Non-circular Research Articles

Noncircular coherent signal direction of arrival estimation for coprime array: A subspace‐based interpolation approach

AbstractTo address the array aperture loss caused by mainstream direction of arrival (DOA) estimation algorithms for coherent signals using matrix interpolation techniques, a non‐circular (NC) coherent signal direction‐finding method that fully utilises covariance matrix information is proposed. The received data is firstly extended by leveraging its NC properties. Then, eigenvalue decomposition is performed on the covariance matrix to extract the signal subspace, which is mapped to a uniform linear array signal subspace via a mapping matrix. The first row of the covariance matrix corresponding to the signal subspace is employed to construct a Toeplitz matrix, and nuclear norm minimisation method is applied to recover the missing information. Finally, to avoid extra NC phase searching, the reduced‐dimension method is applied to obtain the estimation results. Performance analysis and simulation results show that the proposed algorithm achieves improvements in computational complexity, source estimation capability, and estimation accuracy.

Direction of arrival tracking of non-circular signals: probabilistic hypothesis density filter with an improved likelihood function

Abstract Conventional direction of arrival (DOA) tracking methods typically track circular signals with a fixed number of sources without considering time-varying numbers of non-circular (NC) signals DOA tracking. To solve this problem, we propose a DOA tracking method for NC signals with a probabilistic hypothesis density (PDH) filter based on an improved likelihood function. We improve the likelihood function of PHD by taking advantage of the NC signal characteristics, thereby separating the mixed measures and constructing the likelihood function corresponding to each source. In addition, we exponentially weight the likelihood function to improve the accuracy further. Finally, we achieved DOA tracking of the time-varying NC signals using the PHD filtering implemented by sequential Monte Carlo (SMC). Based on simulation findings, the suggested approach outperforms the conventional method in terms of tracking accuracy and can address the issue of changing NC signal numbers throughout the tracking process.

Load transfer model for vertically loaded non-circular piles

The load-transfer (LT) method, involving a series of ‘t-z’ springs to represent pile shaft-soil interaction and a ‘q-z’ spring to represent pile tip-soil interaction, is widely used to model the load–displacement behaviour of vertically loaded piles. While a plethora of LT models have been proposed for piles with circular cross sections, they are not directly applicable to non-circular (NC) cross sections. Therefore, there is a lack of tailored solutions for rectangular, H-shaped and X-shaped piles even though they are common in geotechnical practice. The aim of this paper is to develop a general theoretical framework for the construction of LT curves for NC piles. This framework allows NC piles with any cross section to be considered. The LT curves for NC piles are formulated using a two-parameter hyperbolic model which requires input of (a) the initial elastic LT stiffness and (b) the unit ultimate shaft friction. The elastic stiffness for vertically loaded NC piles with arbitrary cross sections is described using a rigorous semi-analytical (SA) solution involving variational principles. A nonlinear numerical solution for the pile governing equation with the incorporated t-z and q-z curves is achieved using the Runge-Kutta (RK) method. The proposed LT curves are used to construct the pile governing equation and a numerical solution using the Runge-Kutta (RK) method is proposed to predict the responses of NC piles in drained sand. The proposed theoretical LT predictions of pile response are validated through comparisons to elastic–plastic finite element calculations.

An enhanced direct position determination of non-circular sources via sparse Bayesian inference and grid refinement strategy

In this study, an off-grid sparse Bayesian inference (OGSBI) based direct position determination (DPD) algorithm is investigated. Existing SBI-based DPD algorithms are confronted with the challenge of excessive computational loads and lack of consideration for non-circular (NC) signals. To address these limitations, we present an enhanced OGSBI-based DPD algorithm for multiple non-circular sources. By utilizing the conjugate information of the NC signals, we expand the dimensionality of the data matrix to achieve a significant improvement in the localization performance. Additionally, a grid refinement strategy is developed to alleviate the computational loads, which involves an initial search to determine the approximate source locations, followed by fine localization using a denser grid. Moreover, the computational complexity and Cramér-Rao lower bound are derived to provide a comprehensive analysis of the proposed algorithm. Numerical simulations demonstrate the superiority of the proposed algorithm in terms of both localization accuracy and computational efficiency.

Geometric Algebra based 2D-DOA Estimation for Non-circular Signals with an Electromagnetic Vector Array

This paper presents two novel methods based on geometric algebra (GA) to estimate two-dimensional (2D) direction-of-arrival (DOA) of non-circular (NC) signals for uniform rectangular array (URA). Traditional methods treat the received NC signals as a long vector which will inevitably lose orthogonality inside each electromagnetic vector sensor (EMVS) and thus miss some information of second-order statistical properties. Furthermore, the computational complexity will also increase. By contrast, the GA-based estimating signal parameter via rotational invariance techniques (ESPRIT) and propagation method (PM) algorithms are proposed to estimation DOA of NC signals. Taking advantage of GA, the relationship among multidimensional signals can be maintained. First, the six components of the EMVS are represented as a multivector in GA space. Then, we construct the GA-based extended covariance matrix to utilize the signal information more completely. The DOA parameters can be estimated through the ESPRIT and PM principle. The proposed GA-based estimation of signal parameter via rotational invariance techniques for NC signals processing (GANC-ESPRIT) can estimate DOA with high estimation accuracy. The proposed GA-based propagation method for NC signals estimation (GANC-PM) uses linear transformation to calculate angle parameters. Our model has much lower memory requirements and less computational burden compared with long vector models. Simulation results demonstrate the robustness and superiority of the proposed GANC-ESPRIT algorithm in terms of angular resolution. Complexity analysis shows that the proposed GANC-PM algorithm performances better with less computations.

Motor Transfer and Proactive Interference in Cycling With a Noncircular Chainring.

Athletes must transfer their performance when changing equipment due to innovative developments in sports technology. This kind of transfer has received only moderate attention. The aim of this study was to examine whether a mechanical change in sports equipment disturbs an athlete's performance and affects biomechanical and neurophysiological parameters. Therefore, an experiment was conducted in which 36 participants in three groups pedaled at 70 rounds per minute on a cycling ergometer with a circular and a noncircular (NC) chainring. The dependent variables were the total variability of the cadence, torque effectiveness, and muscle cocontraction (electromyographic cocontraction) of four antagonistic acting muscle pairs. Data were recorded during an acquisition phase, a transfer phase, and a retention phase. The results revealed that practice on a circular chainring induces a positive transfer on the NC chainring for total variability without a proactive interference effect. Torque effectiveness did not change within or between groups during the acquisition, transfer, and retention phases. Torque effectiveness and electromyographic cocontraction were not affected when the chainrings were altered from Day 1 to Day 2. During the retention phase, electromyographic cocontraction was higher when using the NC chainring, but the difference was small in absolute terms. The results regarding transfer and proactive interference seem to be strongly dependent on the movement task and the change in sports equipment. Transfer from the circular to NC chainring indicates refined neuromuscular control and improved movement coordination.

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.

Design of relocating sparse nested arrays for DOA estimation of non-circular signals

Recently, the ability of sparse arrays to achieve higher degrees of freedom (DOFs) and to be less sensitive to the mutual coupling (MC) effect has aroused increasing attention. Aiming at this, we investigate the sparse array design problem for direction of arrival (DOA) estimation of non-circular (NC) signals. Then, we propose two novel relocating sparse nested array (RSNA) structures, named RSNA-I and RSNA-II, respectively, and derive simple explicit expressions for their array locations and DOFs. Specifically, the RSNA-I is composed of two subarrays plus a separate sensor. By utilizing the separate sensor to enable the virtual sum and difference co-array (SDCA) with a longer consecutive segment and to alleviate the MC effect. In addition, to further make the array structure more robust to the MC effect, RSNA-II relocates the dense subarray of RSNA-I. Theoretical analysis and numerical simulation results demonstrate that the proposed RSNA-I and RSNA-II achieve a better balance between DOFs and the MC effect, which in turn effectively solves DOA estimation of NC signals and yields better DOA estimation performance.

A Low-Complexity Direction-of-Arrival Estimation Algorithm for Noncircular Signals via Subspace Rotation Technique

In this paper, we investigate the problem of the heavy computational burden of the direction-of-arrival (DoA) estimation for the noncircular (NC) signals. A novel low-complexity direction-of-arrival estimation algorithm for NC signals via subspace rotation technique (SRT) is proposed. The proposed algorithm divides the noise subspace matrix along its row direction into two submatrices, and the SRT is performed to get a new reduced-dimension noise subspace. Then, utilizing the separation of variables and the orthogonality between the reduced-dimension noise subspace and the space spanned by the columns of the extended manifold matrix, a new one-dimensional spectral search function is derived to estimate DoAs. As the size of the block matrices of the noise subspace matrix has a great impact on the computational complexity of the spectral search, the optimal number of rows of the block matrices is determined. The proposed algorithm not only avoids the two-dimensional spectral search but also efficiently removes the redundancy computations in the one-dimensional spectral search. Theoretical analysis and simulation results show that the proposed algorithm can significantly improve the computational efficiency on the premise of ensuring the accuracy of DoA estimation for the NC signals, especially in scenarios where large numbers of sensors are applied.

Joint DOA, range and polarization estimation based on geometric algebra for near-field non-circular source with a symmetric MIMO array

Multiple-input multiple-output (MIMO) radar is a new type of radar with excellent performance in target detection and parameter estimations. The MIMO radar equipped with electromagnetic vector sensors (EMVSs) can record multi-dimensional information of the electromagnetic (EM) signal. Traditional methods based on long vector model cannot preserve orthogonality between the outputs of each component of the EMVS and have high computational complexity. In this paper, a novel geometric algebra (GA)-based localization method is proposed for direction of arrival (DOA), range, and polarization estimation of near-field (NF) non-circular (NC) sources for a symmetric MIMO array. First, the proposed method converts the output of an EMVS into a multi-vector through a GA matrix transformation. Then, the DOA and range parameters can be estimated by searching the one-dimensional (1-D) spectrum based on the GA-based NC rank reduction (GANC-RARE) algorithm. Finally, the polarization can be estimated by employing estimated DOAs and ranges based on the modified GA-based NC multiple signal classification (MGANC-MUSIC) algorithm. The computational complexity analysis demonstrates that our solution improves memory spaces of the covariance matrix and reduces computation efforts of the eigenvalue decomposition (EVD). The simulation results show the superiority and reliability of our proposed method in model error and coherent noise environment.

Improved symmetric flipped nested array for mixed near‐field and far‐field non‐circular sources localization

AbstractAt present, there are few sparse arrays used in the mixed near‐field (NF) and far‐field (FF) localization based on non‐circular (NC) signals. Inspired by the symmetric flipped nested array (SFNA) used in the existing mixed NF and FF NC source, in order to further improve the parameter estimation accuracy of the mixed NF and FF NC signal, an improved symmetric flipped nested array (ISFNA) for mixed NF and FF NC sources localization was developed. First, the uniform subarrays in the SFNA are rearranged, elements are extracted from the uniform subarrays and rearranged into ISFNA. ISFNA is more sparse, the array aperture is larger, and the array degree of freedom (DOF) is higher; second, the formula of the maximum consecutive lags of ISFNA is given; third, a special fourth‐order cumulant is used to eliminate the range parameter and then use a one‐dimensional (1‐D) spectral peak search to obtain all Directions of Arrival (DOAs). By defining the range search, the range can be obtained by bringing in estimated DOAs. Finally, the superiority of the proposed array is proved by simulation.

Off-Grid DOA Estimation for Noncircular Signals via Block Sparse Representation Using Extended Transformed Nested Array

An off-grid direction-of-arrival (DOA) estimation method based on block sparse representation is proposed to localize the strictly noncircular (NC) sources utilizing an extended transformed nested array (ETNA). This novel off-grid DOA estimation algorithm effectively promotes spatial distribution information mining. Furthermore, it is conducive to providing stable signal recovery, which refines the DOA estimation precision with interpolation over a coarse grid. We then combine the above algorithm with the designed ETNA to improve the detection performance. The ETNA is an optimal displacement on the existing TNA, which enlarges the degree of freedom (DOF) and lengthens the maximum contiguous segment from the derived virtual array. Simulation results demonstrate its superiority in estimation performance and DOF.

Direct Position Determination of Noncircular Sources with Multiple Nested Arrays via Weighted Subspace Data Fusion

Direct position determination (DPD) of noncircular (NC) sources for multiple nested arrays (NA) is researched in this study. For noncircular sources, the dimension reduction method is used to decrease the computing complexity and remove the noncircular phase. Furthermore, nested array and noncircular sources extend spatial degree of freedom. Due to inferior stability and noise susceptibility of original algorithm, we propose SNR weighted subspace data fusion (W-SDF) algorithm. Each observation station places a nested array, spatial smoothing technology, and sum and difference co-array are used to deal with the nested array. Simulation results show that under nested array and noncircular sources, the proposed W-SDF algorithm has decreased the complexity of the algorithm and improved the location accuracy, degree of freedom, and resolution.

Direct Position Determination of Non-Circular Sources for Multiple Arrays via Weighted Euler ESPRIT Data Fusion Method

In recent years, direct position determination (DPD) with multiple arrays for non-circular (NC) signals is a hot topic to research. Conventional DPD techniques with spectral peak search methods have high computational complexity and are sensitive to the locations of the observation stations. Besides, there will be loss when the signal propagates in the air, which leads to different received signal-to-noise ratios (SNRs) for each observation station. To attack the problems mentioned above, this paper derives direct position determination of non-circular sources for multiple arrays via weighted Euler estimating signal parameters viarotational invariance techniques (ESPRIT) data fusion (NC-Euler-WESPRIT) method. Firstly, elliptic covariance information of NC signals and Euler transformation are used to extend the received signal. Secondly, ESPRIT is applied to avoid the high-dimensional spectral function search problem of each observation station. Then, we combine the information of all observation stations to construct a spectral function without complex multiplication to reduce the computational complexity. Finally, the data of each observation station is weighted to compensate for the projection error. The consequence of simulation indicates that the proposed NC-Euler-WESPRIT algorithm not only improves the estimation performance, but also greatly reduces the computational complexity compared with subspace data fusion (SDF) technology and NC-ESPRIT algorithm.

Robust DOA estimation of complex correlated signals in non-Gaussian CES distributed models

The sample covariance matrix (SCM) is commonly used in direction-of-arrival (DOA) estimation methods when the noise or observations are circular complex Gaussian (C CG) distributed. However, with a very heavy-tailed non-Gaussian noise model, the SCM-based DOA estimation methods fail to provide an accurate estimate of DOA. This paper presents a numerical analysis of the resolving capability of subspace-based circular (C) and non-circular (NC) multiple signal classification (MUSIC) DOA estimation methods of arbitrarily narrowband correlated signal sources corrupted by circular complex elliptical symmetric (C CES) distributed noise. It evaluates the robustness of these methods for correlated C and NC sources by employing the robust complex M-estimators instead of SCM. It study also the effects of correlation on robust MUSIC-based DOA estimation algorithms accuracy as a function of the magnitude and phase of the correlation coefficients. Simulations results show that the NC MUSIC algorithm which requires fewer sensor elements yields robust estimates of DOA for correlated sources than the C MUSIC algorithm using the M-estimators.

Weighted Direct Position Determination via the Dimension Reduction Method for Noncircular Signals

This work studies the direct position determination (DPD) of noncircular (NC) signals with multiple arrays. Existing DPD algorithms of NC sources ignore the impact of path propagation loss on the performance of the algorithms. In practice, the signal-to-noise ratios (SNRs) of different observation stations are often different and unstable when the NC signal of the same radiation target strikes different observation locations. Besides, NC features of the target signals are applied not only to extend the virtual array manifold but also to bring high-dimensional search. For the sake of addressing the above problems, this study develops a DPD method of NC sources for multiple arrays combing weighted subspace data fusion (SDF) and dimension reduction (RD) search. First, NC features of the target signals are applied to extend the virtual array manifold. Second, we assign a weight to balance the error and obtain higher location accuracy with better robustness. Then, the RD method is used to eliminate the high computational complexity caused by the NC phase search dimension. Finally, the weighted fusion cost function is constructed by using the eigenvalues of the received signal covariance matrixes. It is verified by simulation that the proposed algorithm can effectively improve the location performance, get better robustness, and distinguish more targets compared with two-step location technology and SDF technology. In addition, without losing the estimation performance, the proposed algorithm can significantly reduce the complexity caused by the NC phase search dimension.

Sparse Array Design for DOA Estimation of Non-Circular Signals: Reduced Co-Array Redundancy and Increased DOF

In the past decade, sparse arrays have aroused considerable attention for the ability to provide larger array aperture, more degrees of freedom (DOFs) and better estimation performance compared with the uniform linear array (ULA). In this paper, the problem of sparse array design is investigated for direction of arrival (DOA) estimation of non-circular (NC) signals, where both the difference co-array (DCA) and sum co-array (SCA) of physical array configuration can be utilized. Specifically, we first introduce a sliding array strategy which moves the array along the axis as a whole in order to reduce the co-array redundancy and enhance the consecutive DOFs of the sum-difference co-array (SDCA). Subsequently, we focus on the SCA of the conventional nested array (NA) and propose an approach which increases the DOFs of the SCA by relocating proper sensors. Furthermore, based on the above strategy and approach, we derive an extended sliding nested array (ESNA) with few redundant virtual sensors and an enhanced SCA, which can offer increased consecutive DOFs, enlarged virtual array aperture (VAA) and can thus bring high angle estimation accuracy of NC signals. Finally, analytic solutions and simulation results are given to validate the superiority of the proposed ESNA in terms of DOFs and estimation performance.

DOA Estimation of Non-Circular Source for Large Uniform Linear Array With a Single Snapshot: Extended DFT Method

This letter considers the direction of arrival (DOA) estimation of non-circular (NC) source using a large uniform linear array with a single snapshot. First, by utilizing the waveform characteristics of the NC signal, the proposed algorithm can enlarge the virtual array aperture that is twice the length of the physical array, and resultantly enhance DOA estimation accuracy. Moreover, by multiplying a correction matrix, the proposed algorithm can offset the NC phase error, thereby obtain further improved DOA estimation performance. Furthermore, the advocated algorithm can reduce the computation complexity by using the Newton-Raphson method instead of the traditional grid-search method. Finally, the numerical simulation results are provided to demonstrate the effectiveness and superiority of the proposed algorithm.

Generalized covariance-based ESPRIT-like solution to direction of arrival estimation for strictly non-circular signals under Alpha-stable distributed noise

Direction of arrival (DOA) estimation with high resolution and low computational complexity has been one essential research topic. Meanwhile, many solutions have been proposed to handle parameter estimation problems for strictly non-circular (NC) signals. However, second-order statistics-based solutions usually degrade under fractional lower-order Alpha-stable distributed noise. To solve this problem, three concepts are first defined, and then related properties are proved, including a specific generalized covariance (GC), the non-circularity based on GC, and the extended GC matrix. Based on these concepts and theorems, a novel ESPRIT-like method is derived especially for strictly non-circular sources and termed NC-GC-ESPRIT. Monte-Carlo simulations under four different experimental conditions are executed, involving eleven state-of-the-art DOA approaches as comparison candidates. Thus, the superior performance of the proposed solution is validated through the comparison with these candidate algorithms.

Resolving power of MUSIC-like algorithms for circular or rectilinear correlated sources in CES data models

The concept of threshold array signal-to-noise ratio (ASNR) which is defined as the minimal SNR at which specific high-resolution algorithms are able to resolve two closely spaced far-field sources, allows to quantify and to compare sensors array performance in localizing remote targets. This paper generalizes and extends the expressions of the threshold ASNR given in the literature for the conventional and non-circular (NC) MUSIC direction-of-arrival (DOA) estimation algorithms in the context of uncorrelated stochastic circular or rectilinear Gaussian sources and circular complex Gaussian (C-CG) noise, in a more general stochastic framework. We assume that the sources are correlated with an arbitrary distribution, which is inherent in a context of multipath or smart jammers, and that the noise is circular complex elliptically symmetric (C-CES) distributed, which can model impulsive noise with heavy-tailed distributions. The C-CES and NC-CES distributed observations are also considered to quantify the gain in resolution provided when the sample covariance matrix (SCM) of the observations is replaced by M-estimates of this matrix. Asymptotic approaches and perturbation analysis have been performed to derive closed-form expressions of the mean null spectra of the two considered MUSIC algorithms for both observation models, which allow us to derive, for the first time, general unified explicit analytical expressions of the threshold ASNR along the Cox and the Sharman and Durrani criteria. These expressions allow us to quantify the impact of the non-Gaussianity of noise and observations, as well as of the phase and magnitude of the correlation on the resolution threshold, and to quantify the benefit provided when the SCM is replaced by M-estimates of this covariance matrix for CES observations. Finally, numerical illustrations are included to support our theoretical analysis.