A Unifying View on Blind Source Separation of Convolutive Mixtures based on Independent Component Analysis

In many daily-life scenarios, acoustic sources recorded in an enclosure can only be observed with other interfering sources. Hence, convolutive Blind Source Separation (BSS) is a central problem in audio signal processing. Methods based on Independent Component Analysis (ICA) are especially important in this field as they require only few and weak assumptions and allow for blindness regarding the original source signals and the acoustic propagation path. Most of the currently used algorithms belong to one of the following three families: Frequency Domain ICA (FD-ICA), Independent Vector Analysis (IVA), and TRIple-N Independent component analysis for CONvolutive mixtures (TRINICON). While the relation between ICA, FD-ICA and IVA becomes apparent due to their construction, the relation to TRINICON is not well established yet. This paper fills this gap by providing an in-depth treatment of the common building blocks of these algorithms and their differences, and thus provides a common framework for all considered algorithms

Experimentally testing quantum field theory concepts with spinor Bose gases far from equilibrium

The number of parameters needed to specify the state of a many-body quantum system grows exponentially with the number of its constituents. This fact makes it computationally intractable to exactly describe dynamics and characterize the state on a microscopic level. In this thesis, we employ quantum field theory concepts for experimentally characterizing a spinor Bose gas far from equilibrium. First, we introduce the relevant concepts, which provide efficient descriptions for emerging macroscopic phenomena, in a formulation matching the capabilities of ultracold atomic systems. For our experimental study we employ a 87Rb spinor Bose-Einstein condensate in a quasi-one-dimensional trap geometry. We explore the phase diagram as a function of the effective quadratic Zeeman shift by measuring the fluctuations in the spin degree of freedom and identify three distinct phases. With this knowledge, we study the instabilities which occur after an instantaneous quench across the quantum phase transition separating the different phases. These instabilities allow us to drive the system far from equilibrium in a highly controlled fashion. For long times after the quench we observe universal dynamics associated with the emergence of a non-thermal fixed point. The structure factor of the angular orientation of the transversal spin features rescaling in time and space with universal exponents as well as a universal scaling function. Using the experimental control, we probe the insensitivity of this phenomenon to details of the initial condition. Spatially resolved snapshots of the complex-valued transversal spin field allow for the extraction of one-particle irreducible correlation functions, the building blocks of the quantum effective action. We find a strong suppression of the 4-vertex at low momenta emerging in the highly occupied regime. The introduced concepts together with the presented experimental applicability give new means for studying many-body systems at all stages of their evolution: from the initial instabilities and across transient phenomena far from equilibrium to the eventual thermalization

Adaptive signal processing algorithms for noncircular complex data

The complex domain provides a natural processing framework for a large class of signals encountered in communications, radar, biomedical engineering and renewable energy. Statistical signal processing in C has traditionally been viewed as a straightforward extension of the corresponding algorithms in the real domain R, however, recent developments in augmented complex statistics show that, in general, this leads to under-modelling. This direct treatment of complex-valued signals has led to advances in so called widely linear modelling and the introduction of a generalised framework for the differentiability of both analytic and non-analytic complex and quaternion functions. In this thesis, supervised and blind complex adaptive algorithms capable of processing the generality of complex and quaternion signals (both circular and noncircular) in both noise-free and noisy environments are developed; their usefulness in real-world applications is demonstrated through case studies. The focus of this thesis is on the use of augmented statistics and widely linear modelling. The standard complex least mean square (CLMS) algorithm is extended to perform optimally for the generality of complex-valued signals, and is shown to outperform the CLMS algorithm. Next, extraction of latent complex-valued signals from large mixtures is addressed. This is achieved by developing several classes of complex blind source extraction algorithms based on fundamental signal properties such as smoothness, predictability and degree of Gaussianity, with the analysis of the existence and uniqueness of the solutions also provided. These algorithms are shown to facilitate real-time applications, such as those in brain computer interfacing (BCI). Due to their modified cost functions and the widely linear mixing model, this class of algorithms perform well in both noise-free and noisy environments. Next, based on a widely linear quaternion model, the FastICA algorithm is extended to the quaternion domain to provide separation of the generality of quaternion signals. The enhanced performances of the widely linear algorithms are illustrated in renewable energy and biomedical applications, in particular, for the prediction of wind profiles and extraction of artifacts from EEG recordings

Robust computation of linear models by convex relaxation

Consider a dataset of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called REAPER, that can reliably fit a low-dimensional model to this type of data. This approach parameterizes linear subspaces using orthogonal projectors, and it uses a relaxation of the set of orthogonal projectors to reach the convex formulation. The paper provides an efficient algorithm for solving the REAPER problem, and it documents numerical experiments which confirm that REAPER can dependably find linear structure in synthetic and natural data. In addition, when the inliers lie near a low-dimensional subspace, there is a rigorous theory that describes when REAPER can approximate this subspace.Comment: Formerly titled "Robust computation of linear models, or How to find a needle in a haystack

Multiscale Modelling of Polymer Self-Assembly in Binary Solvent Mixtures

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