Sparse Uniform Linear Array Research Articles

A Novel Nested Array Structure for DOA Based on CCM of Subarrays

An innovative nested array structure that can notably increase the DOF and the linear arrays’ direction of arrival (DOA) estimated performance is proposed. For the conventional nested array, we vectorize the ACM’s output of the whole array and then remove the repeated rows to get a virtual array’s output. In contrast with classical nested arrays, the superiority of the new nested array is that it utilizes the combination of reverse ordering and conjugation of the inner uniform linear array (ULA) for one of the two subarrays and the sparse outer ULA for another to increase the array aperture. This means the outer sparse ULA’s sensor spacing is approximately the length of the entire inner ULA in the classical nested array structure. Yet, it is about twice the length in the new nested array structure. We generate one lengthy consecutive virtual uniform array excluding any redundant virtual sensors by vectorizing the two subarrays’ cross-correlation matrix (CCM), which is mentioned above. With the export signal of the virtual array and its conjugate form of it, an equivalent covariance matrix of full rank is constructed, which is called the Toeplitz matrix, to compensate for the rank lack of the virtual array’s ACM. For the sake of obtaining the DOA of input signals, the conventional DOA estimation method will be implemented on the ACM. The increase in outer sparse ULA’s sensor spacing ensures an increase in virtual array aperture in principle, which means it is capable of increasing the number of DOA estimations, which is called DOF, and optimizing the DOA estimated performance in contrast with the ULA and classical nested arrays. Results from trials certify the preponderance of the proposed structure in the aspect of DOF and the accuracy of DOA estimation.

Self-Calibration for Sparse Uniform Linear Arrays with Unknown Direction-Dependent Sensor Phase by Deploying an Individual Standard Sensor

Calibration of the unknown direction-dependent (DD) sensor phase and aliasing-free directions of arrival (DOA) estimation for sparse linear arrays are difficult tasks. In this work, we deploy an individual standard sensor with a known sensor phase response along the axis of the uncalibrated sparse linear array, a self-calibration method is proposed, in which the unknown DD sensor phase and the aliasing-free DOAs are both estimated. The proposed method is realized with a two-step scheme. In the first step, the sensor phase is eliminated by the Kronecker product of the covariance matrices in two different frequency bins, and the frequency difference satisfies the spatial Nyquist sampling theorem. Then, the DOAs can be robustly estimated without the influences of grating lobes and unknown sensor phase parameters. In the second step, the steering matrix is estimated with the known phase parameters of the deployed standard sensor. Then, the DD sensor phase is extracted from the steering matrix using the DOAs obtained in the first step. Hence, the disadvantages of iteration-based strategies in conventional calibration algorithms (e.g., local minima convergence) can be avoided. The performance of the proposed method is evaluated using simulation data and compared with that of Cramer–Rao bounds.

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.

Optimized Sparse Nested Arrays for DoA Estimation of Non-circular Signals

The recent criteria for designing sparse arrays used in many applications are the number of degrees of freedom (DoFs) and mutual coupling. In this paper, two optimized sparse nested arrays, called OSNA-I and OSNA-II, are proposed for the direction of arrival (DoA) estimation of non-circular signals. The proposed structures aim to enhance the DoFs and mitigate mutual coupling. The OSNA-I and OSNA-II are obtained by rearranging physical sensors of the conventional nested array. Specifically, for the proposed OSNA-I, its dense uniform linear array (ULA) is exactly the same as the dense ULA of the conventional nested array, while the physical sensors of its sparse ULA are located at n(2N1+1)d. For the proposed OSNA-II, it is composed of two subarrays plus one separate sensor that is appropriately placed. The OSNA-II keeps the dense array of the conventional nested array untouched except the sensor at N1d being removed, while its sparse array is deployed according to n(2N1)d. The separate sensor is set apart with the rightmost sensor of the sparse array by a space of 3N1d. The closed-form expressions are derived for the actual sensor locations, number of uniform DoFs and available DoFs, respectively. Besides, it is also proven that OSNA-I and OSNA-II have small weight values due to their sparse structures, which means that OSNA-I and OSNA-II can achieve a good balance between DoFs and weight values. Simulation results demonstrate the superiorities of the proposed structures in terms of robustness and estimation accuracy.

The Limited Aperture Sparse Array for DOA Estimation of Coherent Signals: A Mutual-Coupling-Optimized Array and Coherent DOA Estimation Algorithm

The paper investigates DOA estimation of coherent Signals with the limited aperture sparse array. Mutual coupling between the sensors of the array cannot be ignored in practical radar with a limited aperture of array sensors, which will result in a degradation in the performance of Direction of Arrival (DOA) estimation. This paper proposes a Mutual-coupling-optimized array (MCOA) with a limited aperture in this scenario to reduce the mutual coupling effect. Firstly, we prove the sparse uniform linear array (SULA) has the smallest mutual coupling leakage when the array aperture and the number of sensors is determined. Secondly, we modify the spacing of the array sensors in SULA to make sure that the spacing between all array sensors and the reference sensor are coprime aiming to estimate DOA without spatial aliasing. Thirdly, we give an expression for the array element spacing arrangement with reduced mutual coupling leakage. Finally, the coherent signals are well resolved by the Sparse Bayesian Learning (SBL) algorithm. Numerous simulations are conducted to validate the advantages of the proposed array compared to several sparse arrays for estimating coherent signals in the presence of mutual coupling.

Nested inter-antenna spacing constrained sparse arrays for direction finding

In the design of a sparse array with large uniform degree of freedom (uDOF) for direction finding, the minimum inter-antenna spacing is usually constrained to be no more than half-wavelength. This is not attractive for reducing antenna mutual coupling especially when the antennas are dense. To alleviate antenna mutual coupling, we herein propose a new sparse array structure whose inter-antenna spacing is larger than half-wavelength. This three-level nested inter-antenna spacing constrained (termed NISC) array can be obtained by implicitly imposing the inter-antenna spacing constraint on three sparse uniform linear arrays (ULAs). The NISC difference co-array has holes due to the one-wavelength inter-antenna spacing constraint; however, these holes can be easily filled by adding just one antenna into NISC (termed NISC-A). Both the spatial smoothing subspace schemes and the compressive sensing schemes can be directly applied to the proposed NISC and NISC-A for direction finding. Simulation results are provided to illustrate the behaviors of NISC and NISC-A, in terms of antenna mutual coupling reduction, uDOF increase, and direction finding accuracy.

Symmetric thinned coprime array with reduced mutual coupling for mixed near‐field and far‐field sources localization

As we all know, the non-uniform array is widely used in the mixed near-field (NF) and far-field sources localization. The nested array contains a uniform linear array, it is easy to be affected by mutual coupling, so we propose a symmetric thinned coprime array (STCA) to reduce the mutual coupling effect of mixed source localization. In this paper, the STCA configuration consists of two sparse uniform linear arrays, the first subarray is composed of 2N − 1 sensors, and the sensor element spacing is Md. The second subarray is composed of 2 M + 2 ⌊ M 2 ⌋ sensors, and the sensor element spacing are Nd and Md respectively. The two subarrays form the symmetric coprime array, which shares a reference sensor. Under the same number of physical array sensors, compared with the latest symmetric nested array, although STCA has lower consecutive lags, the array sensors have larger inter-sensor spacing and larger physical array aperture, which can significantly reduce the mutual coupling effect between physical sensors, and better estimation performance of direction-of-arrival (DOA) estimation can be achieved. Finally, simulation results show that STCA can achieve better performance than other symmetric nested arrays under the same array sensors and mutual coupling effects.

Underdetermined DOA estimation for moving array with reduced mutual coupling in unknown nonuniform noise environment

In practice, direction of arrival (DOA) estimation methods will suffer from significant performance degradation due to limited physical sensors, mutual coupling effect and unknown nonuniform noise. To improve the performance of underdetermined DOA estimation in the case, this paper proposes a moving sparse uniform linear array (ULA) for more degrees of freedom (DOF) and lower mutual coupling, and then provides an estimation algorithm with unknown nonuniform noise mitigation. There is a systematic procedure to determine the inter-sensor spacing of the original array and number of shifted arrays for the synthetic ULA, for achieving increased DOF. The construction and analysis of mutual coupling matrix are exploited to show the reduced mutual coupling in the synthetic array. A procedure of nonuniform noise suppression for the synthetic array is given by averaging the noise power from the original array, and then root-MUSIC is employed for DOA estimation with low-complexity. Simulation results are given to demonstrate the superiority of the proposed array and algorithm in terms of DOF enhancement, mutual coupling reduction and nonuniform noise mitigation.

Sparse linear array with low mutual coupling ratio for DOA estimation

In order to reduce the mutual coupling ratio of sparse linear array, a new linear array structure with two sparse uniform linear arrays interleaved nested, is proposed in this paper. The new sparse linear array structure is consisted of two subarrays which have N and M physical sensors, respectively. By setting appropriate interelement spacing and interleaving, the degrees-of-freedom, uniform degrees-of-freedom and array aperture of the proposed sparse array can reach 2 N M + 2 N , 2 ( N + 2 M ) + 1 and max { ( N − 1 ) R 1 , ( M − 1 ) R 2 + d } , respectively. On one hand, the proposed sparse array has closed-form expressions for the sensor locations. On the other hand, through comparative analysis, although the proposed array structure has lower uniform degrees-of-freedom, it can greatly increase the degrees-of-freedom, extend the array aperture, and reduce the mutual coupling ratio between physical sensors, which means better estimation performance of direction-of-arrival estimation can be achieved. Finally, the direction-of-arrival estimation performance of the proposed array structure is verified by numerical simulations.

Symmetric Displaced Coprime Array Configurations for Mixed Near- and Far-Field Source Localization

Conventionally, mixed near-field (NF) and far-field (FF) source localization is performed using a symmetric uniform linear array (ULA). Recently, two symmetric nonuniform linear arrays (NLAs), symmetric nested array (SNA) and compressed SNA (CSNA), have been developed to enhance the positioning performance by their advantages in the degrees of freedom and the array aperture over symmetric ULAs. In this article, we devise a new symmetric NLA, termed symmetric displaced coprime array (SDCA), to locate the NF and FF sources simultaneously. The SDCA configuration consists of three sparse ULAs with a certain displacement, implying there are two displaced coprime arrays with one common subarray. For a given number of sensors, the SDCA configuration is solely determined by a closed-form expression and its consecutive coarray ranges can also be analytically computed. In addition, we derive two optimum SDCA configurations by maximizing the number of the unique and consecutive lags in the difference coarray. Compared with the SNA and the CSNA, the SDCA provides more unique and consecutive virtual sensors, as well as a larger physical array aperture. Numerical experiments are presented to verify the superiorities of the SDCA configuration over the existing symmetric NLAs.

MIMO Radar Array Configuration with Enhanced Degrees of Freedom and Increased Array Aperture

This paper presents a new antenna array design approach for multiple-input multiple-output (MIMO) radar. The geometry exhibits increased degrees of freedom (DOFs) with precise antenna locations for direction-of-arrival (DOA) estimation of sources. The enhancement is realized by using the method of sum coarray of the difference coarray (SCDC). By effectively increasing the inter-element spacing of the transmitting array, a larger hole-free SCDC uniform linear array (ULA) is obtained. This increment provides more DOFs and extended virtual array aperture (VAA) for intended geometry. The transmitting and receiving sensor positions are based on the maximum inter-element spacing constraint (MISC) principle. The MISC composition involves three sparse ULAs with two separate antennas placed appropriately. The mathematical expressions are presented to validate the designed model. Moreover, arbitrary and optimized antenna scenarios for the transmitter and receiver are investigated in terms of DOFs and VAA. The advantages of the proposed configuration are demonstrated by conducting rigorous Monte Carlo simulations. The experimental results of DOFs, VAA, number of resolvable sources, Cram $$\acute{\hbox {e}}$$ r–Rao bound performance, detection, and resolution ability reveal that the proposed sensor array design approach is suitable for MIMO radar which is also capable of estimating the DOAs of multiple sources efficiently in underdetermined scenarios.

DOA Estimation Using Fourth-Order Cumulants in Nested Arrays with Structured Imperfections.

Recently developed super nested array families have drawn much attention owing to their merits on keeping the benefits of the standard nested arrays while further mitigating coupling in dense subarray portions. In this communication, a new mutual coupling model for nested arrays is constructed. Analyzing the structure of the newly formed mutual coupling matrix, a transformation of the distorted steering vector to separate angular information from the mutual coupling coefficients is revealed. By this property, direction of arrival (DOA) estimates can be determined via a grid search for the minimum of a determinant function of DOA, which is induced by the rank reduction property. We also extend the robust DOA estimation method to accommodate the unknown mutual coupling and gain-phase mismatches in the nested array. Compared with the schemes of super nested array families on reducing the mutual coupling effects, the solutions presented in this paper has two advantages: (a) It is applicable to the standard nested arrays without rearranging the configuration to increase the inter-element spacing, alleviating the cross talk in dense uniform linear arrays (ULAs) as well as gain-phase errors in sparse ULA parts; (b) Perturbations in nested arrays are estimated in colored noise, which is significant but rarely discussed before. Simulations results corroborate the superiority of the proposed methods using fourth-order cumulants.

Extended Nested Arrays for Consecutive Virtual Aperture Enhancement

In this letter, we propose an extended nested array (ENA) strategy, which relaxes the constraints on the two-side subarrays of the augmented nested array and allows the sensors further placed from the sparse uniform linear array. To guarantee the continuity of the difference co-array (DCA), several conditions are provided for specific design. Based on these rules, two ENAs, referred to as one-side extended nested array (OS-ENA) and two-side extended nested array (TS-ENA), are developed. The DCAs of both the OS-ENA and TS-ENA are hole-free and possess more consecutive lags than the existing nested-like arrays. Simulations verify the superiorities of the proposed ENAs with the spacial smoothing MUSIC algorithm used.

Sparse nested linear array for direction of arrival estimation

In this paper, we provide an approach to perform direction of arrival (DOA) estimation with the hole-free part of difference coarray, including the physical sparse uniform linear array (ULA) itself. The sparse two-level nested linear array (NLA), whose inter-element spacings are all enlarged by the identical rate compared with the conventional two-level NLA, is selected as the example to present our work due to the typicalness, under-determinedness and the relatively obvious mutual coupling effect. The difference coarray of the sparse two-level NLA is a sparse ULA and the existing methods can not be exploited for it. We derive the sparse uniform linear array discrete Fourier transform (SULA-DFT) algorithm based on the original DFT method to extend its application from only conventional ULA to sparse ULA to solve this problem. With the SULA-DFT, reliable estimation for positive angle within a limited range under a constraint condition between angle and sparsity of array is able to be obtained directly without any ambiguous angle. Compared with the conventional two-level NLA and many other state of the art configurations, the sparse two-level NLA is able to get least mutual coupling effect and largest array aperture and the SULA-DFT often obtain better estimation accuracy with the sparse two-level NLA. On the whole, sparse two-level NLA with more sparsity gets better estimation accuracy. The numerical simulations verify the superiority of the SULA-DFT with the sparse two-level NLA.

DOA Estimation Exploiting Sparse Array Motions

This paper utilizes sparse array motion to increase the numbers of achievable both degrees of freedom (DOFs) and consecutive lags in direction-of-arrival (DOA) estimation problems. We use commonly employed environment-independent sparse array configurations. The design of these arrays is not dependent on the sources in the field of view, but rather aims at achieving desirable difference co-arrays. They include structured coprime and nested arrays, minimum redundancy array (MRA), minimum hole array (MHA), and sparse uniform linear array (SULA). Array motion can fill the holes in the spatial autocorrelation lags associated with a fixed platform and, therefore, increases the number of sources detectable by the same number of array sensors. Quasi-stationarity of the environment is assumed where the source locations and waveforms are considered invariant over array motion of half wavelength. Closed-form expressions of the number of DOFs and consecutive spatial correlation lags for coprime and nested arrays as well as SULA, due to array translation motion, are derived. The number of DOFs and consecutive lags for the specific cases of MRA an 5 avaluated. We show the respective DOA estimation performance based on sparse reconstruction techniques.

MISC Array: A New Sparse Array Design Achieving Increased Degrees of Freedom and Reduced Mutual Coupling Effect

Recently, nested and coprime arrays have attracted considerable interest due to their capability of providing increased array aperture, enhanced degrees of freedom (DOFs), and reduced mutual coupling effect compared to uniform linear arrays (ULAs). These features are critical to improving the performance of direction-of-arrival estimation and adaptive beamforming. In this paper, a new sparse array configuration based on the maximum inter-element spacing constraint (MISC) is proposed. The MISC array configuration generally consists of three sparse ULAs plus two separate sensors that are appropriately placed. The MISC array configurations are designed based on the inter-element spacing set, which, for a given number of sensors, is uniquely determined by a closed-form expression. We also derive closed-form expressions for the number of uniform DOFs of the MISC arrays with any number of sensors. Compared with the existing sparse arrays, the MISC array enjoys two important advantages, namely, providing a higher number of DOFs and reducing the mutual coupling effects. Numerical simulations are conducted to verify the superiority of the MISC array over other sparse arrays.

Combined root‐MUSIC algorithms for multi‐carrier MIMO radar with sparse uniform linear arrays

In this study, sparse uniform linear arrays are used to improve the direction-of-arrival (DOA) estimation accuracy for the multiple-input multiple-output radar. With a suitable choice of multi-carrier frequencies, the DOA estimation ambiguity problem caused by spatial under-sampling can be resolved. Two kinds of search-free DOA estimation algorithms based on root-multiple signal classification (MUSIC) are proposed. The first algorithm directly applies the root-MUSIC to each carrier frequency separately for the true DOAs as well as their spurious DOAs, then, a novel DOA replicas matching algorithm is proposed to obtain the true DOAs from the ambiguous ones. The second algorithm utilises the manifold separation technique (MST) to align the noise subspaces of all multi-carrier frequencies. Using the MST, the separable representation of the array manifold vector of each carrier frequency is obtained, then the root-MUSIC polynomials of all multi-carrier frequencies are combined to construct a new polynomial, the true DOAs can be obtained directly by applying polynomial rooting without any matching processing. The two proposed algorithms are both practical, computationally efficient and robust. Numerical simulations verify the effectiveness of the proposed algorithms in terms of root-mean-squared error.

A Novel Nested Configuration Based on the Difference and Sum Co-Array Concept.

Recently, the concept of the difference and sum co-array (DSCa) has attracted much attention in array signal processing due to its high degree of freedom (DOF). In this paper, the DSCa of the nested array (NA) is analyzed and then an improved nested configuration known as the diff-sum nested array (DsNA) is proposed. We find and prove that the sum set for the NA contains all the elements in the difference set. Thus, there exists the dual characteristic between the two sets, i.e., for the difference result between any two sensor locations of the NA, one equivalent non-negative/non-positive sum result of two other sensor locations can always be found. In order to reduce the redundancy for further DOF enhancement, we develop a new DsNA configuration by moving nearly half the dense sensors of the NA to the right side of the sparse uniform linear array (ULA) part. These moved sensors together with the original sparse ULA form an extended sparse ULA. For analysis, we provide the closed form expressions of the DsNA locations as well as the DOF. Compared with some novel sparse arrays with large aperture such as the NA, coprime array and augmented nested array, the DsNA can achieve a higher number of DOF. The effectiveness of the proposed array is proved by the simulations.

Performance Analysis for Uniform Linear Arrays Exploiting Two Coprime Frequencies

Sparse arrays can achieve a higher number of degrees of freedom (DOFs) compared with uniform linear array (ULA) counterparts. To further reduce the number of physical sensors while keeping a high number of DOFs, a direction of arrival (DOA) estimation algorithm by exploiting coprime frequencies base on a sparse ULA is recently proposed. However, the performance of such approach is not properly analyzed. In this letter, we analyze the Cramer–Rao bound (CRB) as the lower bound of the DOA estimation performance. The difference between the results presented in this letter and the recent CRB results on sparse arrays lies primarily in the additional phases occurred when utilizing different frequencies. It is shown in this letter that the phases affect the covariance matrix of the received data vector and, as a result, change the number of resolvable sources and alter the achieved CRB. We first demonstrate the effect of the additional phases with an example of two closely spaced sources, and the CRB for a sparse ULA exploiting two coprime frequencies is then derived. Numerical simulations are provided to validate the analyses.

Fast DOA Estimation Algorithms for Sparse Uniform Linear Array With Multiple Integer Frequencies

The sparse uniform linear array (SULA) with larger array aperture is adopted to improve the accuracy of direction of arrival (DOA) estimation with a limited number of physical antennas. In the case of single working frequency configuration, spatial aliasing problem occurs due to the spatial under-sampling. However, with a sufficient anti-aliasing condition for multiple frequencies, the unique DOA estimations can be obtained without ambiguity by using multiple integer frequencies. The multiple equivalent array structures of the multiple integer frequencies are analyzed. To perform DOA estimation, in the first part of the paper, a pair of fast DOA estimation methods are proposed. Based on the fact that a real DOA and its ambiguous positions are periodic distributed in the sine domain, the two proposed methods are applied to each equivalent array separately to obtain the real DOA positions as well as their replica positions. Then, an effective matching algorithm is proposed to pick out the real DOAs from the ambiguous ones. The first algorithm is a partial spectral search-based algorithm, and the second algorithm is a search-free-based method. These two algorithms are referred to as matching partial spectral search-based algorithm (matching PPS) and matching root-multiple signal classification algorithm (matching root-MUSIC), respectively. Our third proposed algorithm is a combined root-MUSIC which is applied to the multiple equivalent arrays by taking them as sub-arrays of a common filled ULA. The combined root-MUSIC algorithm utilizes the MUSIC null-spectrum functions of all equivalent arrays to form a new polynomial, and the DOA estimations can be obtained efficiently without any spatial ambiguity by applying polynomial rooting to the new polynomial. Simulations are provided to verify the validity of the proposed algorithms.