Sparsity Promoting Off-grid Methods with Applications in Direction Finding

University of Minnesota Ph.D. dissertation. May 2017. Major: Electrical/Computer Engineering. Advisor: Mostafa Kaveh. 1 computer file (PDF); x, 99 pages.In this dissertation, the problem of directions-of-arrival (DoA) estimation is studied by the compressed sensing application of sparsity-promoting regularization techniques. Compressed sensing can recover high-dimensional signals with a sparse representation from very few linear measurements by nonlinear optimization. By exploiting the sparse representation for the multiple measurement vectors or the spatial covariance matrix of correlated or uncorrelated sources, the DoA estimation problem can be formulated in the framework of sparse signal recovery with high resolution. There are three main topics covered in this dissertation. These topics are recovery methods for the sparse model with structured perturbations, continuous sparse recovery methods in the super-resolution framework, and the off-grid DoA estimation with array self-calibration. These topics are summarized below. For the first topic, structured perturbation in the sparse model is considered. A major limitation of most methods exploiting sparse spectral models for the purpose of estimating directions-of-arrival stems from the fixed model dictionary that is formed by array response vectors over a discrete search grid of possible directions. In general, the array responses to actual DoAs will most likely not be members of such a dictionary. Thus, the sparse spectral signal model with uncertainty of linearized dictionary parameter mismatch is considered, and the dictionary matrix is reformulated into a multiplication of a fixed base dictionary and a sparse matrix. Based on this sparse model, we propose several convex optimization algorithms. However, we are also concerned with the development of a computationally efficient optimization algorithm for off-grid direction finding using a sparse observation model. With an emphasis on designing efficient algorithms, various sparse problem formulations are considered, such as unconstrained formulation, primal-dual formulation, or conic formulation. But, because of the nature of nondifferentiable objective functions, those problems are still challenging to solve in an efficient way. Thus, the Nesterov smoothing methodology is utilized to reformulate nonsmooth functions into smooth ones, and the accelerated proximal gradient algorithm is adopted to solve the smoothed optimization problem. Convergence analysis is conducted as well. The accuracy and efficiency of smoothed sparse recovery methods are demonstrated for the DoA estimation example. In the second topic, estimation of directions-of-arrival in the spatial covariance model is studied. Unlike the compressed sensing methods which discretize the search domain into possible directions on a grid, the theory of super resolution is applied to estimate DoAs in the continuous domain. We reformulate the spatial spectral covariance model into a multiple measurement vectors (MMV)-like model, and propose a block total variation norm minimization approach, which is the analog of Group Lasso in the super-resolution framework and that promotes the group-sparsity. The DoAs can be estimated by solving its dual problem via semidefinite programming. This gridless recovery approach is verified by simulation results for both uncorrelated and correlated source signals. In the last topic, we consider the array calibration issue for DoA estimation, and extend the previously considered single measurement vector model to multiple measurement vectors. By exploiting multiple measurement snapshots, a modified nuclear norm minimization problem is proposed to recover a low-rank matrix with high probability. The definition of linear operator for the MMV model is given, and its corresponding matrix representation is derived so that a reformulated convex optimization problem can be solved numerically. In order to alleviate computational complexity of the method, we use singular value decomposition (SVD) to reduce the problem size. Furthermore, the structured perturbation in the sparse array self-calibration estimation problem is considered as well. The performance and efficiency of the proposed methods are demonstrated by numerical results

Super-resolved localisation in multipath environments

In the last few decades, the localisation problems have been studied extensively. There are still some open issues that remain unresolved. One of the key issues is the efficiency and preciseness of the localisation in presence of non-line-of-sight (NLoS) path. Nevertheless, the NLoS path has a high occurrence in multipath environments, but NLoS bias is viewed as a main factor to severely degrade the localisation performance. The NLoS bias would often result in extra propagation delay and angular bias. Numerous localisation methods have been proposed to deal with NLoS bias in various propagation environments, but they are tailored to some specif ic scenarios due to different prior knowledge requirements, accuracies, computational complexities, and assumptions. To super-resolve the location of mobile device (MD) without prior knowledge, we address the localisation problem by super-resolution technique due to its favourable features, such as working on continuous parameter space, reducing computational cost and good extensibility. Besides the NLoS bias, we consider an extra array directional error which implies the deviation in the orientation of the array placement. The proposed method is able to estimate the locations of MDs and self-calibrate the array directional errors simultaneously. To achieve joint localisation, we directly map MD locations and array directional error to received signals. Then the group sparsity based optimisation is proposed to exploit the geometric consistency that received paths are originating from common MDs. Note that the super-resolution framework cannot be directly applied to our localisation problems. Because the proposed objective function cannot be efficiently solved by semi-definite programming. Typical strategies focus on reducing adverse effect due to the NLoS bias by separating line-of-sight (LoS)/NLoS path or mitigating NLoS effect. The LoS path is well studied for localisation and multiple methods have been proposed in the literature. However, the number of LoS paths are typically limited and the effect of NLoS bias may not always be reduced completely. As a long-standing issue, the suitable solution of using NLoS path is still an open topic for research. Instead of dealing with NLoS bias, we present a novel localisation method that exploits both LoS and NLoS paths in the same manner. The unique feature is avoiding hard decisions on separating LoS and NLoS paths and hence relevant possible error. A grid-free sparse inverse problem is formulated for localisation which avoids error propagation between multiple stages, handles multipath in a unified way, and guarantees a global convergence. Extensive localisation experiments on different propagation environments and localisation systems are presented to illustrate the high performance of the proposed algorithm compared with theoretical analysis. In one of the case studies, single antenna access points (APs) can locate a single antenna MD even when all paths between them are NLoS, which according to the authors’ knowledge is the first time in the literature.Open Acces

Non-uniform Array and Frequency Spacing for Regularization-free Gridless DOA

Gridless direction-of-arrival (DOA) estimation with multiple frequencies can be applied in acoustics source localization problems. We formulate this as an atomic norm minimization (ANM) problem and derive an equivalent regularization-free semi-definite program (SDP) thereby avoiding regularization bias. The DOA is retrieved using a Vandermonde decomposition on the Toeplitz matrix obtained from the solution of the SDP. We also propose a fast SDP program to deal with non-uniform array and frequency spacing. For non-uniform spacings, the Toeplitz structure will not exist, but the DOA is retrieved via irregular Vandermonde decomposition (IVD), and we theoretically guarantee the existence of the IVD. We extend ANM to the multiple measurement vector (MMV) cases and derive its equivalent regularization-free SDP. Using multiple frequencies and the MMV model, we can resolve more sources than the number of physical sensors for a uniform linear array. Numerical results demonstrate that the regularization-free framework is robust to noise and aliasing, and it overcomes the regularization bias

Compact Formulations for Sparse Reconstruction in Fully and Partly Calibrated Sensor Arrays

Sensor array processing is a classical field of signal processing which offers various applications in practice, such as direction of arrival estimation or signal reconstruction, as well as a rich theory, including numerous estimation methods and statistical bounds on the achievable estimation performance. A comparably new field in signal processing is given by sparse signal reconstruction (SSR), which has attracted remarkable interest in the research community during the last years and similarly offers plentiful fields of application. This thesis considers the application of SSR in fully calibrated sensor arrays as well as in partly calibrated sensor arrays. The main contributions are a novel SSR method for application in partly calibrated arrays as well as compact formulations for the SSR problem, where special emphasis is given on exploiting specific structure in the signals as well as in the array topologies

Efficient algorithms and data structures for compressive sensing

Wegen der kontinuierlich anwachsenden Anzahl von Sensoren, und den stetig wachsenden Datenmengen, die jene produzieren, stößt die konventielle Art Signale zu verarbeiten, beruhend auf dem Nyquist-Kriterium, auf immer mehr Hindernisse und Probleme. Die kürzlich entwickelte Theorie des Compressive Sensing (CS) formuliert das Versprechen einige dieser Hindernisse zu beseitigen, indem hier allgemeinere Signalaufnahme und -rekonstruktionsverfahren zum Einsatz kommen können. Dies erlaubt, dass hierbei einzelne Abtastwerte komplexer strukturierte Informationen über das Signal enthalten können als dies bei konventiellem Nyquistsampling der Fall ist. Gleichzeitig verändert sich die Signalrekonstruktion notwendigerweise zu einem nicht-linearen Vorgang und ebenso müssen viele Hardwarekonzepte für praktische Anwendungen neu überdacht werden. Das heißt, dass man zwischen der Menge an Information, die man über Signale gewinnen kann, und dem Aufwand für das Design und Betreiben eines Signalverarbeitungssystems abwägen kann und muss. Die hier vorgestellte Arbeit trägt dazu bei, dass bei diesem Abwägen CS mehr begünstigt werden kann, indem neue Resultate vorgestellt werden, die es erlauben, dass CS einfacher in der Praxis Anwendung finden kann, wobei die zu erwartende Leistungsfähigkeit des Systems theoretisch fundiert ist. Beispielsweise spielt das Konzept der Sparsity eine zentrale Rolle, weshalb diese Arbeit eine Methode präsentiert, womit der Grad der Sparsity eines Vektors mittels einer einzelnen Beobachtung geschätzt werden kann. Wir zeigen auf, dass dieser Ansatz für Sparsity Order Estimation zu einem niedrigeren Rekonstruktionsfehler führt, wenn man diesen mit einer Rekonstruktion vergleicht, welcher die Sparsity des Vektors unbekannt ist. Um die Modellierung von Signalen und deren Rekonstruktion effizienter zu gestalten, stellen wir das Konzept von der matrixfreien Darstellung linearer Operatoren vor. Für die einfachere Anwendung dieser Darstellung präsentieren wir eine freie Softwarearchitektur und demonstrieren deren Vorzüge, wenn sie für die Rekonstruktion in einem CS-System genutzt wird. Konkret wird der Nutzen dieser Bibliothek, einerseits für das Ermitteln von Defektpositionen in Prüfkörpern mittels Ultraschall, und andererseits für das Schätzen von Streuern in einem Funkkanal aus Ultrabreitbanddaten, demonstriert. Darüber hinaus stellen wir für die Verarbeitung der Ultraschalldaten eine Rekonstruktionspipeline vor, welche Daten verarbeitet, die im Frequenzbereich Unterabtastung erfahren haben. Wir beschreiben effiziente Algorithmen, die bei der Modellierung und der Rekonstruktion zum Einsatz kommen und wir leiten asymptotische Resultate für die benötigte Anzahl von Messwerten, sowie die zu erwartenden Lokalisierungsgenauigkeiten der Defekte her. Wir zeigen auf, dass das vorgestellte System starke Kompression zulässt, ohne die Bildgebung und Defektlokalisierung maßgeblich zu beeinträchtigen. Für die Lokalisierung von Streuern mittels Ultrabreitbandradaren stellen wir ein CS-System vor, welches auf einem Random Demodulators basiert. Im Vergleich zu existierenden Messverfahren ist die hieraus resultierende Schätzung der Kanalimpulsantwort robuster gegen die Effekte von zeitvarianten Funkkanälen. Um den inhärenten Modellfehler, den gitterbasiertes CS begehen muss, zu beseitigen, zeigen wir auf wie Atomic Norm Minimierung es erlaubt ohne die Einschränkung auf ein endliches und diskretes Gitter R-dimensionale spektrale Komponenten aus komprimierten Beobachtungen zu schätzen. Hierzu leiten wir eine R-dimensionale Variante des ADMM her, welcher dazu in der Lage ist die Signalkovarianz in diesem allgemeinen Szenario zu schätzen. Weiterhin zeigen wir, wie dieser Ansatz zur Richtungsschätzung mit realistischen Antennenarraygeometrien genutzt werden kann. In diesem Zusammenhang präsentieren wir auch eine Methode, welche mittels Stochastic gradient descent Messmatrizen ermitteln kann, die sich gut für Parameterschätzung eignen. Die hieraus resultierenden Kompressionsverfahren haben die Eigenschaft, dass die Schätzgenauigkeit über den gesamten Parameterraum ein möglichst uniformes Verhalten zeigt. Zuletzt zeigen wir auf, dass die Kombination des ADMM und des Stochastic Gradient descent das Design eines CS-Systems ermöglicht, welches in diesem gitterfreien Szenario wünschenswerte Eigenschaften hat.Along with the ever increasing number of sensors, which are also generating rapidly growing amounts of data, the traditional paradigm of sampling adhering the Nyquist criterion is facing an equally increasing number of obstacles. The rather recent theory of Compressive Sensing (CS) promises to alleviate some of these drawbacks by proposing to generalize the sampling and reconstruction schemes such that the acquired samples can contain more complex information about the signal than Nyquist samples. The proposed measurement process is more complex and the reconstruction algorithms necessarily need to be nonlinear. Additionally, the hardware design process needs to be revisited as well in order to account for this new acquisition scheme. Hence, one can identify a trade-off between information that is contained in individual samples of a signal and effort during development and operation of the sensing system. This thesis addresses the necessary steps to shift the mentioned trade-off more to the favor of CS. We do so by providing new results that make CS easier to deploy in practice while also maintaining the performance indicated by theoretical results. The sparsity order of a signal plays a central role in any CS system. Hence, we present a method to estimate this crucial quantity prior to recovery from a single snapshot. As we show, this proposed Sparsity Order Estimation method allows to improve the reconstruction error compared to an unguided reconstruction. During the development of the theory we notice that the matrix-free view on the involved linear mappings offers a lot of possibilities to render the reconstruction and modeling stage much more efficient. Hence, we present an open source software architecture to construct these matrix-free representations and showcase its ease of use and performance when used for sparse recovery to detect defects from ultrasound data as well as estimating scatterers in a radio channel using ultra-wideband impulse responses. For the former of these two applications, we present a complete reconstruction pipeline when the ultrasound data is compressed by means of sub-sampling in the frequency domain. Here, we present the algorithms for the forward model, the reconstruction stage and we give asymptotic bounds for the number of measurements and the expected reconstruction error. We show that our proposed system allows significant compression levels without substantially deteriorating the imaging quality. For the second application, we develop a sampling scheme to acquire the channel Impulse Response (IR) based on a Random Demodulator that allows to capture enough information in the recorded samples to reliably estimate the IR when exploiting sparsity. Compared to the state of the art, this in turn allows to improve the robustness to the effects of time-variant radar channels while also outperforming state of the art methods based on Nyquist sampling in terms of reconstruction error. In order to circumvent the inherent model mismatch of early grid-based compressive sensing theory, we make use of the Atomic Norm Minimization framework and show how it can be used for the estimation of the signal covariance with R-dimensional parameters from multiple compressive snapshots. To this end, we derive a variant of the ADMM that can estimate this covariance in a very general setting and we show how to use this for direction finding with realistic antenna geometries. In this context we also present a method based on a Stochastic gradient descent iteration scheme to find compression schemes that are well suited for parameter estimation, since the resulting sub-sampling has a uniform effect on the whole parameter space. Finally, we show numerically that the combination of these two approaches yields a well performing grid-free CS pipeline

Direction Finding in Partly Calibrated Arrays Exploiting the Whole Array Aperture

We consider the problem of direction finding using partly calibrated arrays, a distributed subarray with position errors between subarrays. The key challenge is to enhance angular resolution in the presence of position errors. To achieve this goal, existing algorithms, such as subspace separation and sparse recovery, have to rely on multiple snapshots, which increases the burden of data transmission and the processing delay. Therefore, we aim to enhance angular resolution using only a single snapshot. To this end, we exploit the orthogonality of the signals of partly calibrated arrays. Particularly, we transform the signal model into a special multiple-measurement model, show that there is approximate orthogonality between the source signals in this model, and then use blind source separation to exploit the orthogonality. Simulation and experiment results both verify that our proposed algorithm achieves high angular resolution as distributed arrays without position errors, inversely proportional to the whole array aperture