Two-Dimensional Sparse Arrays with Hole-Free Coarray and Reduced Mutual Coupling

Two-dimensional sparse arrays with hole-free difference coarrays, like billboard arrays and open box arrays, can identify O(N^2) uncorrelated source directions (DOA) using N sensors. These arrays contain some dense ULA segments, leading to many sensor pairs separated by λ/2. The DOA estimation performance often suffers degradation due to mutual coupling between such closely-spaced sensor pairs. This paper introduces a new 2D array called the half open box array. For a given N, this array has the same hole-free coarray as an open box array. At the same time, the number of sensor pairs with small separation is significantly reduced

Sparse Array Design via Fractal Geometries

Sparse sensor arrays have attracted considerable attention in various fields such as radar, array processing, ultrasound imaging and communications. In the context of correlation-based processing, such arrays enable to resolve more uncorrelated sources than physical sensors. This property of sparse arrays stems from the size of their difference coarrays, defined as the differences of element locations. Thus, the design of sparse arrays with large difference coarrays is of great interest. In addition, other array properties such as symmetry, robustness and array economy are important in different applications. Numerous studies have proposed diverse sparse geometries, focusing on certain properties while lacking others. Incorporating multiple properties into the design task leads to combinatorial problems which are generally NP-hard. For small arrays these optimization problems can be solved by brute force, however, in large scale they become intractable. In this paper, we propose a scalable systematic way to design large sparse arrays considering multiple properties. To that end, we introduce a fractal array design in which a generator array is recursively expanded according to its difference coarray. Our main result states that for an appropriate choice of the generator such fractal arrays exhibit large difference coarrays. Furthermore, we show that the fractal arrays inherit their properties from their generators. Thus, a small generator can be optimized according to desired requirements and then expanded to create a fractal array which meets the same criteria. This approach paves the way to efficient design of large arrays of hundreds or thousands of elements with specific properties.Comment: 16 pages, 9 figures, 1 Tabl

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.Comment: 4+1 pages, 5 figures, 1 tabl

Sparse Array Signal Processing: New Array Geometries, Parameter Estimation, and Theoretical Analysis

Array signal processing focuses on an array of sensors receiving the incoming waveforms in the environment, from which source information, such as directions of arrival (DOA), signal power, amplitude, polarization, and velocity, can be estimated. This topic finds ubiquitous applications in radar, astronomy, tomography, imaging, and communications. In these applications, sparse arrays have recently attracted considerable attention, since they are capable of resolving O(N2) uncorrelated source directions with N physical sensors. This is unlike the uniform linear arrays (ULA), which identify at most N-1 uncorrelated sources with N sensors. These sparse arrays include minimum redundancy arrays (MRA), nested arrays, and coprime arrays. All these arrays have an O(N2)-long central ULA segment in the difference coarray, which is defined as the set of differences between sensor locations. This O(N2) property makes it possible to resolve O(N2) uncorrelated sources, using only N physical sensors. The main contribution of this thesis is to provide a new direction for array geometry and performance analysis of sparse arrays in the presence of nonidealities. The first part of this thesis focuses on designing novel array geometries that are robust to effects of mutual coupling. It is known that, mutual coupling between sensors has an adverse effect on the estimation of 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 MRA, nested arrays, and coprime arrays, have reduced mutual coupling compared to ULA, but all of these have their own disadvantages. This thesis introduces a new array called the super nested array, which has many of the good properties of the nested array, and at the same time achieves reduced mutual coupling. Many theoretical properties are proved and simulations are included to demonstrate the superior performance of super nested arrays in the presence of mutual coupling. Two-dimensional planar sparse arrays with large difference coarrays have also been known for a long time. These include billboard arrays, open box arrays (OBA), and 2D nested arrays. However, all of them have considerable mutual coupling. This thesis proposes new planar sparse arrays with the same large difference coarrays as the OBA, but with reduced mutual coupling. The new arrays include half open box arrays (HOBA), half open box arrays with two layers (HOBA-2), and hourglass arrays. Among these, simulations show that hourglass arrays have the best estimation performance in presence of mutual coupling. The second part of this thesis analyzes the performance of sparse arrays from a theoretical perspective. We first study the Cramér-Rao bound (CRB) for sparse arrays, which poses a lower bound on the variances of unbiased DOA estimators. 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 applicable in the context of sparse arrays for which the number of identifiable sources exceeds the number of sensors. This thesis derives a new expression for the CRB to fill this gap. Based on the proposed CRB expression, 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 sources could be resolved. 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 thesis 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 empirically known that the coarray structure is susceptible to sensor failures, and the reliability of sparse arrays remains a significant but challenging topic for investigation. This thesis advances a general theory for quantifying such robustness, by studying the effect of sensor failure on the difference coarray. We first present the (k-)essentialness property, which characterizes the combinations of the faulty sensors that shrink the difference coarray. Based on this, the notion of (k-)fragility is proposed to quantify the reliability of sparse arrays with faulty sensors, along with comprehensive studies of their properties. These novel concepts provide quite a few insights into the interplay between the array geometry and its robustness. For instance, for the same number of sensors, it can be proved that ULA is more robust than the coprime array, and the coprime array is more robust than the nested array. Rigorous development of these ideas leads to expressions for the probability of coarray failure, as a function of the probability of sensor failure. The thesis concludes with some remarks on future directions and open problems.</p

Augmented Multi-Subarray Dilated Nested Array with Enhanced Degrees of Freedom and Reduced Mutual Coupling

Sparse linear arrays (SLAs) can be designed in a systematic way, with the ability for underdetermined DOA estimation where a greater number of sources can be detected than that of sensors. In this paper, as the first stage, a new systematic design named multi-subarray dilated nested array (MDNA), whose difference co-array (DCA) can be proved to be hole-free, is firstly proposed by introducing a sparse ULA and multiple identical dense ULAs with appropriate sub-ULA spacings. The MDNA will degenerate into the nested array under specific conditions, and the uniform degrees of freedom (uDOFs) of MDNA is larger than that of its parent nested array. On the basis of MDNA, to reduce the mutual coupling effect, an augmented multi-subarray dilated nested array (AMDNA) is constructed by migrating some elements of the dense segments of MDNA, without reducing the number of uDOFs. Several theoretical properties of the proposed array structures are proved, and simulation results are provided to demonstrate the effectiveness and superiority of the proposed AMDNA over some existing sparse arrays

Novel Algorithms for Analyzing the Robustness of Difference Coarrays to Sensor Failures

Sparse arrays have drawn attention because they can identify O(N²) uncorrelated source directions using N physical sensors, whereasuniform linear arrays (ULA) find at most N−1 sources. The main reason is that the difference coarray, defined as the set of differences between sensor locations, has size of O(N²) for some sparse arrays. However, the performance of sparse arrays may degrade significantly under sensor failures. In the literature, the k-essentialness property characterizes the patterns of k sensor failures that change the difference coarray. Based on this concept, the k-essential family, the k-fragility, and the k-essential Sperner family provide insights into the robustness of arrays. This paper proposes novel algorithms for computing these attributes. The first algorithm computes the k-essential Sperner family without enumerating all possible k-essential subarrays. With this information, the second algorithm finds the k-essential family first and the k-fragility next. These algorithms are applicable to any 1-D array. However, for robust array design, fast computation for the k-fragility is preferred. For this reason, a simple expression associated with the k-essential Sperner family is proposed to be a tighter lower bound for the k-fragility than the previous result. Numerical examples validate the proposed algorithms and the presented lower bound