Difference Coarray Research Articles

A Joint Angle and Frequency Spectrum Estimation Algorithm Using Difference Coarray

The spectrum sensing that jointly estimates direction-of-arrival (DOA) and frequency spectrum is an important issue for cognitive radio. Existing off-grid DOA estimation algorithms using difference coarray require a large number of snapshots to guarantee the estimation accuracy. Meanwhile, inaccurate DOA estimation renders inaccurate frequency spectrum estimation due to coupled estimation process. In order to overcome these disadvantages, we propose a joint angle and frequency spectrum estimation algorithm using difference coarray in this work. The proposed algorithm first transforms the received signal into the coarray domain and then adopts an “estimate and subtract” method to separate multiple signals. Subsequently, the DOA estimate of each source is corrected iteratively based on the proposed simple interpolation method. Finally, the frequency spectra are obtained by estimating the frequency response matrix based on the previous estimated DOAs. The shortage of requiring a large number of snapshots are overcome by utilizing the deterministically orthogonal signal model in which the signals are completely uncorrelated. Simulation results demonstrate the effectiveness of the proposed algorithm for off-grid DOA and frequency spectrum estimation using difference coarray with few snapshots.

Enhanced DOA Estimation With Augmented CADiS by Exploiting Array Motion Strategies

With its increased aperture and enlarged minimum spacing among sensors, coprime array with displaced subarrays (CADiS) receives a great deal of attention among sparse arrays. However, the holes in the difference co-array of CADiS extremely degrade the achievable number of uniform degrees of freedom (uDOFs). In recent years, increasing researches of direction of arrival (DOA) estimation with moving sparse arrays have brought new insights into the releasing the holes problem. In this paper, we first explore the structure of difference co-array generated from array motion, and propose a hole-free motion strategy that can fill all the holes in difference co-array of coprime arrays, and resulting increased number of uDOFs as well as array aperture. Secondly, based on the analysis of hole positions, we propose another motion strategy that needs only one step motion of CADiS to generate a difference co-array with no holes in the central part. We find all the qualified steps length of the one-step motion method and provide the closed-form expression of the corresponding achievable number of uDOFs. Finally, the number of uDOF and DOA estimation accuracy of the proposed motion strategies are compared with several state-of-the-art motion methods. Numerical results show the superiority of the proposed methods.

Investigations on the Performance Comparison of Co-Prime Array with and without interpolation for DOA Estimation

The sparse arrays have recently attracted a lot of attention in the last decade, due to their ability to find more sources than sensors present in an array configuration. There might be some sparse arrays which might have holes in their difference coarray, and different interpolation techniques are used to fill these holes. In this paper, the comparison of estimation performances using RMSE vs SNR plots of co-prime array before and after interpolation are analyzed using both coarray MUSIC and ESPRIT algorithms.

Gridless wideband DOA estimation in nonuniform noise with increased DOFs

This paper presents gridless schemes for direction finding of wideband sources using sparse array with nonuniform noise. According to the frequencies relationship, we obtain multiple equivalent scaled difference coarrays, which are combined later to formulate synthetic virtual array when sources share the same normalized power spectrum in different subbands. After treating the term affected by nonuniform noise as hole, an atomic norm optimization problem to recover the noise-free signal of interpolated synthetic uniform linear array (ULA) is designed. We analyze the estimation error distribution of the noise-independent terms in synthetic virtual array and based on this, impose a constraint to avoid tuning process and effect of noise. Furthermore, it extremely increases the number of detectable sources, even making it possible to exceed the degree of freedoms of difference coarray. For the general case when sources have arbitrary powers, a multi-snapshot atomic norm optimization problem is formulated, where signal from scaled coarrays is considered as partial observation of synthetic ULA for different snapshots. Similarly, a newly constructed constraint is added, with which most of the above advantages are sustained. Besides, we give the semidefinite programming form and the Cramer-Rao bounds. The superiority of the proposed algorithms is demonstrated by simulations.

Symmetric Extended Nested Array for Passive Localization of Mixture Near- and Far-Field Sources

Nowadays, symmetric nonuniform linear arrays (SNLAs) have been considered for mixture near-field (NF) and far-field (FF) source localization since it can provide larger physical array aperture and higher degree-of-freedom (DOF) than uniform linear arrays (ULAs). In this communication, a symmetric extended nested array (SENA) is proposed on the basis of compressed symmetric nested array (CSNA). The closed-form expression for the SENA and its difference coarray (DCA) is presented in detail. For the sake of comparison to the existing SNLAs with sensors, the longer consecutive DCA lags and larger physical array aperture are obtained with optimal design to achieve better estimation results. Numerical results demonstrate that utilizing SS-MUSIC and MUSIC algorithms, the SENA outperforms exiting SNLAs in terms of the DOA and range estimate accuracy for mixture field sources.

Improved Coprime Array Designs Based on Moving Platform for Direction of Arrival Estimation

In this article, we mainly propose two improved coprime arrays (CAs) on array motion, ImCAAM1 and ImCAAM2, for direction of arrival (DOA) estimation, which can generate a longer consecutive difference co-array. Both the designs can solve the problem that part of lags, not surrounded by holes in difference co-array, have invalid use, when considering moving sparse array. The first scheme primarily implements two operations: compressing the interelement spacing of one subarray and staggering a certain distance between two subarrays. Two operations make contributions to increasing consecutive degrees of freedom (cDOF). Whereas the second one is able to further fill the remaining holes located at both the ends of virtual array. With the interelement spacing compress factor, equal to the number of elements of another subarray, and re-staggering the distance between subarrays, we can eventually obtain a longer full-populated virtual array. Finally, numerical simulations are presented to verify the efficacy of the proposed sparse array geometries using the subspace-based DOA estimation algorithm in terms of cDOF, spatial spectrum, and DOA estimation accuracy.

Ternary redundant sparse linear array design with high robustness

Sparse linear array is prone to element failure affected by environmental and other factors in practical application, resulting in the decline of degrees of freedom and the attenuation or even failure of direction of arrival (DOA) estimation performance. To address this problem, a ternary redundant sparse array composed of three uniform linear arrays with different spacings at specific distances is designed, and the analytical expression of its degrees of freedom is derived. The configuration rules of the array are concise, and the difference coarrays of the array are hole-free. Mathematically proves that all virtual array elements can be covered with a weight of not less than 3 except the 4 farthest. The array has lower sensors importance and less generalized k-fragility. The array design takes into account the maximum degrees of freedom, redundancy and sparsity. In case of sensors failure, compared with the other sparse linear arrays, the array has more stable difference coarrays and higher uniform degrees of freedom. Compared with multiple-fold redundancy arrays, the array has lower mutual coupling. Simulation results show that the ternary redundant array has superior robustness and higher DOA estimation accuracy under sensors failure.

Direction of Arrival Estimation of Generalized Nested Array via Difference–Sum Co-Array

To address the weakness that the difference co-array (DCA) only enhances the degrees of freedom (DOFs) to a limited extent, a new configuration called the generalized nested array via difference-sum co-array (GNA-DSCA) is proposed for direction of arrival (DOA) estimation. We consider both the temporal and spatial information of the array output to construct the DSCA model, based on which the DCA and sum co-array (SCA) of the GNA are systematically analyzed. The closed-form expression of the DOFs for the GNA-DSCA is derived under the determined dilation factors. The optimal results show that the GNA-DSCA has a more flexible configuration and more DOFs than the GNA-DCA. Moreover, the larger dilation factors yield significantly wider virtual aperture, which indicates that it is more attractive than the reported DSCA-based sparse arrays. Finally, a hole-filling strategy based on atomic norm minimization (ANM) is utilized to overcome the degradation of the estimation performance due to the non-uniform virtual array, thus achieving accurate DOA estimation. The simulation results verify the superiority of the proposed configuration in terms of virtual array properties and estimation performance.

Interference Suppression Method with MR-FDA-MIMO Radar

In the conventional array radar, the suppressible interference is generally limited by the array element number which is referred to as the degree-of-freedom (DOF) of the array. Similarly, the suppressible number of deceptive interference is limited by the transmit element number of frequency diverse array (FDA) multiple-input multiple-output (MIMO) radar. In this paper, an interference suppression method based on data-independent beamforming is proposed with FDA-MIMO radar by using minimum redundancy array (MRA) design in its transmit port, referred to as MR-FDA-MIMO radar. This breakthroughs the limitation on the number of suppressible deceptive interferences. The MRA is employed to obtain extended DOF by using difference co-array, resulting in suppressible number of order O(2M) with M transmit element number. The beamforming method consists of two manifolds: i) in the receive beamforming stage, the possible target is preserved by using orthogonal subspace projection and the sidelobe barrage interference is suppressed; ii) in the equivalent transmit beamforming stage, possible non-ideal factors are taken into consideration for constructing the covariance matrix to widen the notches corresponding to the predicted deceptive interference. The proposed method outperforms in terms of number of the suppressible deceptive interferences and the signal-to-interference-plus-noise ratio, which is demonstrated with simulation examples.

Direction-of-Arrival Estimation for Nested Acoustic Vector-Sensor Arrays Using Quaternions

There is an increasing interest in direction-of-arrival (DOA) estimation using nested arrays composed of vector sensors. Considering acoustic vector sensors (AVSs) commonly used in underwater applications, this paper proposes a novel algorithm, called augmented nested quaternion-MUSIC (ANQ-MUSIC), to perform DOA estimation. By judiciously arranging the multi-component outputs of AVSs, we model the received signals from the entire nested AVS array as a quaternion observation vector in a compact way to reduce the computational complexity. Next, we formulate a quaternion-based difference co-array (QDCA) model via vectorizing the quaternion covariance matrix (QCM). Based on the obtained insights from the QDCA model, we derive a suitable QCM, which is constructed by applying the spatial smoothing technique. Finally, classical quaternion-MUSIC is logically introduced to estimate the DOA parameters. In simulations, we take into account non-uniform received noise and inter-component correlated noise, which may occur in practical underwater environments. The results demonstrate that the proposed method shows superiority in angular resolution and achieves a desirable trade-off between estimate accuracy and computational burden, besides showing robust performance in the above test scenarios.

Sparse Billboard and T-Shaped Arrays for Two-Dimensional Direction of Arrival Estimation

In two-dimensional direction of arrival (2D-DOA) estimation, planar arrays can estimate the elevation and azimuth angles simultaneously. However, many planar array topologies such as billboard, L-shaped, T-shaped, and 2D nested arrays suffer from mutual coupling that results from the small separation between the physical sensors (antennas), which limits the estimation capability of the sensor array. In an attempt to reduce mutual coupling between sensors, this article proposes sparse billboard and T-shaped arrays in which the number of closely separated sensors is significantly reduced. In addition to extending the CRB for fourth order coarray, this article also derives closed-form expressions for the sensor locations and the number of consecutive lags or the uniform degrees of freedom (uDOF), in the fourth-order difference coarray (FODC). Simulation results demonstrate the robustness of the proposed sparse arrays against mutual coupling.

Half-Inverted Array Design Scheme for Large Hole-Free Fourth-Order Difference Co-Arrays

Half-Inverted Array Design Scheme for Large Hole-Free Fourth-Order Difference Co-Arrays

Particle-Velocity Coarray Augmentation for Direction Finding with Acoustic Vector Sensors

In this paper, the problem of passive direction finding is addressed using an acoustic vector sensor array (AVS), which may be deployed either in free space or near a reflecting boundary. Building upon the \(4 \times 1\) vector field measured by an AVS, the particle-velocity coarray augmentation (PVCA) is proposed to admit the underdetermined direction finding using the spatial difference coarray derived from the vectorization of the array covariance matrix. Unlike the widely used spatial coarray Toeplitz recovery technique, the PVCA is applicable to arbitrary array geometries and imposes no reduction of the spatial difference coarray aperture. For the array located at or near a reflecting boundary, the PVCA allows resolving up to 13 sources, while for the array located in free space, the PVCA can identify 9 sources at most. By applying to the systematically designed nonuniform arrays, such as coprime arrays and nested arrays, the PVCA can be coupled with the spatial smoothing technique to get the number of resolvable sources multiplied. Finally, the effectiveness of the PVCA is verified by numerical simulations.

Analysis of the fourth-order co-array for a mixture of circular and noncircular signals

A detailed analysis of the degrees of freedom (DOFs) (and therefore the maximum number of signals to be estimated) of the fourth-order sum and difference co-arrays for direction of arrival (DOA) estimation in the presence of circular, strictly noncircular and nonstrictly noncircular signals is presented. There are different ways in combining noncircularity, fourth-order cumulants and sparse arrays to increase the DOFs of the system for DOA estimation. However, there are some confusions or a lack of clarity in the combination. In this work, we aim to fill the gap and clarify some relevant issues by providing a detailed analysis for the fourth-order co-array for a mixture of circular, strictly noncircular and nonstrictly noncircular signals based on a general signal model, including consideration of the noncircular phases of signals in DOA estimation, the fourth-order co-array aperture by considering all the fourth-order cumulants, the difference in the number of signals to be resolved for the strictly and nonstrictly noncircular signals, and the general analysis of DOFs for a mixture of circular, strictly noncircular and nonstrictly noncircular signals based on either uniform or sparse arrays. Furthermore, the expansion and shift scheme with one sub-array being a nested array and another one being a stamp array is proposed, which provides the most DOFs among considered sparse array construction schemes.

Efficient DOA and Polarization Estimation for Dual-Polarization Synthetic Nested Arrays

To obtain a larger aperture for improving parameter estimation performance, we present a novel polarization sensitive array with dual-polarization synthetic nested structure, and then propose a low-complexity method for direction of arrival (DOA) and polarization estimation with two steps. There is a systematic procedure to determine the number and positions of antennas in each single-polarization subarray used for the synthetic nested array, in order to achieve more degrees of freedom (DOFs) and larger aperture for difference coarray with limited physical antennas. The increased DOFs of cross- and self-difference coarrays are quantitatively provided. To handle underdetermined parameter estimation with the resulting DOFs, the proposed method first estimates initial DOA and polarization state by using root-MUSIC and oblique projection operator, based on hole-free cross-difference coarray from cross-correlation matrix between subarrays. In the second step, holes in self-difference coarray, which is from autocorrelation matrix of single-polarization subarray, are filled with initial estimation. By exploiting both hole-free self- and cross-difference coarrays, an efficient search-based algorithm with initially estimated parameters is explored for refined estimation. The Cramér–Rao bound for the array and computational complexity of the method are provided. Simulation results are given to demonstrate the superior performance of the proposed array and method.

K-Level Extended Sparse Array Design for Direction-of-Arrival Estimation

Sparse arrays based on the concept of a sum-difference coarray (SDCA) have increased degrees of freedom and enlarged effective array aperture compared to those only considering a difference coarray. Nevertheless, there still exist a number of overlapping virtual sensors between the difference coarray and the sum coarray, yielding high coarray redundancy. In this paper, we propose a k-level extended sparse array configuration consisting of one sparse subarray with k-level expansion and one uniform linear subarray. By systematically analyzing the inherent structure of the k-level extended sparse array, the closed-form expressions for sensor locations, uniform DOF and coarray redundancy ratio (CARR) are derived. Moreover, with the utilization of a k-level extended strategy, the proposed array remains a hole-free property and achieves low coarray redundancy. According to the proposed sparse array, the spatial and temporal information of the incident sources are jointly exploited for underdetermined direction-of-arrival estimation. The theoretical propositions are proven and numerical simulations are performed to demonstrate the superior performance of the proposed array.

Synthetic sparse nested array with extended aperture and reduced mutual coupling for DOA estimation

To obtain a larger aperture for improving direction of arrival (DOA) estimation performance, we present a synthetic sparse nested array consisting of three subarrays with dual-polarization antennas, and then propose a cascade method composing of three steps for super-resolution angle estimation with range of 180 degree. There is a systematic procedure to determine the number and positions of antennas in each single-polarization subarray used for the synthetic sparse nested array, in order to efficiently reduce mutual coupling and obtain increased degrees of freedom and aperture for difference coarray with limited physical antennas. By exploiting the configuration and fully utilizing the aperture of the proposed array, two sparse uniform linear arrays are achieved in difference coarray domain, for giving coarse estimation and fine but ambiguous estimation in the first and second steps, respectively. Then an automatic pair matching based on pseudo ambiguous angles is provided to obtain unique DOA estimation in the third step. Theoretical analysis and simulation results are given to demonstrate the superiority of the proposed array and method for high-accuracy DOA estimation in the presence of mutual coupling.

Underdetermined DOA Estimation Using Arbitrary Planar Arrays via Coarray Manifold Separation

Conventional direction-of-arrival (DOA) estimation algorithms like MUSIC only allow localization of fewer number of sources than the number of physical sensors. In this paper, underdetermined azimuth localization (localizing more sources than the number of sensors) using arbitrary planar arrays has been proposed, using only second-order statistics of the received data. To achieve this, we utilize the difference coarray of the actual array and express the elements of the array covariance matrix as the signal received by the virtual sensors of the coarray. We explore the structure and geometry of the difference coarray of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -element planar array and show that the coarray can provide an increased degree-of-freedom (DOF) of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(N^{2})$</tex-math></inline-formula> which enables underdetermined localization. Then, we extend the manifold separation (MS) technique to the coarray to express the coarray steering matrix in terms of a Vandermonde structured matrix by designing a signal independent coarray characteristic matrix. As the signal model of a coarray is a single snapshot model, the Vandermonde structure enables us to perform a spatial smoothing type operation to restore the rank of the coarray covariance matrix. This allows us to propose a novel subspace-based algorithm, which we call the coarrayMS-MUSIC, to perform underdetermined source localization using arbitrary planar arrays. We have also introduced the polynomial rooting version of our algorithm called the coarrayMS-rootMUSIC. Finally, we have conducted extensive numerical simulations to verify the effectiveness and usefulness of the proposed methods.

Sparse convolutional array for DOA estimation

Array signal processing plays an important role in many areas. Besides the Uniform Linear Array, there are many sparse array that have been proposed, for example, Minimum-redundancy array, Co-prime array, nested array, etc. However, most of the array structures have certain disadvantages. A new type of nonlinear sparse sensor array called sparse convolutional array is illustrated which can reduce the number of physical sensors while remain a decent performance in DOA estimation. The sparse convolutional array contains three groups of physical sensors and is able to form a hole-free difference co-array. By adding sensors on two sides instead of the center, the proposed array shows improved performance compared to some other approaches while reminds few physical sensors. The array geometrical structure is illustrated, and the numerical result is provided. We also extended this array structure to two-dimensional case, and the performance is illustrated.

A New Sparse Optimal Array Design Based on Extended Nested Model for High-Resolution DOA Estimation

In this paper, we propose a new strategy with increased element spacings placed on two-side subarrays and increase the physical sparse sensors array aperture. In order to ensure the continuity of difference co-array, four conditions are proposed. Based on these conditions, a new array structure named TS2-ENA is derived. Compared with other nested-like arrays with a fixed number of elements, our proposed structure possesses a higher degree of freedom, closed-form expression and outstanding performance. Simulation results show the superiority of the new configuration with the spacial smoothing MUSIC algorithm conducted.