Minimum Redundancy Arrays Research Articles

Sparse Active Rectangular Array With Few Closely Spaced Elements

Sparse sensor arrays offer a cost effective alternative to uniform arrays. By utilizing the co-array, a sparse array can match the performance of a filled array, despite having significantly fewer sensors. However, even sparse arrays can have many closely spaced elements, which may deteriorate the array performance in the presence of mutual coupling. This paper proposes a novel sparse planar array configuration with few unit inter-element spacings. This Concentric Rectangular Array (CRA) is designed for active sensing tasks, such as microwave or ultra-sound imaging, in which the same elements are used for both transmission and reception. The properties of the CRA are compared to two well-known sparse geometries: the Boundary Array and the Minimum-Redundancy Array (MRA). Numerical searches reveal that the CRA is the MRA with the fewest unit element displacements for certain array dimensions.

Joint DOA and Frequency Estimation With Sub-Nyquist Sampling in the Sparse Array System

In this letter, we apply previous array receiver architecture which employs time-domain sub-Nyquist sampling techniques to jointly estimate frequency and direction-of-arrival(DOA) of narrowband far-field signals. Herein, a more general situation is taken into consideration, where there may be more than one signal in a subband. We build time-space union model, analyze the identification of the model, and give the maximum signal number which can be classified. We also proof that the Cramer-Rao Bound (CRB) is lower than that of which employs Nyquist sampling. Simulation results verify the capacity to estimate the number of sources. Meanwhile, simulations show that our estimation performance closely matches the CRB and is superior for more sources than sensors, especially when the minimum redundancy array (MRA) is employed.

Degree-of-Freedom Strengthened Cascade Array for DOD-DOA Estimation in MIMO Array Systems.

In spatial spectrum estimation, difference co-array can provide extra degrees-of-freedom (DOFs) for promoting parameter identifiability and parameter estimation accuracy. For the sake of acquiring as more DOFs as possible with a given number of physical sensors, we herein design a novel sensor array geometry named cascade array. This structure is generated by systematically connecting a uniform linear array (ULA) and a non-uniform linear array, and can provide more DOFs than some exist array structures but less than the upper-bound indicated by minimum redundant array (MRA). We further apply this cascade array into multiple input multiple output (MIMO) array systems, and propose a novel joint direction of departure (DOD) and direction of arrival (DOA) estimation algorithm, which is based on a reduced-dimensional weighted subspace fitting technique. The algorithm is angle auto-paired and computationally efficient. Theoretical analysis and numerical simulations prove the advantages and effectiveness of the proposed array structure and the related algorithm.

A Unified Array Geometry Composed of Multiple Identical Subarrays With Hole-Free Difference Coarrays for Underdetermined DOA Estimation

In this paper, we propose a unified array geometry, dubbed generalized nested subarray (GNSA), for the underdetermined direction-of-arrival estimation. The GNSA is composed of multiple, identical subarrays, which can be a minimum redundancy array (MRA), a (super) nested array, a uniform linear array (ULA), or any other linear arrays with hole-free difference coarrays (DCAs). By properly design the spacings between subarrays, the resulting DCA of the GNSA can also be a hole-free (filled) ULA. When the subarray is an MRA and meanwhile its sensors’ positions also follow an MRA configuration, a nested MRA (NMRA) is constructed. This NMRA can provide up to $\mathcal {O}(M^{2}N^{2})$ degrees of freedom (DOFs) using only $MN$ physical sensors. In order to fully utilize the increased DOF, we develop a new DOA estimation algorithm, which consists of a dimensional reduction matrix to exploit the data of all virtual elements, a Toeplitz matrix to decorrelate the equivalent coherent sources, and a root-MUSIC method to mitigate the computational workload. This new algorithm can achieve better DOA estimation performance than traditional spatial smoothing MUSIC algorithm with lower computational complexity. Numerical simulation results demonstrate the superiorities of the proposed array geometry in resolving more sources than sensors, DOA estimation performance, and the angular resolution.

Joint DOA and frequency estimation with sub‐Nyquist sampling for more sources than sensors

Multiple antenna systems with sub-Nyquist sampling technique have been applied to jointly estimate frequency and direction-of-arrival of narrowband far-field signals. Unlike the current some techniques, where only one signal can be handled in some certain subband, the authors propose a novel technique to deal with the case of multiple signals. The authors first build a new time-space union model for this case and analyse the identification of the model, and then provide the maximum number of signals which can be classified. The authors also prove that the Cramer-Rao Bound (CRB) of the model is lower than the CRB based on Nyquist sampling. Simulation results verify the identification capacity and show that the proposed algorithm has a superior estimation performance close to CRB especially in the case of the minimum redundancy array case.

Comparing the effect of aperture extension on the peak sidelobe level of sparse arrays.

The reduced number of sensors in sparse arrays create high peak sidelobe levels (PSLs). This letter compares the PSLs of minimum redundancy arrays (MRAs), minimum hole arrays (MHAs), and co-prime sensor arrays (CSAs) (conventionally beamformed and product processed) with fully populated uniform linear arrays (ULAs) as a function of aperture using both numerical simulations and experimental data. This letter finds that PSLs of MRAs, MHAs, and conventionally processed CSAs are much higher than the ULA PSL and are largely insensitive to aperture extension. In contrast, CSA product processing decreases the PSL with increasing aperture, eventually matching the ULA PSL.

Subarray processing with coprime and minimum redundancy arrays

A coprime sensor array (CSA) refers to a non-uniform linear array consisting of two undersampled uniform linear arrays (ULAs) with coprime numbers of elements, respectively. Its beam pattern can be enhanced by multiplying beampatterns of individual subarrays. This technique, called product processing, provides improved resolution because its sum coarray length is extended from the product of beampatterns. Nevertheless, the extended coarray inevitably includes holes that result in higher peak sidelobes than the full ULA case. In this paper, we propose a technique to reduce peak sidelobes through the combination with minimum redundancy arrays (MRAs) and an additional subarray. Two ULAs comprising the coprime array are thinned by the MRA technique, and the reduced number of sensors is then utilized to construct the third subarray. The third array suppresses peak sidelobes by placing nulls at the peak sidelobe locations. The characteristics of the proposed technique are analyzed in terms of white noise gain (WNG) and maximum sidelobe level (MSL). It is demonstrated that the proposed technique can improve the MSL with similar number of sensors, thereby reducing the estimation error in the presence of multiple sources.

Correlation Subspaces: Generalizations and Connection to Difference Coarrays

Direction-of-arrival (DOA) estimation finds applications in many areas of science and engineering. In these applications, sparse arrays such as minimum redundancy arrays, nested arrays, and coprime arrays can be exploited to resolve $O(N^2)$ uncorrelated sources using $N$ physical sensors. Recently, it has been shown that correlation subspaces , which reveal the structure of the covariance matrix, help to improve some existing DOA estimators. However, the bases, the dimension, and other theoretical properties of correlation subspaces remain to be investigated. This paper proposes generalized correlation subspaces in one and multiple dimensions. This leads to new insights into correlation subspaces and DOA estimation with prior knowledge. First, it is shown that the bases and the dimension of correlation subspaces are fundamentally related to difference coarrays , which were previously found to be important in the study of sparse arrays. Furthermore, generalized correlation subspaces can handle certain forms of prior knowledge about source directions. These results allow one to derive a broad class of DOA estimators with improved performance. It is demonstrated through examples that using sparse arrays and generalized correlation subspaces, DOA estimators with source priors exhibit better estimation performance than those without priors, in extreme cases like low SNR and limited snapshots.

Hourglass Arrays and Other Novel 2-D Sparse Arrays With Reduced Mutual Coupling

Linear [one-dimensional (1-D)] sparse arrays such as nested arrays and minimum redundancy arrays have hole-free difference coarrays with $O(N^2)$ virtual sensor elements, where $N$ is the number of physical sensors. The hole-free property makes it easier to perform beamforming and DOA estimation in the coarray domain which behaves like an uniform linear array. The $O(N^2)$ property implies that $O(N^2)$ uncorrelated sources can be identified. For the 2-D case, planar sparse arrays with hole-free coarrays having $O(N^2)$ elements have also been known for a long time. These include billboard arrays, open box arrays (OBA), and 2-D nested arrays. Their merits are similar to those of the 1-D sparse arrays mentioned above, although identifiability claims regarding $O(N^2)$ sources have to be handled with more care in 2-D. This paper introduces new planar sparse arrays with hole-free coarrays having $O(N^2)$ elements just like the OBA, with the additional property that the number of sensor pairs with small spacings such as $\lambda /2$ decreases, reducing the effect of mutual coupling. The new arrays include half-open box arrays, half-open box arrays with two layers, and hourglass arrays. Among these, simulations show that hourglass arrays have the best estimation performance in presence of mutual coupling.

Array Interpolation Using Covariance Matrix Completion of Minimum-Size Virtual Array

Sparse arrays increase aperture and resolution capability for direction-of-arrival estimation. Spatial smoothing step in coarray MUSIC algorithm prevents us from using the full coarray; so, there is a limitation for aperture $N_a$ and degrees of freedom that are determined by minimum redundancy arrays. Interpolation methods can reduce this limitation and make it possible to use the full coarray of partially augmentable arrays. In interpolation methods based on matrix completion techniques, we should complete an $N_a\times N_a$ Toeplitz matrix. For this purpose, a semidefinite programming (SDP) with $O({N_a}^6)$ complexity should be solved. In this letter, we introduce a set that contains arrays that have the equal aperture with original array and have filled coarray. By selecting the array that have minimum number of sensors in this set, we formulate a new structured matrix completion problem. Numerical examples indicate that the proposed structured matrix completion not only decreases the complexity of SDP problem but also, in many instances, increases the estimation accuracy and probability of resolution.

Wideband source enumeration and direction-of-arrival estimation using sparse array periodogram averaging in low snapshot scenarios

Sparse arrays, such as minimum redundancy arrays and coprime arrays, often exploit the second-order statistics of the propagating field to localize more sources than sensors by constructing an augmented covariance matrix (ACM) from the estimated spatial correlation. The source localization performance largely depends on the number of snapshots available, which might be limited in many acoustical environments due to the propagation speed, large array aperture, and non-stationary field. This paper proposes a new approach for wideband source enumeration and direction-of-arrival (DOA) estimation on any sparse array geometry. The proposed algorithm decomposes the wideband signals into multiple disjoint frequency bands, computes the narrowband spatial periodograms and averages them to reinforce the sources’ spatial spectral information. The spatial correlation estimated from the wideband periodogram populates the diagonals of a Hermitian Toeplitz ACM. This ACM then goes through eigenvalue decomposition, where i...

Super Nested Arrays: Linear Sparse Arrays With Reduced Mutual Coupling—Part II: High-Order Extensions

In array processing, mutual coupling between sensors has an adverse effect on the estimation of parameters (e.g., DOA). Sparse arrays such as nested arrays, coprime arrays, and minimum redundancy arrays (MRA) have reduced mutual coupling compared to uniform linear arrays (ULAs). These arrays also have a difference coarray with $O\left({N}^{2}\right)$ virtual elements, where $N$ is the number of physical sensors, and can therefore resolve $O\left({N}^{2}\right)$ uncorrelated source directions. But these well-known sparse arrays have disadvantages: MRAs do not have simple closed-form expressions for the array geometry; coprime arrays have holes in the coarray; and nested arrays contain a dense ULA in the physical array, resulting in significantly higher mutual coupling than coprime arrays and MRAs. In a companion paper, a sparse array configuration called the (second-order) super nested array was introduced, which has many of the advantages of these sparse arrays, while removing most of the disadvantages. Namely, the sensor locations are readily computed for any $N$ (unlike MRAs), and the difference coarray is exactly that of a nested array, and therefore hole-free. At the same time, the mutual coupling is reduced significantly (unlike nested arrays). In this paper, a generalization of super nested arrays is introduced, called the $Q$ th-order super nested array. This has all the properties of the second-order super nested array with the additional advantage that mutual coupling effects are further reduced for $Q > 2$ . Many theoretical properties are proved and simulations are included to demonstrate the superior performance of these arrays.

Super Nested Arrays: Linear Sparse Arrays With Reduced Mutual Coupling—Part I: Fundamentals

In array processing, mutual coupling between sensors has an adverse effect on the estimation of parameters (e.g., DOA). While there are methods to counteract this through appropriate modeling and calibration, they are usually computationally expensive, and sensitive to model mismatch. On the other hand, sparse arrays, such as nested arrays, coprime arrays, and minimum redundancy arrays (MRAs), have reduced mutual coupling compared to uniform linear arrays (ULAs). With $N$ denoting the number of sensors, these sparse arrays offer $O({N}^{2})$ freedoms for source estimation because their difference coarrays have $O({N}^{2})$ -long ULA segments. But these well-known sparse arrays have disadvantages: MRAs do not have simple closed-form expressions for the array geometry; coprime arrays have holes in the coarray; and nested arrays contain a dense ULA in the physical array, resulting in significantly higher mutual coupling than coprime arrays and MRAs. This paper introduces a new array called the super nested array, which has all the good properties of the nested array, and at the same time achieves reduced mutual coupling. There is a systematic procedure to determine sensor locations. For fixed $N$ , the super nested array has the same physical aperture, and the same hole-free coarray as does the nested array. But the number of sensor pairs with small separations ( $\lambda /2,2\times \lambda /2$ , etc.) is significantly reduced. Many theoretical properties are proved and simulations are included to demonstrate the superior performance of these arrays. In the companion paper, a further extension called the $Q$ th-order super nested array is developed, which further reduces mutual coupling.

Mutual coupling effect and compensation in non-uniform arrays for direction-of-arrival estimation

In this paper, we investigate the effect of mutual coupling on direction-of-arrival (DOA) estimation using non-uniform arrays. We compare and contrast the DOA estimation accuracy in the presence of mutual coupling for three different non-uniform array geometries, namely, minimum redundancy arrays (MRAs), nested arrays, and co-prime arrays, and for two antenna types, namely dipole antennas and microstrip antennas. We demonstrate through numerical simulations that the mutual coupling, if unaccounted for, can, in general, lead to performance degradation, with the MRA faring better against mutual coupling than the other two non-uniform structures for both antenna types. We also propose two methods that can compensate for the detrimental effects of mutual coupling, leading to accurate and reliable DOA estimation. Supporting numerical simulation results are provided which show the effectiveness of the proposed compensation methods.

Cramér–Rao bounds for coprime and other sparse arrays, which find more sources than sensors

The Cramér–Rao bound (CRB) offers a lower bound on the variances of unbiased estimates of parameters, e.g., directions of arrival (DOA) in array processing. While there exist landmark papers on the study of the CRB in the context of array processing, the closed-form expressions available in the literature are not easy to use in the context of sparse arrays (such as minimum redundancy arrays (MRAs), nested arrays, or coprime arrays) for which the number of identifiable sources D exceeds the number of sensors N. Under such situations, the existing literature does not spell out the conditions under which the Fisher information matrix is nonsingular, or the condition under which specific closed-form expressions for the CRB remain valid. This paper derives a new expression for the CRB to fill this gap. The conditions for validity of this expression are expressed as the rank condition of a matrix defined based on the difference coarray. The rank condition and the closed-form expression lead to a number of new insights. For example, it is possible to prove the previously known experimental observation that, when there are more sources than sensors, the CRB stagnates to a constant value as the SNR tends to infinity. It is also possible to precisely specify the relation between the number of sensors and the number of uncorrelated sources such that these conditions are valid. In particular, for nested arrays, coprime arrays, and MRAs, the new expressions remain valid for D=O(N2), the precise detail depending on the specific array geometry.

Sparsity-Based Direction Finding of Coherent and Uncorrelated Targets Using Active Nonuniform Arrays

In this letter, direction-of-arrival (DOA) estimation of a mixture of coherent and uncorrelated targets is performed using sparse reconstruction and active nonuniform arrays. The data measurements from multiple transmit and receive elements can be considered as observations from the sum coarray corresponding to the physical transmit/receive arrays. The vectorized covariance matrix of the sum coarray observations emulates the received data at a virtual array whose elements are given by the difference coarray of the sum coarray (DCSC). Sparse reconstruction is used to fully exploit the significantly enhanced degrees-of-freedom offered by the DCSC for DOA estimation. Simulated data from multiple-input multiple-output minimum redundancy arrays and transmit/receive co-prime arrays are used for performance evaluation of the proposed sparsity-based active sensing approach.

Fourth-Order Cumulants Beamforming for Direction of Arrival Estimation

With the rapid development of DOA (direction-of-arrival) estimation methods, the existing beamforming technologies still have the problem of a way to balance high spatial resolution and robust performance. The aperture extension characteristics of fourth-order cumulants is used to raise the spatial resolution of the array in this paper. By using the structure of minimum redundancy array, the calculation of the fourth-order cumulants matrix can be reduced. Simulations of the properties of fourth–order cumulants beamforming method are analyzed in this paper, including the resolution analysis, multi-sources analysis and error analysis. Simulation results show that the fourth-order cumulants beamforming method has a higher spatial resolution than the conventional beamforming method without increasing the number of the array sensors.

Sparse Array Design for Azimuthal Direction-of-Arrival Estimation

We propose a novel approach to the design of array geometries for azimuthal direction-of-arrival (DOA) estimation. The proposed array design is related to that of minimum redundancy arrays, but the array sensors are not required to lie on a uniform grid. Based on the proposed array geometry design, we develop a subspace-based DOA estimation technique, which allows to estimate the DOAs of more sources than sensors, using only second-order statistics of the received data. This DOA estimation technique is related to the covariance augmentation technique, but in contrast to the latter technique, it provides nonambiguous DOA estimates for the full 360° azimuth field-of-view.

Nested Arrays: A Novel Approach to Array Processing With Enhanced Degrees of Freedom

A new array geometry, which is capable of significantly increasing the degrees of freedom of linear arrays, is proposed. This structure is obtained by systematically nesting two or more uniform linear arrays and can provide O(N2) degrees of freedom using only N physical sensors when the second-order statistics of the received data is used. The concept of nesting is shown to be easily extensible to multiple stages and the structure of the optimally nested array is found analytically. It is possible to provide closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors. This cannot be done for existing classes of arrays like minimum redundancy arrays which have been used earlier for detecting more sources than the number of physical sensors. In minimum-input-minimum-output (MIMO) radar, the degrees of freedom are increased by constructing a longer virtual array through active sensing. The method proposed here, however, does not require active sensing and is capable of providing increased degrees of freedom in a completely passive setting. To utilize the degrees of freedom of the nested co-array, a novel spatial smoothing based approach to DOA estimation is also proposed, which does not require the inherent assumptions of the traditional techniques based on fourth-order cumulants or quasi stationary signals. As another potential application of the nested array, a new approach to beamforming based on a nonlinear preprocessing is also introduced, which can effectively utilize the degrees of freedom offered by the nested arrays. The usefulness of all the proposed methods is verified through extensive computer simulations.

Broadband Interference Suppression Performance of Minimum Redundancy Arrays

The broadband interference suppression performance of the minimum redundancy array (MRA) used in the linearly constrained minimum variance (LCMV) beamformer is studied in this paper. The ideal continuous-time, infinite-length filter instead of the realistic finite-order digital filter is assumed in the array processor. The expressions of the power spectrum of the beamformer output in different scenarios are derived to compute the mean output power. Several original properties of the performance of the MRA compared with that of the uniform linear array (ULA) are obtained by mathematical proofs and numerical studies.