Fourth-order Difference Co-array Research Articles

An enhanced expanding and shift sparse array based on the coprime array and nested array

AbstractIn recent years, sparse arrays based on the fourth‐order difference coarray (FODCA) have received increased research attention due to their small number of array elements and large consecutive virtual arrays. The expanding and shift structure based on the nested array (NA) and coprime array (CPA) (EAS‐NA‐CPA) is the representative one. The EAS‐NA‐CPA comprises two sparse subarrays, wherein the NA is expanded and shifted according to the second‐order difference coarray of the CPA. Although the number of consecutive elements is large, there are many redundant elements in the FODCA of the EAS‐NA‐CPA. To decrease the number of redundant elements and increase the number of consecutive elements in the FODCA, the authors proposed the enhanced EAS‐NA‐CPA (EEAS‐NA‐CPA). Compared with the original EAS‐NA‐CPA, the EEAS‐NA‐CPA increases the adjacent physical sensor spacing further by using more fourth‐order cross‐correlation coarrays in the FODCA, which are not considered in the original EAS‐NA‐CPA. By performing a theoretical analysis of the FODCA provided by the proposed EEAS‐NA‐CPA, the sensor spacing in the expanded NA was further increased to obtain a larger number of consecutive elements compared with the original EAS‐NA‐CPA. Furthermore, the relationship between the number of physical sensors and corresponding number of consecutive elements provided by the FODCA was explicitly expressed. Experimental simulations were conducted to verify the effectiveness of the proposed sparse array for the fourth‐order cumulant‐based direction of arrival estimation.

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.

A Sum-Difference Expansion Scheme for Sparse Array Construction Based on the Fourth-Order Difference Co-Array

A Sum-Difference Expansion Scheme for Sparse Array Construction Based on the Fourth-Order Difference Co-Array

Direction of arrival estimation of non‐Gaussian signals for nested arrays: Applying fourth‐order difference co‐array and the successive method

Herein, we estimate the direction of arrival (DOA) of non-Gaussian signals for nested arrays (NAs) by implementing the fourth-order difference co-array (FODC) and successive methods. In particular, considering the property of the fourth-order cumulant (FOC), we first construct the FODC of the NA, which can obtain O(N4) virtual elements using N physical sensors, whereas conventional FOC methods can only obtain O(N2) virtual elements. In addition, the closed-form expression of FODC is presented to verify the enhanced degrees of freedom (DOFs). Subsequently, we exploit the vectorized FOC (VFOC) matrix to match the FODC of the NA. Notably, the VFOC matrix is a single snapshot vector, and the initial DOA estimates can be obtained via the discrete Fourier transform method under the underdetermined correlation matrix condition, which utilizes the complete DOFs of the FODC. Finally, fine estimates are obtained through the spatial smoothing-Capon method with partial spectrum searching. Numerical simulation verifies the effectiveness and superiority of the proposed method.

A Design Method of Sparse Array with High Degrees of Freedom Based on Fourth-Order Cumulants

Recently, the design of sparse linear array for direction of arrival (DOA) estimation of non-Gaussian signals has attracted considerable interest due to the fact that the fourth-order difference coarray offered by non-Gaussian significantly increases the aperture of a virtual linear array, which improves the performance of DOA estimation. In this paper, a super four-level nested array (S-FL-NA) configuration based on fourth-order cumulants (FOC) is proposed. The S-FL-NA consists of uniform linear arrays which have different interelement spacing. The proposed array configuration is designed based on interelement spacing, which, for a given number of sensors, is uniquely determined by a closed-form expression. We also derive the closed-form expression for the degrees of freedom (DOFs) of the proposed array. The optimal distribution of the number of sensors in each uniform linear array of the proposed array is given for an arbitrary number of sensors. Compared with the existing sparse arrays, the proposed array can provide a higher number of degrees of freedom and a larger physical array aperture. In addition, to improve the calculation speed of the fourth-order cumulant matrix, we simplify the FOC matrix by removing some redundancy. Numerical simulations are conducted to verify the superiority of the S-FL-NA over other sparse arrays.

The Generalized Compressed Nested Array for the Construction of Fourth-order Difference Co-array

The Generalized Compressed Nested Array for the Construction of Fourth-order Difference Co-array

The fourth-order difference co-array construction by expanding and shift nested array: Revisited and improved

The expanding and shift scheme with two nested arrays (EAS-NA-NA) is easy to construct and effective for achieving large number of degrees of freedom (DOFs), which is one of the representative sparse arrays exploiting the fourth-order difference co-array (FODC). In this letter, it is found that by suitably enlarging the sensor spacing of the expanded nested array in the original EAS-NA-NA, EAS-NA-NA with larger spacing (EAS-NA-NALS) can be built and the DOFs available for direction of arrival (DOA) estimation can be remarkably increased. The proposed EAS-NA-NALS can achieve the longest consecutive virtual array among the well-known FODC-based sparse arrays when the number of physical sensors is large. Simulation experiments are conducted to verify the DOA estimation performance of EAS-NA-NALS and several representative sparse arrays are taken for comparison. Experimental results show that EAS-NA-NALS can achieve more DOFs than other sparse arrays, thus acquiring the improved performance for DOA estimation.

On the Size and Redundancy of the Fourth-Order Difference Co-Array

In array processing, the fourth-order difference co-array of sparse arrays with $N$ sensors admits to resolve $\mathcal {O}(N^4)$ source directions with proper assumptions. This $\mathcal {O}(N^4)$ property is relevant to the size of the central uniform linear array (ULA) segment of the fourth-order difference co-array. However, among the existing sparse arrays, the fundamental limits for the sizes of the fourth-order difference co-array remain a topic for further study. This paper characterizes the lower and upper bounds of the size of the fourth-order difference co-array. It is proved that the ULA and the exponential array achieve the lower and upper bounds, respectively. We also propose the fourth-order redundancy to quantify the efficiency of the ULA segment in the fourth-order difference co-array. The fourth-order redundancy owns a provable lower bound depending only on $N$ , providing further insights into the size of the central ULA segment of the fourth-order difference co-array. These bounds are validated through numerical examples.

Generalized L-Shaped Nested Array Concept Based on the Fourth-Order Difference Co-Array

In this paper, a generalized L-shaped nested array based on the fourth-order difference co-array is proposed for two-dimensional (2D) directions’ estimation. The new structure framework makes full use of the physical sensor locations to form a virtual uniform rectangular array (URA) as large as possible. As it utilizes the fourth-order difference instead of the traditional second-order difference result, this structure framework can acquire a much higher degree-of-freedom (DOF) than the existing 2D sparse arrays. The proposed structures have two advantages. One is that the subarrays can be chosen as any nested-class arrays, which makes the sparse array design more flexible. We can choose arbitrary subarray structures for DOF enhancement purposes. Another advantage is that the relative position of two subarrays can be set as any integral multiple of half wavelength. This means that two subarrays can be located as far as possible so that the relative influence between two physical subarrays can be ignored. The DOFs of several typical generalized L-shaped nested arrays (GLNAs) are compared in this paper. By setting the subarrays as different types and the relative position as a special value, a special GLNA is presented. Simulations show that GLNAs have obvious superiority in 2D direction-of-arrival estimation.

An improved expanding and shift scheme for the construction of fourth-order difference co-arrays

Abstract An improved expanding and shift (IEAS) scheme for efficient fourth-order difference co-array construction is proposed. Similar to the previously proposed expanding and shift (EAS) scheme, it consists of two sparse sub-arrays, but one of them is modified and shifted according to a new rule. Examples are provided with the second sub-array being a two-level nested array (IEAS-NA), as such a choice can generate more fourth-order difference lags (FODLs), although with the same number of consecutive lags. Furthermore, the array aperture of IEAS-NA is always greater than the corresponding EAS structure, which helps improving the DOA estimation result. Simulations results are provided to show the improved performance by the proposed new scheme.

Sparse Method for Direction of Arrival Estimation Using Denoised Fourth-Order Cumulants Vector.

Fourth-order cumulants (FOCs) vector-based direction of arrival (DOA) estimation methods of non-Gaussian sources may suffer from poor performance for limited snapshots or difficulty in setting parameters. In this paper, a novel FOCs vector-based sparse DOA estimation method is proposed. Firstly, by utilizing the concept of a fourth-order difference co-array (FODCA), an advanced FOCs vector denoising or dimension reduction procedure is presented for arbitrary array geometries. Then, a novel single measurement vector (SMV) model is established by the denoised FOCs vector, and efficiently solved by an off-grid sparse Bayesian inference (OGSBI) method. The estimation errors of FOCs are integrated in the SMV model, and are approximately estimated in a simple way. A necessary condition regarding the number of identifiable sources of our method is presented that, in order to uniquely identify all sources, the number of sources K must fulfill . The proposed method suits any geometry, does not need prior knowledge of the number of sources, is insensitive to associated parameters, and has maximum identifiability , where M is the number of sensors in the array. Numerical simulations illustrate the superior performance of the proposed method.

Off-Grid DOA Estimation Aiding Virtual Extension of Coprime Arrays Exploiting Fourth Order Difference Co-Array With Interpolation

In this paper, a novel array structure exploiting coprime arrays is proposed which can be very proficient to determine the number of consecutive lags in proportion with the number of array elements. The proposed method comprises novel array structure by configuring three subarrays positioned in alignment with some prescribed values. By increasing array elements in third subarray while keeping other subarrays fixed, explicit number of consecutive lags could be obtained proportionately. The proposed method offers maximization of consecutive lags in remarkable number by calculating the fourth order difference co-array unifying interpolation. The forth order difference co-array is achieved by exploiting the second order difference co-array twice. The consideration of third subarray in addition with two coprime subarrays leads to a novel array structure which can significantly enhance degrees of freedom. An effective interpolation technique nuclear norm minimization is considered to fill the holes subsisting in the virtual co-array in order to exploit full virtual co-array length. This interpolation method uses convex framework which is trackable and very simple to implement yielding a freedom of fixing any predefined extra tuning parameter. The proposed method in this paper is named as VEFODCI which stands for virtual extension of coprime arrays by exploiting fourth order difference co-array with interpolation. The sparse Bayesian learning is used for direction of arrival (DOA) estimation exploiting the proposed novel array structure by imposing interpolation to fill the holes. The array geometry of sensor distributions proves that the proposed method is less susceptible from mutual coupling effect. The simulation results stipulate that the proposed method is performing DOA estimation accurately even in lower angular separation of sources by achieving larger number of consecutive lags than the state of the art.

An Expanding and Shift Scheme for Constructing Fourth-Order Difference Coarrays

An expanding and shift (EAS) scheme for efficient fourth-order difference co-array construction is proposed. It consists of two sparse sub-arrays, where one of them is modified and shifted according to the analysis provided. The number of consecutive lags of the proposed structure at the fourth order is consistently larger than two previously proposed methods. Two effective construction examples are provided with the second sparse sub-array chosen to be a two-level nested array, as such a choice can increase the number of consecutive lags further. Simulations are performed to show the improved performance by the proposed method in comparison with existing structures.

Extension of Co-Prime Arrays Based on the Fourth-Order Difference Co-Array Concept

An effective sparse array extension method for maximizing the number of consecutive lags in the fourth-order difference co-array is proposed, leading to a novel enhanced sparse array structure based on co-prime arrays (CPAs) with significantly increased number of degrees of freedom (DOFs). One method to exploit the increased DOFs based on nonstationary signals is also proposed, with simulation results provided to demonstrate the effectiveness of the proposed structure.

Direction-of-arrival estimation of quasi-stationary signals using two-level Khatri-Rao subspace and four-level nested array

The Khatri-Rao (KR) subspace method is a high resolution method for direction-of-arrival (DOA) estimation. Combined with 2q level nested array, the KR subspace method can detect O(N2q) sources with N sensors. However, the method cannot be applicable to Gaussian sources when q is equal to or greater than 2 since it needs to use 2q-th order cumulants. In this work, a novel approach is presented to conduct DOA estimation by constructing a fourth order difference co-array. Unlike the existing DOA estimation method based on the KR product and 2q level nested array, the proposed method only uses second order statistics, so it can be employed to Gaussian sources as well as non-Gaussian sources. By exploiting a four-level nested array with N elements, our method can also identify O(N4) sources. In order to estimate the wideband signals, the proposed method is extended to the wideband scenarios. Simulation results demonstrate that, compared to the state of the art KR subspace based methods, the new method achieves higher resolution.