and eigenvectors is proposed to combine with traditional denoising algorithms, including fractional low-order moment (FLOM), phased fractional low-order moment (PFLOM), and correntropy-based correlation (CRCO) methods. It can improve the direction of arrival (DOA) estimation of signals in impulsive noise. Firstly, the algorithm performs eigenvalue decomposition on the received covariance matrix to obtain eigenvectors and eigenvalues, and then the Toeplitz matrix is created according to the eigenvectors corresponding to its eigenvalues. Secondly, the spatial averaging method is used to obtain an unbiased estimate of the Toeplitz matrix, which is then weighted and added based on the corresponding eigenvalues. Next, the noise subspace of the Toeplitz matrix is reconstructed to obtain the one that has less angle information. Finally, the DOA of the coherent signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. The improved method based on the Toeplitz matrix can not only suppress the effect of impulsive noise but can also solve the problem of aperture loss due to its decoherence. A series of simulations have shown that they have better performances than other algorithms.

An asymptotically exact estimate of the median noise eigenvalue of sample covariance matrices

The Dominant Mode Rejection beamformer [Abraham & Owsley (1990)] averages the noise subspace eigenvalues of the sample covariance matrix (SCM) to estimate the background noise power. This noise power estimate from snapshot deficient SCMs can be unreliable for large arrays in time-varying environments with limited snapshots. Median filtering offers robustness to outliers and subspace mismatch for these challenging problems. Recent numerical experiments [Campos Anchieta & Buck (2022)] identified a simple regression relating the median sample eigenvalue to the true background power for snapshot deficient SCMs. However, the complicated expression of the Marchenko-Pastur (MP) distribution for the noise eigenvalues of the SCM thwarted prior attempts to find a closed form estimator for the MP distribution median. This talk exploits a coordinate transform to obtain a closed-form median of the MP distribution that is exact for the leading term in the power series expansion of the distribution. The new median estimator is more accurate than the previous numerical regression when the ratio of sensors to snapshots is 2 or more. Given the central role of SCM eigenvalues in principal component analysis and constant false alarm rate detectors, the new median expression should find application in other data science algorithms for underwater acoustics. [Work supported by ONR Code 321US.]

Enhancing ship noise reduction: a deep-learning based noise extraction model

In active sonar detection systems, the self-radiated noise from the platform itself is a primary factor that interferes with system performance. The platform self-noise exhibits non-Gaussian characteristics, comprising line spectrum and continuous spectrum, where line spectrum exhibits higher intensity and greater frequency stability. To reduce the ship noise, this study introduces a deep-learning based time-domain model with an encoder-separator-decoder architecture. In detail, the encoder and decoder modules utilize learnable convolutional layers to generate distinguishable features with a symmetrical structure. For separator module, we observe that noise components prominently predominate within the noisy signal in low signal-to-noise ratio (SNR) scenarios. The extraction of noise components is typically more tractable than of signal components. Therefore, in contrast to conventional frameworks that focus on extracting the signal subspace, this module is dedicated to extracting the noise subspace. Besides, the modified scale invariant SNR (SI-SNR) is proposed as the loss function for model optimization, which yields better performance than non-noise extraction structure. The robustness of the model is validated on the DeepShip dataset with four ship categories, demonstrating that the model consistently outperforms baselines, including matched filter, in all ship categories.

Improving the Direction of Arrival Estimation Using the Parasitic Subspace Generated by Active-Parasitic Antenna (APA) Arrays

The improvement in Direction of Arrival (DOA) estimation when the received signals impinge on Active-Parasitic Antenna (APA) arrays will be studied in this work. An APA array consists of several active antennas; others are parasitic antennas. The responses to the received signals are measured at the loaded terminals of the active element. The terminals of the parasitic element are shorted. The effect of the received signals on the parasites, i.e., the induced short-circuit current, is mutually coupled to the active elements. Eigen decomposition of the covariance matrix of the measurements of the APA array generates a third subspace in addition to the traditional signal and noise subspaces generated by the all-active antenna receiving array. This additional subspace, the parasitic subspace, is accompanied by very small eigenvalues (approaching zero). Hence, a complete orthogonality between this subspace and the column space of the steering matrix of the array can be obtained. As a result, better resolution in estimating the DOA can be achieved. Several simulations in conjunction with the MUSIC algorithm, which have been conducted in this work, depict that the APA array outperforms the all-active antenna array as a direction finder, regardless of the size of the array, the number of active elements, or the number of measurement snapshots. Furthermore, super-resolution DOA estimation can be achieved when a subset of the parasitic subspace is used as if the measurement were noiseless. Also, the APA array contributes to very small RMSE values over a wide range of S/N of the received signals.

Direct position tracking method for non‐circular signals with distributed passive arrays via first‐order approximation

AbstractIn this study, a direct position tracking method for non‐circular (NC) signals using distributed passive arrays is proposed. First, we calculate the initial positions of sources using a direct position determination (DPD) approach; next, we transform the tracking into a compensation problem. The offsets of the adjacent time positions are calculated using a first‐order Taylor expansion. The fusion calculation of the noise subspace is performed according to the NC characteristics. Because the proposed method uses the signal information from the previous iteration, it can realize automatic data associations. Compared with traditional DPD and two‐step localization methods, our novel process has lower computational complexity and provides higher accuracy. Moreover, its performance is better than that of the traditional tracking methods. Numerous simulation results support the superiority of our proposed method.

Model order estimation based on the correntropy of observation eigenvalues

This paper presents a novel approach for estimating the model order in the presence of observation errors. The proposed method is based on the correntropy estimation of eigenvalues in the observation space. By utilizing the bootstrap method to resample the observations, more precise estimations can be obtained. The next step in the algorithm involves partitioning the observation space, generated by the covariance matrix of mixtures, into two subspaces: the signal subspace and the noise subspace. The order of the model, corresponding to the dimension of the signal subspace, is determined using a correntropy estimator based on kernel functions. Theoretical analysis demonstrates the consistency of the proposed algorithm. Comparative evaluations against existing methods in the literature highlight the superior performance of this information-theoretic approach.

Robustness Meets Low-Rankness: Unified Entropy and Tensor Learning for Multi-View Subspace Clustering

In this paper, we develop the weighted error entropy-regularized tensor learning method for multi-view subspace clustering (WETMSC), which integrates the noise disturbance removal and subspace structure discovery into one unified framework. Unlike most existing methods which focus only on the affinity matrix learning for the subspace discovery by different optimization models and simply assume that the noise is independent and identically distributed (i.i.d.), our WETMSC method adopts the weighted error entropy to characterize the underlying noise by assuming that noise is independent and piecewise identically distributed (i.p.i.d.). Meanwhile, WETMSC constructs the self-representation tensor by storing all self-representation matrices from the view dimension, preserving high-order correlation of views based on the tensor nuclear norm. To solve the proposed nonconvex optimization method, we design a half-quadratic (HQ) additive optimization technology and iteratively solve all subproblems under the alternating direction method of multipliers framework. Extensive comparison studies with state-of-the-art clustering methods on real-world datasets and synthetic noisy datasets demonstrate the ascendancy of the proposed WETMSC method.

Super-resolution DOA estimation methods based on element multiplication and summation

In this article, we propose two direction-of-arrival (DOA) estimation methods to further improve the resolution and performance of subspace-based algorithms. While most subspace-based algorithms employ either noise or signal subspaces, we utilize the signal and noise subspaces jointly. First, we project the steering vectors to the estimated signal and noise subspaces respectively to obtain the projection vectors, then, by investigating the numerical characteristics behind the inner products between the steering vectors and the estimated eigenvectors of the sample covariance matrix, an initial estimator is proposed based on the multiplication of elements of the projection vectors instead of the summation. Though the initial estimator achieves higher resolution than most existing subspace-based methods, the appearance of pseudo peaks, especially in high SNR, may affect the estimation accuracy. By analyzing the main elements in the initial objective function that cause pseudo peaks, two improved algorithms incorporating element summation are proposed to balance the resolution capacity and estimation accuracy, which are especially suitable for small sample size, low SNR and closely spaced sources. The numerical simulations verify the effectiveness and superiority of the proposed algorithms.

A Sparse Bayesian Learning Method for Direction of Arrival Estimation in Underwater Maneuvering Platform Noise

The underwater maneuvering platform generates self-noise when sailing, which shows spatial directionality to the arrays fixed on the platform. In this paper, it is called spatially colored noise (SCN). The direction of arrival (DOA) estimation results are often influenced by this self-noise, leading to a decrease in estimation accuracy and to the appearance of spurious peaks. To resolve this problem, a sparse Bayesian learning (SBL) method adapted to underwater maneuvering platform noise is proposed in this paper. The SBL framework with unknown SCN is established first. Then, the SCN covariance matrix is estimated by projecting the received data covariance matrix into the noise subspace, and the DOA estimation results are finally obtained through multiple iterations. The simulation results show that the proposed method avoids spurious peaks, and compared to the existing methods, the proposed method achieves a higher accuracy in the case of low SNRs and small snapshot numbers. The sea trial data processing results show that the proposed method provides lower and flatter noise spectrum levels without spurious peaks.

Smoothed Sparse Blind DOA Estimation for MC-CDMA Systems Based on Tensor Decomposition

We propose a blind direction of arrival (DOA) estimation method for multicarrier code-division multiple access (MC-CDMA) systems, which does not require knowledge of the spreading codes of transmitters. Our method involves reconstructing a fourth-order tensor with observations from the antenna array at the receiver. We then apply higher-order singular value decomposition (HOSVD) to separate the signal subspace from the noise subspace. To enhance the sparsity of the solution, we develop a smoothed ℓ0-norm sparse reconstruction method by designing continuous functions with reweighted vectors. Finally, DOA estimates are obtained by searching the spectrum of the sparse solution using an iterative method that ensures fast convergence. Numerical simulations show its superiority over previous work.

DOA estimation based on smoothed sparse reconstruction with time-modulated linear arrays

A novel direction of arrival (DOA) estimation method based on smoothed sparse reconstruction with time-modulated linear arrays (TMLA) is proposed in this paper. With the analysis of the noise variation after time modulation, a closed-form expression for the noise covariance under the optimized unidirectional phase center motion time sequence scheme is derived. Then a reweighted continuous function is designed for the smooth approximation to ℓ0-norm based on the orthogonality between the signal subspace and the noise subspace. Simulation results demonstrate that, compared with conventional DOA estimation method with time-modulated arrays, the proposed method possesses higher estimation accuracy and lower computational complexity, especially for the scenario with small snapshot and low signal-to-noise ratio.

An Improved Two-Stage Spherical Harmonic ESPRIT-Type Algorithm

Sensor arrays are gradually becoming a current research hotspot due to their flexible beam control, high signal gain, robustness against extreme interference, and high spatial resolution. Among them, spherical microphone arrays with complex rotational symmetry can capture more sound field information than planar arrays and can convert the collected multiple speech signals into the spherical harmonic domain for processing through spherical modal decomposition. The subspace class direction of arrival (DOA) estimation algorithm is sensitive to noise and reverberation, and its performance can be improved by introducing relative sound pressure and frequency-smoothing techniques. The introduction of the relative sound pressure can increase the difference between the eigenvalues corresponding to the signal subspace and the noise subspace, which is helpful to estimate the number of active sound sources. The eigenbeam estimation of signal parameters via the rotational invariance technique (EB-ESPRIT) is a well-known subspace-based algorithm for a spherical microphone array. The EB-ESPRIT cannot estimate the DOA when the elevation angle approaches 90°. Huang et al. proposed a two-step ESPRIT (TS-ESPRIT) algorithm to solve this problem. The TS-ESPRIT algorithm estimates the elevation and azimuth angles of the signal independently, so there is a problem with DOA parameter pairing. In this paper, the DOA parameter pairing problem of the TS-ESPRIT algorithm is solved by introducing generalized eigenvalue decomposition without increasing the computation of the algorithm. At the same time, the estimation of the elevation angle is given by the arctan function, which increases the estimation accuracy of the elevation angle of the algorithm. The robustness of the algorithm in a noisy environment is also enhanced by introducing the relative sound pressure into the algorithm. Finally, the simulation and field-testing results show that the proposed method not only solves the problem of DOA parameter pairing, but also outperforms the traditional methods in DOA estimation accuracy.

Monopulse Parameter Estimation for FDA-MIMO Radar under Mainlobe Deception Jamming

Multiple input multiple output with frequency diversity array (FDA-MIMO) radar has unique advantages in mainlobe deception jamming suppression and target location. However, if the training sample contains the target signal, it will lead to poor jamming suppression performance and large target measurement error. To deal with the problem, a method of coarse target location in the time domain is proposed based on the cumulative sampling analysis. Taking full advantages of the strongest correlation characteristic between the expected steering vector and the true target, the feature vector and feature value corresponding to the true target are found after feature decomposition. The time domain location of the target is roughly estimated during the cumulative sampling analysis from near to far. Then, a pure jamming training sample can be obtained by avoiding the location. Noise subspace projection algorithm is used to measure the angle and range of the target while suppressing mainlobe jamming. The simulation results show that the proposed method can roughly estimate the target location in the time domain when the mainlobe deception jamming completely covers the target. Compared with conventional methods, the performance of jamming suppression and target localization error are closer to the performance of ideal sampling.

Joint Direction of Arrival-Polarization Parameter Tracking Algorithm Based on Multi-Target Multi-Bernoulli Filter

This paper presents a tracking algorithm for joint estimation of direction of arrival (DOA) and polarization parameters, which exhibit dynamic behavior due to the movement of signal source carriers. The proposed algorithm addresses the challenge of real-time estimation in multi-target scenarios with an unknown number. This algorithm is built upon the Multi-target Multi-Bernoulli (MeMBer) filter algorithm, which makes use of a sensor array called Circular Orthogonal Double-Dipole (CODD). The algorithm begins by constructing a Minimum Description Length (MDL) principle, taking advantage of the characteristics of the polarization-sensitive array. This allows for adaptive estimation of the number of signal sources and facilitates the separation of the noise subspace. Subsequently, the joint parameter Multiple Signal Classification (MUSIC) spatial spectrum function is employed as the pseudo-likelihood function, overcoming the limitations imposed by unknown prior information constraints. To approximate the posterior distribution of MeMBer filters, Sequential Monte Carlo (SMC) method is utilized. The simulation results demonstrate that the proposed algorithm achieves excellent tracking accuracy in joint DOA-polarization parameter estimation, whether in scenarios with known or unknown numbers of signal sources. Moreover, the algorithm demonstrates robust tracking convergence even under low Signal-to-Noise Ratio (SNR) conditions.