Coprime Array Research Articles

Optimum sparse array configuration for DOA estimation on moving platforms

Sparse array motion can efficiently expand the numbers of achievable degrees of freedom (DOFs) and consecutive lags, improving direction-of-arrival (DOA) estimation. Sparse arrays on a moving platform benefit from motion translation that introduces new sensor positions, which collectively with the original positions can increase the number of spatial autocorrelation lags and lead to full array augmentability. This property has been recently used for the case of environment-independent sparse array configurations, such as those defined by nested and co-prime arrays. In this paper, we consider environment-dependent sparse arrays (EDSAs) design using Cramer-Rao bound (CRB) as the metric of optimality for DOA estimation. The CRB is derived for a sparse array on a moving platform, where the number of identifiable uncorrelated sources exceeds the number of sensors. The CRB expression is used to solve for the sparse array configuration by applying the Genetic algorithm. Simulation results are provided to validate the effectiveness of the proposed EDSA design.

A novel direction finding algorithm for distributed sources under impulsive noise environments

We extend the bearing estimation algorithm for distributed sources to the impulsive noise scenario that can be modeled as an alpha-stable distributed process. Since the distributed source parameter estimator (DSPE) method is optimized based on the Gaussian model, its performance is severely degraded in the presence of impulsive noise. In this paper, a derivative of the error function-based operator (DEOP) is derived, and a novel direction-of-arrival (DOA) estimation algorithm, namely, DEOP-DSPE, is proposed for distributed sources to suppress the impulsive noise. Meanwhile, to make more effective use of the sensors in the array, we also propose the conjugate co-prime array, which can reduce the number of physical sensors without decreasing the number of estimated target sources. Simulation results demonstrate that the DEOP-DSPE algorithm outperforms the existing direction finding methods for distributed sources in the presence of impulsive noise and illustrate the effectiveness of the conjugate co-prime array.

Improved 2D Coprime Array Structure with the Difference and Sum Coarray Concept

Recently, the difference and sum (diff-sum) coarray has attracted much attention in one-dimensional direction-of-arrival estimation for its high degrees-of-freedom (DOFs). In this paper, we utilize both the spatial information and the temporal information to construct the diff-sum coarray for planar sparse arrays. The diff-sum coarray contains both the difference coarray and the sum coarray, which provides much higher DOFs than the difference coarray alone. We take a planar coprime array consisting of two uniform square subarrays as the array model. To fully use the aperture-extending ability of the diff-sum coarray, we propose two novel configurations to improve the planar coprime array. The first configuration compresses the inter-element spacing of one subarray and results in a larger consecutive area in the coarray. The second configuration rearranges the two subarrays and introduces a proper separation between them, which can significantly reduce the redundancy of the diff-sum coarray and increase the DOFs. Besides, we derive the closed-form expressions of the central consecutive ranges in the coarrays of the proposed array configurations. Simulations verify the superiority of the proposed array configurations.

Achievable degree-of-freedom for three-level prime arrays

Due to their large aperture size, coprime arrays are good candidate in the presence of mutual coupling as most of the elements are separated by more than half-wavelength. However, the number of consecutive lags in their difference coarray is limited. Combining three uniform linear subarrays where the number of elements in the subarrays are pairwise coprime integers overcomes this limitation. The resultant array is referred to as three-level prime array (3LPA). The 3LPA has a closed form expression for antenna locations and can improve the DOF using smaller aperture size compared with coprime arrays. In this paper, we derive an approximate expression for the achievable DOF for the 3LPA configuration. Simulation results confirm the validity of the derived expressions compared with the exact DOF.

Particle Filter Algorithm for DOA Tracking Using Co-Prime Array

In this letter, the problem of direction of arrival (DOA) tracking using co-prime array is studied. A novel particle filter (PF)-based DOA tracking algorithm is proposed to recursively estimate the DOAs based on the measurement model and the state transition model. A modified likelihood function model based on the propagator method (PM) pseudo-spectrum is devised to update the involved particles under lower SNR or less samples condition. Compared to the existing techniques, such as the spatial smoothing multiple signal classification method (SS-MUSIC), compressive sensing (CS) and conventional likelihood PF (CL-PF) methods, the proposed algorithm can exhibit improved performance in term of estimating accuracy and computational cost. Simulation studies are carried out to show the superiority of the proposed PF algorithm.

Regular Sparse Array Direction of Arrival Estimation in One Dimension

Traditionally, regularly spaced antenna arrays follow the spatial Nyquist criterion to guarantee an unambiguous analysis. We present a novel technique that makes use of two sparse non-Nyquist regularly spaced antenna arrays, where one of the arrays is just a shifted version of the other. The method offers several advantages over the use of traditional dense Nyquist-spaced arrays, while maintaining a comparable algorithmic complexity for the analysis. Among the advantages we mention: an improved resolution for the same number of receivers and reduced mutual coupling effects between the receivers, both due to the increased separation between the antennas. Because of a shared structured linear system of equations between the two arrays, as a consequence of the shift between the two, the analysis of both is automatically paired, thereby avoiding a computationally expensive matching step as is required in the use of so-called co-prime arrays. In addition, an easy validation step allows to automatically detect the precise number of incoming signals, which is usually considered a difficult issue. At the same time, the validation step improves the accuracy of the retrieved results and eliminates unreliable results in the case of noisy data. The performance of the proposed method is illustrated with respect to the influence of noise as well as to the effect of mutual coupling.

Channel estimation based on the PSS-MUSIC for millimeter-wave MIMO systems equipped with co-prime arrays

The millimeter-wave channel estimation problem can be solved by estimating the path directions and the path gains. The previous schemes are nearly all based on the uniform linear arrays (ULAs). However, compared with the ULAs, the co-prime array can use fewer array elements to realize larger array aperture, which is beneficial to improve the performance of arrival direction estimation contributing to the estimation performance of angle of arrival (AOA) and angle of departures (AOD). Encouraged by the property of co-prime arrays, a novel channel estimation scheme is proposed for two-dimensional (2-d) co-prime arrays. For each path direction, multiple peaks are generated in the spatial spectrum of each subarray and by selecting over any limited sector to search for an arbitrary peak, which can recover the rest peaks. Then by comparing the peaks of the two subarrays, the common peaks are the correct path direction. Compared with the total-spectrum search (TSS) method, this method effectively reduces the complexity. Simulation results show that the proposed algorithm can reduce the computational complexity and maintain accurate channel estimation.

Robust Direction-of-Arrival Estimation for Coprime Array in the Presence of Miscalibrated Sensors

Coprime arrays have been widely adopted for direction-of-arrival (DOA) estimation since it can achieve an increased number of degrees of freedom (DOF). To utilize all information received by the coprime array, array interpolation methods are developed, which construct a virtual uniform linear array (ULA) with the same aperture from the non-uniform coprime array. However, the conventional non-robust DOA estimation algorithms for coprime arrays, including the interpolation based methods, suffer from degraded performance or even failed operation when some sensors are miscalibrated. In this paper, a novel maximum correntropy criterion (MCC) based virtual array interpolation algorithm for robust DOA estimation is developed to address this problem. The proposed approach treats the miscalibrated sensor observations as outliers, and by exploiting the property of MCC, the interpolated virtual array covariance matrix is reconstructed via nuclear norm minimization (NNM) with less influence of these outliers. In this manner, the robust DOA estimation is enabled by the robustly reconstructed covariance matrix. Simulation results demonstrate that the proposed algorithm can effectively the mitigate effect of the miscalibrated sensors while maintaining the enhanced DOF offered by coprime arrays.

Padded Coprime Arrays for Improved DOA Estimation: Exploiting Hole Representation and Filling Strategies

As a generalized coprime array structure, the coprime array with displaced subarrays (CADiS) allows a large minimum inter-element spacing by introducing a specific displacement between two sparse subarrays. While this structure can effectively reduce mutual coupling, the holes in its difference co-array greatly decrease the achievable number of uniform degrees of freedom (DOFs). In this paper, we first provide a complete characterization for the hole locations in the difference co-array generated by a tailored CADiS (tCADiS) as the union of four subsets of locations related via simple symmetry properties. We then introduce two representation approaches for the hole locations, revealing that the latter can be generated from the differences between sensor locations in the subarray of tCADiS and a small uniform linear array, referred to as a padded subarray. Subsequently, we propose novel padded coprime arrays (PCAs) by incorporating the padded subarray into tCADiS to enlarge the consecutive segments in the difference co-array. This not only contributes to increase the number of available uniform DOFs, but also helps mitigating the mutual coupling by limiting the number of sensor pairs with small separations. Finally, numerical simulation results are provided to demonstrate the superiority of PCAs over existing sparse array configurations in terms of DOF, mutual coupling and DOA estimation accuracy.

Interpolation-Based Direction-of-Arrival Estimation for Coprime Arrays via Covariance Matrix Fitting

Coprime arrays can highly increase degree-of-freedom (DOF) by exploiting the equivalent virtual signal. However, since the corresponding virtual array constructed by the coprime array is always a non-uniform linear array (non-ULA), most existing direction-of-arrival (DOA) estimation algorithms fail to utilize all received information and result in performance degradation. To address this issue, we propose a novel interpolation approach for coprime arrays to convert the virtual array into a ULA with which all received information can be efficiently utilized. In this paper, we consider a weighted covariance matrix fitting criterion to formulate a semi-definite programming (SDP) problem with respect to the interpolated virtual signal. After that, we can reconstruct a Hermitian Toeplitz covariance matrix corresponding to the interpolated ULA in a gridless manner, and the number of detectable targets is ulteriorly increased with the reconstructed covariance matrix. The proposed approach is hyperparameter-free so that the tedious process of selecting regularization parameters is avoided. Numerical experiments validate the superiority of the proposed interpolation-based DOA estimation algorithm in terms of DOF characteristic, resolution ability and estimation accuracy compared with several existing techniques.

Improving performance of direction-of-arrival estimation using sparse array in passive radar

A method based on sparse array is applied to direction-of-arrival estimation of passive radar in this paper to increase the number of resolvable sources and improve the direction-of-arrival estimation performance for coprime array. The virtual symmetric non-uniform linear array of coprime array based on passive radar signal model is introduced. Considering the impact of direct wave, extensive cancellation algorithm is used to cancel the direct wave, with the conventional MUSIC with spatial smoothing algorithm and virtual aperture filling applied on the sparse array of passive radar; the resolution of target is low in the low signal-noise-ratio. To effectively improve the estimation of the target under the low signal-noise-ratio, a noise subspace reconstruction method is proposed. The proposed direction-of-arrival estimation method can improve the direction-of-arrival estimation performance of passive radar. The simulations are provided to demonstrate the effectiveness of the proposed method.

Stochastic and Deterministic Framework for Modeling the Effect of Antennas Location Error on DOA Estimation

This article focuses on the estimation of the direction of Arrival based on the interpolation approaches of the sparse array. Such as the minimum redundancy array, co-prime array, and the nested array have attracted much attention, especially for a non-linear array antenna. We analyze the effect of antennas locations errors on overall estimation performance for deterministic and stochastic error models and to improve the estimator performance in the case of uncorrelated sources of signals in spatially-smoothed MUSIC scenarios. In two scenarios: we determine the Cramer-Rao bound in the case of the antennas locations are perfect, and in other cases, we take into account antennas locations errors. For the first scenario, the Cramer-Rao bound converges as the SNR increase. For the second scenario, the Signal-to-noise ratio increases to infinity, to achieve the same Cramer-Rao bound as in the first scenarios. There is a significant gap between the Cramer-Rao bound values in a situation where antennas locations errors are considered, and when other locations errors are not considered. This disparity shows that unknown antenna locations errors have a significant impact on the estimated achievable performance of the direction of Arriva of scattered linear matrices.

Direction-of-Arrival Estimation of Coherent Signals via Coprime Array Interpolation

In this letter, we devise an efficient approach for estimating the directions-of-arrival (DOAs) of coherent signals using coprime arrays. Specifically, we firstly derive an augmented uniform linear array (ULA) via coprime array interpolation. Subsequently, we define a Toeplitz matrix that is formed using the correlation information of the interpolated ULA theoretical outputs, and recover the Toeplitz matrix by solving a nuclear norm minimization problem. After the low-rank Toeplitz matrix is recovered, the coherent signals are well resolved by the MUSIC algorithm. Simulation results demonstrate the advantages of the proposed approach over various existing methods when dealing with coherent signals.

Efficient Gridless 2-D Direction-of-Arrival Estimation for Coprime Array Based on Decoupled Atomic Norm Minimization

This paper presents an efficient gridless sparse reconstruction algorithm for the coprime planar array in two-dimensional (2-D) direction-of-arrival (DOA) estimation problem. According to the equivalent second-order statistic signals derived from the covariance matrix of the coprime planar array, we construct a virtual 2-D difference co-array extended from the coprime line arrays along two directions. The virtual array has a double-sized array aperture leading to an increased number of degree-of-freedoms (DOFs). To address the discontinuity of the virtual planar array, and to reduce the computation complexity for the increased array size, decoupled atomic norm minimization approach is investigated to interpolate the missing sensors without discarding any virtual sensors. The problem of decoupled atomic norm minimization can be solved by semidefinite programming with significantly lower computational cost. Besides, the ratio of the number of missing sensors to full sensors in the interpolated virtual uniform array is smaller than that of the physical coprime array, which further improves the recovery accuracy of decoupled atomic norm minimization algorithm. The numerical examples are provided to demonstrate the practical ability of the proposed method in terms of DOF, computational complexity, and DOA estimation error.

STAP for Airborne Radar Based on Dual-Frequency Space-Time Coprime Sampling Structure

To overcome the space-time adaptive processing (STAP) performance loss caused by discarding the non-uniform parts of difference coarray and copulse for coprime sampling structure under the condition of single-frequency operation, this paper proposes a new STAP algorithm to improve the precision of filter weight vector estimation by the dual-frequency operation. By selecting a single proper additional operation frequency, we can obtain the different coarray and copulse, thus fill two missed virtual sensors and pulses of the difference structures in the single-frequency operation simultaneously. Compared with the single-frequency operation, the dual-frequency operation method can acquire the bigger uniform linear array (ULA) and coherent processing interval (CPI) pulse train to improve the system degrees of freedom (DOF), and result in higher angle-Doppler resolution. In addition, the coprime array has the little mutual coupling effect because of the larger inter-sensors spacing. Therefore, the resulting method can alleviate mutual coupling and enhance the system DOF.

A Novel DOA Estimation for Distributed Sources in an Impulsive Noise Environment

Direction of arrival (DOA) estimation has emerged as a promising technology for many military and civilian applications, such as radio astronomy and wireless communications. However, it remains very challenging to estimate DOA for distributed sources within impulsive noise environments. As is widely known, the utilization of the subspace technology in array signal processing can enable high-resolution DOA estimation. In many instances, most of these algorithms assume that the noise is Gaussian. Considering the impulsive noise environments, the derivative of the error function is derived as the iteratively weighted factor in the objective function of low-rank decomposition. Two iterative algorithms are proposed to compute the signal subspace to implement DOA estimation for coherently distributed sources. To further improve the DOA estimation accuracy and increase the number of estimated sources, the virtual data samples are reconstructed from the extended co-prime array. Simulation results confirm that the two proposed impulsive noise suppression algorithms provide better performance than the existing algorithms. The virtual data samples that are reconstructed from the extended co-prime array can be utilized as the effective inputs for subspace-like algorithms.

A Real-Valued Toeplitz Matrix Method for DOA Estimation

Based on the optimized coprime array (OpCA), we propose a real-valued Toeplitz matrix method. This article analyzes the feasibility of converting the manifold matrix of a virtual domain constructed from a physical array into a real-valued matrix, and the unitary estimation of signal parameters via rotational invariance technique is introduced to transform the equivalent received signal rearranged Toeplitz matrix into a real-valued matrix and obtain directions of arrival (DOAs). Moreover, the proposed method offers higher estimation accuracy. Simulations are demonstrated to prove the validity of this method.

DOA Estimation for Unfolded Coprime Arrays: Successive-MUSIC Algorithm

In this paper, we investigate the problem of direction of arrival (DOA) estimation of unfolded coprime arrays with successive-MUSIC algorithm which cascades multiple signal classification (MUSIC) algorithm and rotational invariance techniques (ESPRIT).We propose An algorithm with low computational complexity and high estimation accuracy, which first use ESPRIT algorithm for initial estimation, then according to the results of initial estimation narrow the search scope, finally, more accurate estimation results are obtained by MUSIC algorithm. Compared with the traditional MUSIC algorithm, this algorithm only needs to conduct a small range of angle search, which greatly reduces the algorithm complexity. Compared with the traditional ESPRIT algorithm, this algorithm also effectively improves the angle estimation performance of the algorithm. Moreover, comparing with traditional coprime linear arrays, the enhanced degrees of freedom (DOFs) can be achieved due to larger array aperture, Simulations are presented to validate the superiority of the proposed algorithm.

Editorial

Chris Toumazou & Ian White 2019 has been a year of exciting development for Electronics Letters. The subjects of our published papers continued to diversify, reflecting the expansion of its editorial board – the board now has in excess of 240 members, with 21 new members joining in 2019. 40 countries are represented, and we are certainly hoping to expand further, given that over sixty countries are numbered amongst the 4000 submissions to the journal. The submissions span the entire scope of electrical and electronic engineering, from implantable antibacterial electronics to distinguishing LEDs based on spectral features; or from optical self-injection to machine learning in a variety of forms. A solution for the ubiquitous problem of DOA estimation for co-prime arrays won the 2019 IET Premium Award: ‘Improved DOA estimation algorithm for co-prime linear arrays using root-MUSIC algorithm’, by Dong Zhang, Yongshun Zhang, Guimei Zhang, Cunqian Feng and Jun Tang, [Electronics Letters, 2017, 53, (18), pp. 1277–1279]. This paper proposes an improved algorithm, which provides a solution to the matching error problem. The algorithm also makes full use of signal space and noise space to improve the estimation accuracy with low complexity. The judges were impressed by the mathematical derivation of the proposed algorithm and the completeness of its description. The submission has proved popular and has been cited many times, reflecting the necessity of the solution and the effectiveness of the method. We have had some fantastic contributions from members of our editorial board. One such contribution comes from board member Alexander Wong along with Mohammad Javad Shafiee, Brendan Chwyl and Francis Li in their submission “GenSynth: a generative synthesis approach to learning generative machines for generate efficient neural networks”. [Electronics Letters, 2019, 55, (18), pp. 986–989]. Wong et al gave an insight into their work on GenSynth, which can automatically generate highly efficient deep neural networks. This generative synthesis is a powerful approach for accelerating and improving the building of deep neural networks for on-device edge scenarios. We look forwards to hearing more about future work involving GenSynth and how it can be applied to image and signal processing tasks. Electronics Letters’ wide scope means we publish a wide variety of interesting research. An example of such a submission was published towards the end of 2019: ‘Implantable cathodic voltage controlled electrical stimulator’ by K. Weeks, C. Clark, M. Tobias, E. McDermott, T.Duquin, M. Ehrensberger and A. H. Titus, [Electronics Letters, 2019, 55, (23), pp. 1215–1217]. This fascinating work presents an implantable device for use with orthopaedic devices. The iCVCES (cathodic voltage controlled electrical stimulation) device could be used in the future to reduce the rate of periprosthetic joint infection, the complications of which are currently projected to cost the US economy over $55 billion by 2030. One of the biggest benefits of the device is that it can be implanted without requiring wiring to the external environment, meaning it can remain in a patient for an extended period of time. In addition, it can wirelessly monitor the patient and transfer data via Bluetooth to a computer. Antennas and Propagation is one our most popular categories year-round. This year has seen a great number of interesting papers on novel antenna and component design, for example: ‘Balanced triple-mode substrate integrated waveguide bandpass filter’, by Qing Liu, Dalong Lv, Dongfang Zhou and Dewei Zhang. [Electronics Letters, 2019, 55, (15), pp. 843–845]. The proposed substrate integrated waveguide bandpass filter boasts a high common-mode suppression, a compact size and a controllable bandwidth. The device could see use in in a number of radiofrequency and microwave applications, especially in the emerging 5G market. It's always fantastic to see so many innovative designs in this field. These highlights, as well as the hundreds of papers published each year by Electronics Letters, represent the great journey ahead for the journal and for the subject as a whole. Finally, we would like to thank the editorial and production teams at the IET, the editorial board, and of course the many thousands of peer-reviewers who make the journal possible. That the reviewers and our board members take the time out of their busy schedules and roles to support the community is hugely appreciated, and is one of the reasons that the community, not just Electronics Letters, is flourishing.

Direction of Arrival Estimation and Robust Adaptive Beamforming With Unfolded Augmented Coprime Array

An augmented coprime array systematically employs two sparse subarrays to produce a large-scale difference co-array with attractive merits, such as enhanced degrees of freedom (DOFs) and enlarged array aperture, whereas the interleaved subarrays are susceptible to mutual coupling. In this paper, we propose an unfolded augmented coprime array (UACA) obtained by careful crafting of small sparse subarrays to fill the holes in the difference co-array generated by unfolding operation. Specifically, UACA can significantly reduce the number of sensor pairs with small spacing and hence inherently weaken the mutual coupling effect. Meanwhile, an increase of the DOFs and improved direction of arrival (DOA) estimation accuracy can be achieved in the presence of mutual coupling. As an application of UACA, we propose a decoupled interference-plus-noise covariance matrix (INCM) reconstruction method for robust adaptive beamforming (RAB) with UACA. Therein, mutual coupling coefficients are estimated based on the remodeled contaminated steering vector and the noise subspace. The estimated mutual coupling matrix is utilized to reconstruct the decoupled covariance matrix which in turn is used to obtain refined DOA estimates, interferer power estimates, and the desired INCM. Extensive simulation results are provided to verify the effectiveness of UACA and the decoupled INCM reconstruction method for RAB.