Progressive developments in computing and sensor technologies during the past decades have enabled the formulation of increasingly advanced problems in statistical inference and signal processing. The thesis is concerned with statistical estimation methods, and is divided into three parts with focus on two different areas: sensor fusion and sparse signal processing. The first part introduces the well-established Bayesian, Fisherian and least-squares estimation frameworks, and derives new estimators. Specifically, the Bayesian framework is applied in two different classes of estimation problems: scenarios in which (i) the signal covariances themselves are subject to uncertainties, and (ii) distance bounds are used as side information. Applications include localization, tracking and channel estimation. The second part is concerned with the extraction of useful information from multiple sensors by exploiting their joint properties. Two sensor configurations are considered here: (i) a monocular camera and an inertial measurement unit, and (ii) an array of passive receivers. New estimators are developed with applications that include inertial navigation, source localization and multiple waveform estimation. The third part is concerned with signals that have sparse representations. Two problems are considered: (i) spectral estimation of signals with power concentrated to a small number of frequencies,and (ii) estimation of sparse signals that are observed by few samples, including scenarios in which they are linearly underdetermined. New estimators are developed with applications that include spectral analysis, magnetic resonance imaging and array processing. / <p>QC 20130426</p>

Optimization theories and algorithms are used to efficiently find optimal solutions under constraints. In the era of “Big Data”, the amount of data is skyrocketing,and this overwhelms conventional techniques used to solve large scale and distributed optimization problems. By taking advantage of structural information in data representations, this thesis offers convex and non-convex optimization solutions to various large scale optimization problems such as super-resolution, sparse signal processing,hypothesis testing, machine learning, and treatment planning for brachytherapy. Super-resolution: Super-resolution aims to recover a signal expressed as a sum of a few Dirac delta functions in the time domain from measurements in the frequency domain. The challenge is that the possible locations of the delta functions are in the continuous domain [0,1). To enhance recovery performance, we considered deterministic and probabilistic prior information for the locations of the delta functions and provided novel semidefinite programming formulations under the information. We also proposed block iterative reweighted methods to improve recovery performance without prior information. We further considered phaseless measurements, motivated by applications in optic microscopy and x-ray crystallography. By using the lifting method and introducing the squared atomic norm minimization, we can achieve super-resolution using only low frequency magnitude information. Finally, we proposed non-convex algorithms using structured matrix completion. Sparse signal processing: L1 minimization is well known for promoting sparse structures in recovered signals. The Null Space Condition (NSC) for L1 minimization is a necessary and sufficient condition on sensing matrices such that a sparse signal can be uniquely recovered via L1 minimization. However, verifying NSC is a non-convex problem and known to be NP-hard. We proposed enumeration-based polynomial-time algorithms to provide performance bounds on NSC, and efficient algorithms to verify NSC precisely by using the branch and bound method. Hypothesis testing: Recovering statistical structures of random variables is important in some applications such as cognitive radio. Our goal is distinguishing two different types of random variables among n>>1 random variables. Distinguishing them via experiments for each random variable one by one takes lots of time and efforts. Hence, we proposed hypothesis testing using mixed measurements to reduce sample complexity. We also designed efficient algorithms to solve large scale problems. Machine learning: When feature data are stored in a tree structured network having time delay in communication, quickly finding an optimal solution to the regularized loss minimization is challenging. In this scenario, we studied a communication-efficient stochastic dual coordinate ascent and its convergence analysis. Treatment planning: In the Rotating-Shield Brachytherapy (RSBT) for cancer treatment, there is a compelling need to quickly obtain optimal treatment plans to enable clinical usage. However, due to the degree of freedom in RSBT, finding optimal treatment planning is difficult. For this, we designed a first order dose optimization method based on the alternating direction method of multipliers, and reduced the execution time around 18 times compared to the previous research.

The first part of this thesis develops methods for UWB communication. To meet the stringent regulatory body constraints, the physical layer signaling technique of the UWB transceiver should be optimally designed. We propose two signaling schemes which are variants of pulse position modulation (PPM) signaling for impulse radio (IR) UWB communication. We also discuss the detectors for the signaling schemes and evaluate the performance of these detectors.  IR-UWB can be used for precise range measurements as it provides a very high time resolution. This enables accurate time of arrival (TOA) estimations from which precise range values can be derived. We propose methods which use range information to arrive at optimal schedules for an all-to-all broadcast problem. Results indicate that throughput can be increased on average by three to ten times for typical network configurations compared to the traditional methods. Next, we discuss hypothesis testing in the context of UWB transceivers. We show that, when multiple detector outputs from a hardware platform are available, fusing the results from them can yield better performance in hypothesis testing than relying on a single detector output. In the second part of this thesis, the emphasis is placed on localization and joint estimation of location and communication parameters. Here, we focus on estimating the TOA of the signal. The wide bandwidth of the UWB signal requires high speed analog to digital converts (ADC) which makes the cost of the digital transceivers prohibitively high. To address this problem, we take two different strategies. In the first approach, we propose a multichannel receiver with each channel having a low-cost energy detector operating at a sub-Nyquist rate. In the second approach, we consider a compressive sampling based technique. Here, we propose a new acquisition front end, using which the sampling rate of the ADC can be significantly reduced. We extended the idea of compressive sampling based TOA estimation towards joint estimation of TOA and PPM symbols. Here, two signaling methods along with the algorithms are proposed based on the dynamicity of the target. They provide similar performance to the ML based estimation, however with a significant savings in the ADC resources. / <p>QC 20161205</p>

Fusion frames and block sparse recovery are of interest in signal processing and communication applications. In these applications it is required that the fusion frame have some desirable properties. One such requirement is that the fusion frame be tight and its subspaces form an optimal packing in a Grassmannian manifold. Such fusion frames are called Grassmannian fusion frames. Grassmannian frames are known to be optimal dictionaries for sparse recovery as they have minimum coherence. By analogy Grassmannian fusion frames are potential candidates as optimal dictionaries in block sparse processing. The present work intends to study fusion frames in finite dimensional vector spaces assuming a specific structure useful in block sparse signal processing. The main focus of our work is the design of Grassmannian fusion frames and their implication in block sparse recovery. We will consider burst error correction as an application of block sparsity and fusion frame concepts. We propose two new algebraic methods for designing Grassmannian fusion frames. The first method involves use of Fourier matrix and difference sets to obtain a partial Fourier matrix which forms a Grassmannian fusion frame. This fusion frame has a specific structure and the parameters of the fusion frame are determined by the type of difference set used. The second method involves constructing Grassmannian fusion frames from Grassmannian frames which meet the Welch bound. This method uses existing constructions of optimal Grassmannian frames. The method, while fairly general, requires that the dimension of the vector space be divisible by the dimension of the subspaces. A lower bound which is an analog of the Welch bound is derived for the block coherence of dictionaries along with conditions to be satisfied to meet the bound. From these results we conclude that the matrices constructed by us are optimal for block sparse recovery from block coherence viewpoint. There is a strong relation between sparse signal processing and error control coding. It is known that burst errors are block sparse in nature. So, here we attempt to solve the burst error correction problem using block sparse signal recovery methods. The use of Grassmannian fusion frames which we constructed as optimal dictionary allows correction of maximum possible number of errors, when used in conjunction with reconstruction algorithms which exploit block sparsity. We also suggest a modification to improve the applicability of the technique and point out relationship with a method which appeared previously in literature. As an application example, we consider the use of the burst error correction technique for impulse noise cancelation in OFDM system. Impulse noise is bursty in nature and severely degrades OFDM performance. The Grassmannian fusion frames constructed with Fourier matrix and difference sets is ideal for use in this application as it can be easily incorporated into the OFDM system.

Compressed Sensing (CS) is a new paradigm in signal processing which exploits the sparse or compressible nature of the signal to significantly reduce the number of measurements, without compromising on the signal reconstruction quality. Recently, many algorithms have been reported in the literature for efficient sparse signal reconstruction. Nevertheless, it is well known that the performance of any sparse reconstruction algorithm depends on many parameters like number of measurements, dimension of the sparse signal, the level of sparsity, the measurement noise power, and the underlying statistical distribution of the non-zero elements of the signal. It has been observed that a satisfactory performance of the sparse reconstruction algorithm mandates certain requirement on these parameters, which is different for different algorithms. Many applications are unlikely to fulfil this requirement. For example, imaging speed is crucial in many Magnetic Resonance Imaging (MRI) applications. This restricts the number of measurements, which in turn affects the medical diagnosis using MRI. Hence, any strategy to improve the signal reconstruction in such adverse scenario is of substantial interest in CS. Interestingly, it can be observed that the performance degradation of the sparse recovery algorithms in the aforementioned cases does not always imply a complete failure. That is, even in such adverse situations, a sparse reconstruction algorithm may provide partially correct information about the signal. In this thesis, we study this scenario and propose a novel fusion framework and an iterative framework which exploit the partial information available in the sparse signal estimate(s) to improve sparse signal reconstruction. The proposed fusion framework employs multiple sparse reconstruction algorithms, independently, for signal reconstruction. We first propose a fusion algorithm viz. FACS which fuses the estimates of multiple participating algorithms in order to improve the sparse signal reconstruction. To alleviate the inherent drawbacks of FACS and further improve the sparse signal reconstruction, we propose another fusion algorithm called CoMACS and variants of CoMACS. For low latency applications, we propose a latency friendly fusion algorithm called pFACS. We also extend the fusion framework to the MMV problem and propose the extension of FACS called MMV-FACS. We theoretically analyse the proposed fusion algorithms and derive guarantees for performance improvement. We also show that the proposed fusion algorithms are robust against both signal and measurement perturbations. Further, we demonstrate the efficacy of the proposed algorithms via numerical experiments: (i) using sparse signals with different statistical distributions in noise-free and noisy scenarios, and (ii) using real-world ECG signals. The extensive numerical experiments show that, for a judicious choice of the participating algorithms, the proposed fusion algorithms result in a sparse signal estimate which is often better than the sparse signal estimate of the best participating algorithm. The proposed fusion framework requires toemploy multiple sparse reconstruction algorithms for sparse signal reconstruction. We also propose an iterative framework and algorithm called {IFSRA to improve the performance of a given arbitrary sparse reconstruction algorithm. We theoretically analyse IFSRA and derive convergence guarantees under signal and measurement perturbations. Numerical experiments on synthetic and real-world data confirm the efficacy of IFSRA. The proposed fusion algorithms and IFSRA are general in nature and does not require any modification in the participating algorithm(s).