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

Design of Nonredundant Sparse Planar Arrays With Reduced Mutual Coupling

Most existing sparse planar arrays cannot fully realize their potential in terms of the degrees-of-freedom (DOFs) due to redundancies in their co-array generation. Meanwhile, the small interelement spacing in conventional planar arrays may cause serious mutual coupling effects. In this article, a series of designs for nonredundant sparse planar arrays in different application scenarios are proposed. They can obtain the maximum possible DOFs under the constraints of area and the number of array elements. We first present the rule for nonredundancy design for planar arrays, which is the basic criterion for the following optimization problems. According to generalized disjunctive programming, we establish systematic solutions for array designs by two mixed-integer linear programming optimization approaches. Then, three classes of nonredundant planar arrays are designed to achieve minimum area, predetermined area, and reduced mutual coupling, respectively. In particular, the nonredundant planar arrays with reduced mutual coupling can be designed to both avoid small interelement spacing and obtain the minimum area, which makes them more robust to mutual coupling conditions. Simulation results are provided to demonstrate the superiority of the proposed planar array configurations for direction-of-arrival estimation

A New Restriction on Low-Redundancy Restricted Array and Its Good Solutions

In array signal processing, a fundamental problem is to design a sensor array with low-redundancy and reduced mutual coupling, which are the main features to improve the performance of direction-of-arrival (DOA) estimation. For a $N$-sensor array with aperture $L$, it is called low-redundancy (LR) if the ratio $R=N(N-1)/(2L)$ is approaching the Leech's bound $1.217\leq R_{opt}\leq 1.674$ for $N\rightarrow\infty$; and the mutual coupling is often reduced by decreasing the numbers of sensor pairs with the first three smallest inter-spacings, denoted as $\omega(a)$ with $a\in\{1,2,3\}$. Many works have been done to construct large LRAs, whose spacing structures all coincide with a common pattern ${\mathbb D}=\{a_1,a_2,\ldots,a_{s_1},c^\ell,b_1,b_2,\ldots,b_{s_2}\}$ with the restriction $s_1+s_2=c-1$. Here $a_i,b_j,c$ denote the spacing between adjacent sensors, and $c$ is the largest one. The objective of this paper is to find some new arrays with lower redundancy ratio or lower mutual coupling compared with known arrays. In order to do this, we give a new restriction for ${\mathbb D}$ to be $s_1+s_2=c$ , and obtain 2 classes of $(4r+3)$-type arrays, 2 classes of $(4r+1)$-type arrays, and 1 class of $(4r)$-type arrays for any $N\geq18$. Here the $(4r+i)$-Type means that $c\equiv i\pmod4$. Notably, compared with known arrays with the same type, one of our new $(4r+1)$-type array and the new $(4r)$-type array all achieves the lowest mutual coupling, and their uDOFs are at most 4 less for any $N\geq18$; compared with SNA and MISC arrays, the new $(4r)$-type array has a significant reduction in both redundancy ratio and mutual coupling. We should emphasize that the new $(4r)$-type array in this paper is the first class of arrays achieving $R<1.5$ and $\omega(1)=1$ for any $N\geq18$

Array Signal Processing Based on Traditional and Sparse Arrays

Array signal processing is based on using an array of sensors to receive the impinging signals. The received data is either spatially filtered to focus the signals from a desired direction or it may be used for estimating a parameter of source signal like direction of arrival (DOA), polarization and source power. Spatial filtering also known as beamforming and DOA estimation are integral parts of array signal processing and this thesis is aimed at solving some key probems related to these two areas. Wideband beamforming holds numerous applications in the bandwidth hungry data traffic of present day world. Several techniques exist to design fixed wideband beamformers based on traditional arrays like uniform linear array (ULA). Among these techniques, least squares based eigenfilter method is a key technique which has been used extensively in filter and wideband beamformer design. The first contribution of this thesis comes in the form of critically analyzing the standard eigenfilter method where a serious flaw in the design formulation is highlighted which generates inconsistent design performance, and an additional constraint is added to stabilize the achieved design. Simulation results show the validity and significance of the proposed method. Traditional arrays based on ULAs have limited applications in array signal processing due to the large number of sensors required and this problem has been addressed by the application of sparse arrays. Sparse arrays have been exploited from the perspective of their difference co-array structures which provide significantly higher number of degrees of freedoms (DOFs) compared to ULAs for the same number of sensors. These DOFs (consecutive and unique lags) are utilized in the application of DOA estimation with the help of difference co-array based DOA estimators. Several types of sparse arrays include minimum redundancy array (MRA), minimum hole array (MHA), nested array, prototype coprime array, conventional coprime array, coprime array with compressed interelement spacing (CACIS), coprime array with displaced subarrays (CADiS) and super nested array. As a second contribution of this thesis, a new sparse array termed thinned coprime array (TCA) is proposed which holds all the properties of a conventional coprime array but with \ceil*{\frac{M}{2}} fewer sensors where $M$ is the number of sensors of a subarray in the conventional structure. TCA possesses improved level of sparsity and is robust against mutual coupling compared to other sparse arrays. In addition, TCA holds higher number of DOFs utilizable for DOA estimation using variety of methods. TCA also shows lower estimation error compared to super nested arrays and MRA with increasing array size. Although TCA holds numerous desirable features, the number of unique lags offered by TCA are close to the sparsest CADiS and nested array and significantly lower than MRA which limits the estimation error performance offered by TCA through (compressive sensing) CS-based methods. In this direction, the structure of TCA is studied to explore the possibility of an array which can provide significantly higher number of unique lags with improved sparsity for a given number of sensors. The result of this investigation is the third contribution of this thesis in the form of a new sparse array, displaced thinned coprime array with additional sensor (DiTCAAS), which is based on a displaced version of TCA. The displacement of the subarrays generates an increase in the unique lags but the minimum spacing between the sensors becomes an integer multiple of half wavelength. To avoid spatial aliasing, an additional sensor is added at half wavelength from one of the sensors of the displaced subarray. The proposed placement of the additional sensor generates significantly higher number of unique lags for DiTCAAS, even more than the DOFs provided by MRA. Due to its improved sparsity and higher number of unique lags, DiTCAAS generates the lowest estimation error and robustness against heavy mutual coupling compared to super nested arrays, MRA, TCA and sparse CADiS with CS-based DOA estimation