Spatial Compressive Sensing for MIMO Radar

We study compressive sensing in the spatial domain to achieve target localization, specifically direction of arrival (DOA), using multiple-input multiple-output (MIMO) radar. A sparse localization framework is proposed for a MIMO array in which transmit and receive elements are placed at random. This allows for a dramatic reduction in the number of elements needed, while still attaining performance comparable to that of a filled (Nyquist) array. By leveraging properties of structured random matrices, we develop a bound on the coherence of the resulting measurement matrix, and obtain conditions under which the measurement matrix satisfies the so-called isotropy property. The coherence and isotropy concepts are used to establish uniform and non-uniform recovery guarantees within the proposed spatial compressive sensing framework. In particular, we show that non-uniform recovery is guaranteed if the product of the number of transmit and receive elements, MN (which is also the number of degrees of freedom), scales with K(log(G))^2, where K is the number of targets and G is proportional to the array aperture and determines the angle resolution. In contrast with a filled virtual MIMO array where the product MN scales linearly with G, the logarithmic dependence on G in the proposed framework supports the high-resolution provided by the virtual array aperture while using a small number of MIMO radar elements. In the numerical results we show that, in the proposed framework, compressive sensing recovery algorithms are capable of better performance than classical methods, such as beamforming and MUSIC.Comment: To appear in IEEE Transactions on Signal Processin

Global optimization methods for localization in compressive sensing

The dissertation discusses compressive sensing and its applications to localization in multiple-input multiple-output (MIMO) radars. Compressive sensing is a paradigm at the intersection between signal processing and optimization. It advocates the sensing of “sparse” signals (i.e., represented using just a few terms from a basis expansion) by using a sampling rate much lower than that required by the Nyquist-Shannon sampling theorem (i.e., twice the highest frequency present in the signal of interest). Low-rate sampling reduces implementation’s constraints and translates into cost savings due to fewer measurements required. This is particularly true in localization applications when the number of measurements is commensurate to antenna elements. The theory of compressive sensing provides precise guidance on how the measurements should be acquired, and which optimization algorithm should be used for signal recovery. The first part of the dissertation addresses the application of compressive sensing for localization in the spatial domain, specifically direction of arrival (DOA), using MIMO radar. A sparse localization framework is proposed for a MIMO array in which transmit and receive elements are placed at random. This allows for a dramatic reduction in the number of elements needed, while still attaining performance comparable to that of a filled (Nyquist) array. By leveraging properties of structured random matrices, a bound on the coherence of the resulting measurement matrix is obtained, and conditions under which the measurement matrix satisfies the so-called isotropy property are detailed. The coherence and isotropy concepts are used to establish uniform and non-uniform recovery guarantees within the proposed spatial compressive sensing framework. In particular, it is shown that non-uniform recovery is guaranteed if the product of the number of transmit and receive elements, MN (which is also the number of degrees of freedom), scales with K (log G)2, where K is the number of targets and G is proportional to the array aperture and determines the angle resolution. In contrast with a filled virtual MIMO array where the product MN scales linearly with G, the logarithmic dependence on G in the proposed framework supports the high-resolution provided by the virtual array aperture while using a small number of MIMO radar elements. The second part of the dissertation focuses on the sparse recovery problem at the heart of compressive sensing. An algorithm, dubbed Multi-Branch Matching Pursuit (MBMP), is presented which combines three different paradigms: being a greedy method, it performs iterative signal support estimation; as a rank-aware method, it is able to exploit signal subspace information when multiple snapshots are available; and, as its name foretells, it possesses a multi-branch structure which allows it to trade-off performance (e.g., measurements) for computational complexity. A sufficient condition under which MBMP can recover a sparse signal is obtained. This condition, named MB-coherence, is met when the columns of the measurement matrix are sufficiently “incoherent” and when the signal-to-noise ratio is sufficiently high. The condition shows that successful recovery with MBMP is guaranteed for dictionaries which do not satisfy previously known conditions (e.g., coherence, cumulative coherence, or the Hanman relaxed coherence). Finally, by leveraging the MBMP algorithm, a framework for target detection from a set of compressive sensing radar measurements is established. The proposed framework does not require any prior information about the targets’ scene, and it is competitive with respect to state-of-the-art detection compressive sensing algorithms

The University Defence Research Collaboration In Signal Processing

This chapter describes the development of algorithms for automatic detection of anomalies from multi-dimensional, undersampled and incomplete datasets. The challenge in this work is to identify and classify behaviours as normal or abnormal, safe or threatening, from an irregular and often heterogeneous sensor network. Many defence and civilian applications can be modelled as complex networks of interconnected nodes with unknown or uncertain spatio-temporal relations. The behavior of such heterogeneous networks can exhibit dynamic properties, reflecting evolution in both network structure (new nodes appearing and existing nodes disappearing), as well as inter-node relations. The UDRC work has addressed not only the detection of anomalies, but also the identification of their nature and their statistical characteristics. Normal patterns and changes in behavior have been incorporated to provide an acceptable balance between true positive rate, false positive rate, performance and computational cost. Data quality measures have been used to ensure the models of normality are not corrupted by unreliable and ambiguous data. The context for the activity of each node in complex networks offers an even more efficient anomaly detection mechanism. This has allowed the development of efficient approaches which not only detect anomalies but which also go on to classify their behaviour

Communications-Inspired Projection Design with Application to Compressive Sensing

We consider the recovery of an underlying signal x \in C^m based on projection measurements of the form y=Mx+w, where y \in C^l and w is measurement noise; we are interested in the case l < m. It is assumed that the signal model p(x) is known, and w CN(w;0,S_w), for known S_W. The objective is to design a projection matrix M \in C^(l x m) to maximize key information-theoretic quantities with operational significance, including the mutual information between the signal and the projections I(x;y) or the Renyi entropy of the projections h_a(y) (Shannon entropy is a special case). By capitalizing on explicit characterizations of the gradients of the information measures with respect to the projections matrix, where we also partially extend the well-known results of Palomar and Verdu from the mutual information to the Renyi entropy domain, we unveil the key operations carried out by the optimal projections designs: mode exposure and mode alignment. Experiments are considered for the case of compressive sensing (CS) applied to imagery. In this context, we provide a demonstration of the performance improvement possible through the application of the novel projection designs in relation to conventional ones, as well as justification for a fast online projections design method with which state-of-the-art adaptive CS signal recovery is achieved.Comment: 25 pages, 7 figures, parts of material published in IEEE ICASSP 2012, submitted to SIIM

Structured random measurements in signal processing

Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This article surveys the theory in three scenarios: compressed sensing (sparse recovery), low rank matrix recovery, and phaseless estimation. The random measurement matrices to be considered include random partial Fourier matrices, partial random circulant matrices (subsampled convolutions), matrix completion, and phase estimation from magnitudes of Fourier type measurements. The article concludes with a brief discussion of the mathematical techniques for the analysis of such structured random measurements.Comment: 22 pages, 2 figure

Adaptive MIMO Radar for Target Detection, Estimation, and Tracking

We develop and analyze signal processing algorithms to detect, estimate, and track targets using multiple-input multiple-output: MIMO) radar systems. MIMO radar systems have attracted much attention in the recent past due to the additional degrees of freedom they offer. They are commonly used in two different antenna configurations: widely-separated: distributed) and colocated. Distributed MIMO radar exploits spatial diversity by utilizing multiple uncorrelated looks at the target. Colocated MIMO radar systems offer performance improvement by exploiting waveform diversity. Each antenna has the freedom to transmit a waveform that is different from the waveforms of the other transmitters. First, we propose a radar system that combines the advantages of distributed MIMO radar and fully polarimetric radar. We develop the signal model for this system and analyze the performance of the optimal Neyman-Pearson detector by obtaining approximate expressions for the probabilities of detection and false alarm. Using these expressions, we adaptively design the transmit waveform polarizations that optimize the target detection performance. Conventional radar design approaches do not consider the goal of the target itself, which always tries to reduce its detectability. We propose to incorporate this knowledge about the goal of the target while solving the polarimetric MIMO radar design problem by formulating it as a game between the target and the radar design engineer. Unlike conventional methods, this game-theoretic design does not require target parameter estimation from large amounts of training data. Our approach is generic and can be applied to other radar design problems also. Next, we propose a distributed MIMO radar system that employs monopulse processing, and develop an algorithm for tracking a moving target using this system. We electronically generate two beams at each receiver and use them for computing the local estimates. Later, we efficiently combine the information present in these local estimates, using the instantaneous signal energies at each receiver to keep track of the target. Finally, we develop multiple-target estimation algorithms for both distributed and colocated MIMO radar by exploiting the inherent sparsity on the delay-Doppler plane. We propose a new performance metric that naturally fits into this multiple target scenario and develop an adaptive optimal energy allocation mechanism. We employ compressive sensing to perform accurate estimation from far fewer samples than the Nyquist rate. For colocated MIMO radar, we transmit frequency-hopping codes to exploit the frequency diversity. We derive an analytical expression for the block coherence measure of the dictionary matrix and design an optimal code matrix using this expression. Additionally, we also transmit ultra wideband noise waveforms that improve the system resolution and provide a low probability of intercept: LPI)