Statistical Performance Analysis of Sparse Linear Arrays

Direction-of-arrival (DOA) estimation remains an important topic in array signal processing. With uniform linear arrays (ULAs), traditional subspace-based methods can resolve only up to M-1 sources using M sensors. On the other hand, by exploiting their so-called difference coarray model, sparse linear arrays, such as co-prime and nested arrays, can resolve up to O(M^2) sources using only O(M) sensors. Various new sparse linear array geometries were proposed and many direction-finding algorithms were developed based on sparse linear arrays. However, the statistical performance of such arrays has not been analytically conducted. In this dissertation, we (i) study the asymptotic performance of the MUtiple SIgnal Classification (MUSIC) algorithm utilizing sparse linear arrays, (ii) derive and analyze performance bounds for sparse linear arrays, and (iii) investigate the robustness of sparse linear arrays in the presence of array imperfections. Based on our analytical results, we also propose robust direction-finding algorithms for use when data are missing. We begin by analyzing the performance of two commonly used coarray-based MUSIC direction estimators. Because the coarray model is used, classical derivations no longer apply. By using an alternative eigenvector perturbation analysis approach, we derive a closed-form expression of the asymptotic mean-squared error (MSE) of both estimators. Our expression is computationally efficient compared with the alternative of Monte Carlo simulations. Using this expression, we show that when the source number exceeds the sensor number, the MSE remains strictly positive as the signal-to-noise ratio (SNR) approaches infinity. This finding theoretically explains the unusual saturation behavior of coarray-based MUSIC estimators that had been observed in previous studies. We next derive and analyze the Cramér-Rao bound (CRB) for general sparse linear arrays under the assumption that the sources are uncorrelated. We show that, unlike the classical stochastic CRB, our CRB is applicable even if there are more sources than the number of sensors. We also show that, in such a case, this CRB remains strictly positive definite as the SNR approaches infinity. This unusual behavior imposes a strict lower bound on the variance of unbiased DOA estimators in the underdetermined case. We establish the connection between our CRB and the classical stochastic CRB and show that they are asymptotically equal when the sources are uncorrelated and the SNR is sufficiently high. We investigate the behavior of our CRB for co-prime and nested arrays with a large number of sensors, characterizing the trade-off between the number of spatial samples and the number of temporal samples. Our analytical results on the CRB will benefit future research on optimal sparse array designs. We further analyze the performance of sparse linear arrays by considering sensor location errors. We first introduce the deterministic error model. Based on this model, we derive a closed-form expression of the asymptotic MSE of a commonly used coarray-based MUSIC estimator, the spatial-smoothing based MUSIC (SS-MUSIC). We show that deterministic sensor location errors introduce a constant estimation bias that cannot be mitigated by only increasing the SNR. Our analytical expression also provides a sensitivity measure against sensor location errors for sparse linear arrays. We next extend our derivations to the stochastic error model and analyze the Gaussian case. We also derive the CRB for joint estimation of DOA parameters and deterministic sensor location errors. We show that this CRB is applicable even if there are more sources than the number of sensors. Lastly, we develop robust DOA estimators for cases with missing data. By exploiting the difference coarray structure, we introduce three algorithms to construct an augmented covariance matrix with enhanced degrees of freedom. By applying MUSIC to this augmented covariance matrix, we are able to resolve more sources than sensors. Our method utilizes information from all snapshots and shows improved estimation performance over traditional DOA estimators

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

Theory and Algorithms for Reliable Multimodal Data Analysis, Machine Learning, and Signal Processing

Modern engineering systems collect large volumes of data measurements across diverse sensing modalities. These measurements can naturally be arranged in higher-order arrays of scalars which are commonly referred to as tensors. Tucker decomposition (TD) is a standard method for tensor analysis with applications in diverse fields of science and engineering. Despite its success, TD exhibits severe sensitivity against outliers —i.e., heavily corrupted entries that appear sporadically in modern datasets. We study L1-norm TD (L1-TD), a reformulation of TD that promotes robustness. For 3-way tensors, we show, for the first time, that L1-TD admits an exact solution via combinatorial optimization and present algorithms for its solution. We propose two novel algorithmic frameworks for approximating the exact solution to L1-TD, for general N-way tensors. We propose a novel algorithm for dynamic L1-TD —i.e., efficient and joint analysis of streaming tensors. Principal-Component Analysis (PCA) (a special case of TD) is also outlier responsive. We consider Lp-quasinorm PCA (Lp-PCA) for

Enhanced High-Resolution Imaging through Multiple-Frequency Coarray Augmentation

In imaging, much attention is paid to increasing the resolution capabilities of a system. Increasing resolution allows for high-accuracy source location and the ability to discriminate between two closely-spaced objects. In conventional narrowband techniques, resolution is fundamentally limited by the size of the aperture. For apertures consisting of individual elements, direction-of-arrival techniques allow for high-resolution images of point sources. The main limiting factor on conventional high-resolution imaging is the number of elements in the aperture. For both passive and active imaging, to resolve K point sources/targets, there must be at least K+1 elements receiving radiation. In active imaging, when these targets reflect coherently - the more difficult case in imaging - an additional constraint is that at least K of the elements must also be transmitting radiation to illuminate the targets. For small arrays consisting of only a few elements, this constraint can be problematic. In this dissertation, we focus on improving resolution by using multiple frequencies in both passive and active imaging, especially for small arrays. Using multiple frequencies increases the size of the coarray, which is the true limiting factor for resolution of an imaging system when virtual arrays are considered. For passive imaging, we show that the number of sources that can be resolved is limited only by the bandwidth available for certain types of sources. In active imaging, we develop a frequency-averaging method that permits resolution of K coherent point targets with fewer than K transmitting and receiving elements. These methods are investigated primarily for linear arrays, but planar arrays are also briefly examined. Another resolution improvement method researched in this work is a retransmission scheme for active imaging using classical beamforming techniques. In this method, the coarray is extended not by using multiple frequencies, but by retransmitting the received data back into the scene as a second transmission and processing the returns. It is known that when this method is used to image multiple targets, the resulting image is contaminated by crossterms. We investigate methods to reduce the crossterms

Laboratory and Field Experimental Study of Underwater Inflatable Co-prime Sonar Array (UICSA)

This paper discusses the design and initial testing of a novel hydrophone array system dubbed the Underwater Inflatable Co-prime Sonar Array (UICSA). The UICSA will be a crucial component of an underwater deployable sensing network that can be rapidly deployed using compact autonomous underwater vehicles (AUVs). The UICSA initially is packed in a compact container to fit the payload space of an AUV. After deployment, the UICSA expands to its predetermined full length to acquire sensing data for source localization.  More specifically, the mechanical compression of the UICSA is achieved through a non-rigid array support structure, which consists of flexible inflatable segments between adjoining hydrophones that are folded in order to package the UICSA for deployment. The system exploits compression in hydrophone layouts by utilizing a sparse array configuration, namely the co-prime array since it requires fewer hydrophones than a uniform linear array of the same length to estimate a given number of sources. With two-way compression, the storage, handling, and transportation of the compactly designed UICSA is convenient, particularly for the AUVs with limited payload space. The deployment concept and process are discussed, as well as the various UICSA designs of different support structures are described. A comparison of the various mechanical designs is presented and a novel hybrid-based expansion prototype is documented in detail. Laboratory study results of the UICSA prototype are presented that include water-swollen material tests in a pressurized environment and water tank validation of the inflation process. The UICSA prototype also has been deployed in the Harbor Branch channel to validate the performance, the related field test details and source localization results