Linear Subarrays Research Articles

Source Estimation Using Coprime Array: A Sparse Reconstruction Perspective

Direction-of-arrival (DOA), power, and achievable degrees-of-freedom (DOFs) are fundamental parameters for source estimation. In this paper, we propose a novel sparse reconstruction-based source estimation algorithm by using a coprime array. Specifically, a difference coarray is derived from a coprime array as the foundation for increasing the number of DOFs, and a virtual uniform linear subarray covariance matrix sparse reconstruction-based optimization problem is formulated for DOA estimation. Meanwhile, a modified sliding window scheme is devised to remove the spurious peaks from the reconstructed sparse spatial spectrum, and the power estimation is enhanced through a least squares problem. Simulation results demonstrate the effectiveness of the proposed algorithm in terms of DOA estimation and power estimation as well as the achievable DOFs.

Direction of Arrival Estimation of Quasi-Stationary Signals Using Unfolded Coprime Array

Direction of arrival (DOA) estimation of quasi-stationary signals (QSS) is discussed in this paper, and an algorithm using unfolded coprime array is proposed. Coprime array consists of two uniform linear subarrays with coprime relationship, and the two subarrays of the unfolded coprime array in this paper are arranged along the positive axis and negative axis, respectively. Through the vectorization of the low dimensional cross-covariance matrix and the further exploiture of the non-circularity within the second-order statistics of QSS, 2MN-1 degrees of freedom can be achieved with M+N physical elements. The uniqueness of the DOA estimation based on the non-continuous virtual elements generated from the unfolded coprime array is proved, and unitary polynomial root finding technique is employed to estimate the DOA. The proposed algorithm has low complexity, and it can obtain better DOA estimation performance and handle more sources than Khatri–rao subspace approach and original coprime array-based method. Simulation results verify the effectiveness of the proposed approach.

Sparse Extension Array Geometry for DOA Estimation With Nested MIMO Radar

The two-level nested array geometry, which systematically nests two uniform linear subarrays, is proved to offer $O(N^{2})$ degrees of freedom (DOFs) with only $N$ sensors. In this paper, a novel sparse extension array geometry for nested multiple-input multiple-output radar is proposed to provide $O(N^{4})$ DOFs with $N$ sensors. In the proposed geometry, both the transmitter and receiver are equipped with the two-level nested arrays, where we particularly extend the inter-element spacing of the transmitter with a sparse extension factor, leading to a great increase of DOF. Furthermore, we derive the closed-form expressions for the sensor locations and the available DOFs. Spatial smoothing-based MUSIC algorithm is employed to validate the effectiveness and superiority of the proposed sparse extension array for direction of arrival estimation.

A Conical Patch Antenna Array for Agile Point-to-Point Communications in the 5.2-GHz Band

We study here a 12-element conical antenna array made of four linear subarrays angularly spaced by 90 degrees and with three probe-fed radiating elements per subarray. A prototype has been designed and manufactured. The measured radiation characteristics of the elementary radiating element (active pattern) and of the subarray are found in good agreement with the simulation results. The proposed array configuration is considered as a promising solution to rotate and steer the antenna beam for ensuring point-to-point communication at 5.2 GHz

A Low-Complexity ESPRIT-Based DOA Estimation Method for Co-Prime Linear Arrays.

The problem of direction-of-arrival (DOA) estimation is investigated for co-prime array, where the co-prime array consists of two uniform sparse linear subarrays with extended inter-element spacing. For each sparse subarray, true DOAs are mapped into several equivalent angles impinging on the traditional uniform linear array with half-wavelength spacing. Then, by applying the estimation of signal parameters via rotational invariance technique (ESPRIT), the equivalent DOAs are estimated, and the candidate DOAs are recovered according to the relationship among equivalent and true DOAs. Finally, the true DOAs are estimated by combining the results of the two subarrays. The proposed method achieves a better complexity–performance tradeoff as compared to other existing methods.

DOA estimation exploiting a uniform linear array with multiple co-prime frequencies

The co-prime array, which utilizes a co-prime pair of uniform linear sub-arrays, provides a systematical means for sparse array construction. By choosing two co-prime integers M and N, O(MN) co-array elements can be formed from only O(M+N) physical sensors. As such, a higher number of degrees-of-freedom (DOFs) is achieved, enabling direction-of-arrival (DOA) estimation of more targets than the number of physical sensors. In this paper, we propose an alternative structure to implement co-prime arrays. A single sparse uniform linear array is used to exploit two or more continuous-wave signals whose frequencies satisfy a co-prime relationship. This extends the co-prime array and filtering to a joint spatio-spectral domain, thereby achieving high flexibility in array structure design to meet system complexity constraints. The DOA estimation is obtained using group sparsity-based compressive sensing techniques. In particular, we use the recently developed complex multitask Bayesian compressive sensing for group sparse signal reconstruction. The achievable number of DOFs is derived for the two-frequency case, and an upper bound of the available DOFs is provided for multi-frequency scenarios. Simulation results demonstrate the effectiveness of the proposed technique and verify the analysis results.

A Universal Two-Dimensional Direction of Arrival Estimation Method Without Parameter Match

In this paper, based on the subspace method, a novel algorithm, which does not require match procedure, is proposed to estimate 2-D direction of arrival (DOA) with the L-shaped array. The novel train of thought for DOA estimation includes the following two steps: 1) using the 1-D uniform linear subarray (ULSA) along the $z$ axes, obtain the elevation angle estimates based on the polynomial root method and 2) with the elevation angle estimates and using the 2-D ULSA along the $x$ axes, achieve the estimation of the azimuth angles based on the subspace method. Finally, the proposed algorithm is extended to adapt to the more general array structure. The proposed algorithm obtains much higher detection probability of successful pair matching than the existing algorithms. Simulation results illustrate the validity of the proposed algorithm.

An evaluation of reverse-time imaging of clustering earthquakes

Clustering earthquakes refer to the seismic events that occur closely with each other in time and space. Because their overlapping waveform records make it difficult to pick the first arrivals, the hypocenters of clustering earthquakes cannot be determined accurately by traveltime location methods. Here we apply a reverse-time imaging (RTI) method to map clustering earthquakes. Taking the advantage of directly using waveforms, the RTI method is capable to map either a single small earthquake or some densely distributed clustering earthquakes beneath a 2-D seismic array. In 3-D case the RTI method is successfully applied to locate the long-offset doublet earthquakes using the data from a set of sparsely distributed surface stations. However, for the same acquisition geometry, the RTI encounters challenges in mapping densely distributed clustering earthquakes. While it is obvious that improving the mapping of clustering earthquakes requires a denser receiver network with wider range of illumination angles, it is necessary to verify the actual resolution of the RTI method with synthetic data. In our study area in the Three Gorges region, our tests in 3-D case suggest that some events beneath the linear aligned sub-arrays have reasonable resolution.

Broadband implementation of coprime linear microphone arrays for direction of arrival estimation.

Coprime arrays represent a form of sparse sensing which can achieve narrow beams using relatively few elements, exceeding the spatial Nyquist sampling limit. The purpose of this paper is to expand on and experimentally validate coprime array theory in an acoustic implementation. Two nested sparse uniform linear subarrays with coprime number of elements ( M and N) each produce grating lobes that overlap with one another completely in just one direction. When the subarray outputs are combined it is possible to retain the shared beam while mostly canceling the other superfluous grating lobes. In this way a small number of microphones ( N+M-1) creates a narrow beam at higher frequencies, comparable to a densely populated uniform linear array of MN microphones. In this work beampatterns are simulated for a range of single frequencies, as well as bands of frequencies. Narrowband experimental beampatterns are shown to correspond with simulated results even at frequencies other than the arrays design frequency. Narrowband side lobe locations are shown to correspond to the theoretical values. Side lobes in the directional pattern are mitigated by increasing bandwidth of analyzed signals. Direction of arrival estimation is also implemented for two simultaneous noise sources in a free field condition.

Direction-of-arrival estimation for noncircular sources via structured least squares–based esprit using three-axis crossed array

A structured least squares (SLS)-based ESPRIT algorithm for direction-of-arrival (DOA) estimation of strictly second-order noncircular signals using a crossed array is devised in this paper. Unlike the conventional ESPRIT for noncircular signals, which employs the least squares or total least squares, the proposed solution is able to exploit the overlapping structure of the subarray configurations to efficiently solve the shift-invariant equations (SIEs), ending up with the SLS solution for a three-axis crossed array. Moreover, an additional constraint requiring the SIEs of the three linear subarrays directed along the x-, y-, and z-axes to share the same set of eigenvectors is applied to solve the rank-deficiency problem. Consequently, the proposed approach provides more accurate DOA estimates, and its superiority over existing ESPRIT schemes is demonstrated via computer simulations.

Experimental validation of a coprime linear microphone array for high-resolution direction-of-arrival measurements.

Coprime linear microphone arrays allow for narrower beams with fewer sensors. A coprime microphone array consists of two staggered uniform linear subarrays with M and N microphones, where M and N are coprime with each other. By applying spatial filtering to both subarrays and combining their outputs, M+N-1 microphones yield M⋅N directional bands. In this work, the coprime sampling theory is implemented in the form of a linear microphone array of 16 elements with coprime numbers of 9 and 8. This coprime microphone array is experimentally tested to validate the coprime array theory. Both predicted and measured results are discussed. Experimental results confirm that narrow beampatterns as predicted by the coprime sampling theory can be obtained by the coprime microphone array.

Generalized Coprime Array Configurations for Direction-of-Arrival Estimation

A coprime array uses two uniform linear subarrays to construct an effective difference coarray with certain desirable characteristics, such as a high number of degrees-of-freedom for direction-of-arrival (DOA) estimation. In this paper, we generalize the coprime array concept with two operations. The first operation is through the compression of the inter-element spacing of one subarray and the resulting structure treats the existing variations of coprime array configurations as well as the nested array structure as its special cases. The second operation exploits two displaced subarrays, and the resulting coprime array structure allows the minimum inter-element spacing to be much larger than the typical half-wavelength requirement, making them useful in applications where a small interelement spacing is infeasible. The performance of the generalized coarray structures is evaluated using their difference coarray equivalence. In particular, we derive the analytical expressions for the coarray aperture, the achievable number of unique lags, and the maximum number of consecutive lags for quantitative evaluation, comparison, and design of coprime arrays. The usefulness of these results is demonstrated using examples applied for DOA estimations utilizing both subspace-based and sparse signal reconstruction techniques.

2-D DOA estimation in case of unknown mutual coupling for multipath signals

In this paper, two dimensional (2-D) direction-of-arrival (DOA) estimation problem in case of unknown mutual coupling and multipath signals is investigated for antenna arrays. A new technique is proposed which uses a special array structure consisting of parallel uniform linear array (PULA). PULA structure is complemented with auxiliary antennas in order to have a structured mutual coupling matrix (MCM). MCM has a symmetric banded Toeplitz structure which allows the application of the ESPRIT algorithm for 2-D paired DOA estimation. The advantage of the PULA structure is exploited by dividing it into overlapping linear sub-arrays (triplets) and spatial smoothing is employed to mitigate multipath signals. Closed form expressions are presented for search-free, paired and unambiguous 2-D DOA estimation. Two algorithms PULA-1 and PULA-2 are proposed to effectively solve the problem. Several simulations are done and the accuracy of the proposed solution is shown.

Pair-Matching Method by Signal Covariance Matrices for 2D-DOA Estimation

The two-dimensional direction-of-arrival (2D-DOA) estimation for the L-shaped array is usually divided into two one-dimensional DOA (1D-DOA) estimation for the two linear subarrays, and then the elevation and azimuth angles are obtained from the two correctly matched 1D-DOAs. This letter presents a novel method for pair-matching of the two 1D-DOAs. The key points of the proposed method are: 1) construct two signal covariance matrices corresponding to the two estimated 1D-DOAs; 2) introduce an objective function, defined over the two signal source covariance matrices, to find the matched 1D-DOAs; 3) by introducing a permutation matrix for the signal covariance matrices, the optimal pair-matching is obtained with an efficient method. Simulations under different scenarios, with low signal-to-noise ratio (SNR), snapshot deficiency, and uncorrelated or coherent signal settings, show the robustness of the proposed method.

A search-free DOA estimation algorithm for coprime arrays

Recently, coprime arrays have been in the focus of research because of their potential in exploiting redundancy in spanning large apertures with fewer elements than suggested by theory. A coprime array consists of two uniform linear subarrays with inter-element spacings Mλ/2 and Nλ/2, where M and N are coprime integers and λ is the wavelength of the signal. In this paper, we propose a fast search-free method for direction-of-arrival (DOA) estimation with coprime arrays. It is based on the use of methods that operate on the uniform linear subarrays of the coprime array and that enjoy many processing advantages. We first estimate the DOAs for each uniform linear subarray separately and then combine the estimates from the subarrays. For combining the estimates, we propose a method that projects the estimated point in the two-dimensional plane onto one-dimensional line segments that correspond to the entire angular domain. By doing so, we avoid the search step and consequently, we greatly reduce the computational complexity of the method. We demonstrate the performance of the method with computer simulations and compare it with that of the FD-root MUSIC method.

Direction-of-Arrival Estimation in Subarrays-Based Linear Sparse Arrays with Gain/Phase Uncertainties

Sparse arrays are of great interest in practical applications owing to their enlarged array apertures, which enable us to achieve better performance in direction finding. A kind of linear sparse arrays (LSpAs) composed of multiple uniform linear subarrays (ULSAs) with gain/phase uncertainties is considered. The problem of direction-of-arrival (DOA) estimation using such kinds of arrays is addressed. In particular two different cases are studied. In the first case all ULSAs are well calibrated, whereas there are unknown gains/phases among them. Using a new ESPRIT-like method, we show that the DOAs can be estimated in closed form. Interestingly, it is found that the unknown gains/phases can also be estimated in closed form by taking advantage of the subspace principle. In the second case we consider a more general situation where some of the ULSAs themselves are suffering from unknown sensor gain/phase responses. Again, we exploit the ESPRIT algorithm and propose a new approach to DOA and gain/phase estimation using this class of arrays. In our proposed methods the unknown parameters (i.e., DOAs and gains/phases) are estimated in closed form without performing spectral search. Representative numerical results are demonstrated to assess the effectiveness of the methods.

Frequency Invariant Uniform Concentric Circular Arrays with Directional Elements

A new approach for designing frequency invariant (FI) uniform concentric circular arrays (UCCAs) with directional elements is proposed, and their applications to direction-of-arrival (DOA) estimation and adaptive beamforming are studied. By treating the sensors along the radial direction of the UCCA as linear subarrays and using appropriately designed beamformers, each subarray is transformed to a virtual element with appropriate directivity. Consequently, the whole UCCA can be viewed as a virtual uniform circular array (UCA) with desired element directivity for broadband processing. By extending the approach for designing FI-UCAs, the frequency dependency of the phase modes of the virtual UCA is compensated to facilitate broadband DOA and adaptive beamforming. Both the linear array beamformers (LABFs) and compensation filters can be designed separately using second- order cone programming (SOCP). Moreover, a new method to tackle the possible noise amplification problem in such large arrays by imposing additional norm constraints on the design of the compensation filters is proposed. The advantages of this decoupled approach are 1) the complicated design problem of large UCCAs can be decoupled into simpler problems of designing the LABFs and compensation filters, and 2) directional elements, which are frequently encountered, can be treated readily under the proposed framework. Numerical examples are provided to demonstrate the effectiveness and improvement of the proposed methods in DOA estimation, adaptive beamforming, and elevation control over the conventional FI-UCCA design method.

Subarray Design Diagnostics for the Suppression of Undesirable Grating Lobes

This paper investigates various subarray pattern distortions that can cause the appearance of grating lobes with the aim to provide an insightful and valuable first-order diagnostic tool for antenna array designers. In typical subarray configurations, the emergence of undesirable grating lobes is a possibility due to distortions in the subarray array factor which can be caused by feeding errors, mutual coupling between elements and/or feed lines, or other discrepancies. This paper will focus on systematic errors that can arise in subarray designs that feature common element excitation schemes. Distortions will be categorized by amplitude and phase errors separately, demonstrating their distinct manifestations in each excitation scheme. Recognition of these manifestations can lead to better subarray designs and save significant time in the development of large array antennas. The concepts introduced here will be supported by analytical array factor calculations for 1 × N linear subarrays, full-wave simulations of a 1 × 4 subarray and 1 × 16 array, and representative measurements of two 1 × 16 arrays at Ku-band.

Microstrip Antenna Array With Series Feeding Network Designed With the Use of Slot-Coupled Three-Way Power Divider

A novel concept of a series feeding network of a microstrip antenna array has been shown. The proposed feeding network utilizes a four-port slot coupler as a three-way power divider, which is composed of two microstrip lines appropriately coupled through a slot within a common ground plane. The proposed power divider is used for simultaneous power distribution between two consecutive linear subarrays and between two 4 × 1 linear arrays constituting a single 8 × 1 linear subarray, where equal-amplitude and out-of-phase signals are required. Such a solution allows for realization of antenna arrays, in which all linear subarrays designed with the use of the “through-element” series feeding technique are fed at their centers from single transmission lines. The theoretical analysis as well as measurement results of the 8 × 8 antenna array operating within 10.5-GHz frequency range are shown.

Decoupled Location Parameter Estimation of 3-D Near-Field Sources in a Uniform Circular Array using the Rank Reduction Algorithm

An algorithm is presented for estimating the 3-D location (i.e., azimuth angle, elevation angle, and range) of multiple sources with a uniform circular array (UCA) consisting of an even number of sensors. Recently the rank reduction (RARE) algorithm for partly-calibrated sensor arrays was developed. This algorithm is applicable to sensor arrays consisting of several identically oriented and calibrated linear subarrays. Assuming that a UCA consists of M sensors, it can be divided into M/2 identical linear subarrays composed of two facing sensors. Based on the structure of the subarrays, the steering vectors are decomposed into two parts: range-independent 2-D direction-of-arrival (DOA) parameters, and range-relevant 3-D location parameters. Using this property we can estimate range-independent 2-D DOAs by using the RARE algorithm. Once the 2-D DOAs are available, range estimation can be obtained for each source by defining the 1-D MUSIC spectrum. Despite its low computational complexity, the proposed algorithm can provide an estimation performance almost comparable to that of the 3-D MUSIC benchmark estimator.