Time-varying parametric modelling and time-dependent spectral characterisation with applications to EEG signals using multi-wavelets

A new time-varying autoregressive (TVAR) modelling approach is proposed for nonstationary signal processing and analysis, with application to EEG data modelling and power spectral estimation. In the new parametric modelling framework, the time-dependent coefficients of the TVAR model are represented using a novel multi-wavelet decomposition scheme. The time-varying modelling problem is then reduced to regression selection and parameter estimation, which can be effectively resolved by using a forward orthogonal regression algorithm. Two examples, one for an artificial signal and another for an EEG signal, are given to show the effectiveness and applicability of the new TVAR modelling method

Overcoming Inter-Subject Variability in BCI Using EEG-Based Identification

The high dependency of the Brain Computer Interface (BCI) system performance on the BCI user is a well-known issue of many BCI devices. This contribution presents a new way to overcome this problem using a synergy between a BCI device and an EEG-based biometric algorithm. Using the biometric algorithm, the BCI device automatically identifies its current user and adapts parameters of the classification process and of the BCI protocol to maximize the BCI performance. In addition to this we present an algorithm for EEG-based identification designed to be resistant to variations in EEG recordings between sessions, which is also demonstrated by an experiment with an EEG database containing two sessions recorded one year apart. Further, our algorithm is designed to be compatible with our movement-related BCI device and the evaluation of the algorithm performance took place under conditions of a standard BCI experiment. Estimation of the mu rhythm fundamental frequency using the Frequency Zooming AR modeling is used for EEG feature extraction followed by a classifier based on the regularized Mahalanobis distance. An average subject identification score of 96 % is achieved

Advanced tensor based signal processing techniques for wireless communication systems and biomedical signal processing

Many observed signals in signal processing applications including wireless communications, biomedical signal processing, image processing, and machine learning are multi-dimensional. Tensors preserve the multi-dimensional structure and provide a natural representation of these signals/data. Moreover, tensors provide often an improved identifiability. Therefore, we benefit from using tensor algebra in the above mentioned applications and many more. In this thesis, we present the benefits of utilizing tensor algebra in two signal processing areas. These include signal processing for MIMO (Multiple-Input Multiple-Output) wireless communication systems and biomedical signal processing. Moreover, we contribute to the theoretical aspects of tensor algebra by deriving new properties and ways of computing tensor decompositions. Often, we only have an element-wise or a slice-wise description of the signal model. This representation of the signal model does not reveal the explicit tensor structure. Therefore, the derivation of all tensor unfoldings is not always obvious. Consequently, exploiting the multi-dimensional structure of these models is not always straightforward. We propose an alternative representation of the element-wise multiplication or the slice-wise multiplication based on the generalized tensor contraction operator. Later in this thesis, we exploit this novel representation and the properties of the contraction operator such that we derive the final tensor models. There exist a number of different tensor decompositions that describe different signal models such as the HOSVD (Higher Order Singular Value Decomposition), the CP/PARAFAC (Canonical Polyadic / PARallel FACtors) decomposition, the BTD (Block Term Decomposition), the PARATUCK2 (PARAfac and TUCker2) decomposition, and the PARAFAC2 (PARAllel FACtors2) decomposition. Among these decompositions, the CP decomposition is most widely spread and used. Therefore, the development of algorithms for the efficient computation of the CP decomposition is important for many applications. The SECSI (Semi-Algebraic framework for approximate CP decomposition via SImultaneaous matrix diagonalization) framework is an efficient and robust tool for the calculation of the approximate low-rank CP decomposition via simultaneous matrix diagonalizations. In this thesis, we present five extensions of the SECSI framework that reduce the computational complexity of the original framework and/or introduce constraints to the factor matrices. Moreover, the PARAFAC2 decomposition and the PARATUCK2 decomposition are usually described using a slice-wise notation that can be expressed in terms of the generalized tensor contraction as proposed in this thesis. We exploit this novel representation to derive explicit tensor models for the PARAFAC2 decomposition and the PARATUCK2 decomposition. Furthermore, we use the PARAFAC2 model to derive an ALS (Alternating Least-Squares) algorithm for the computation of the PARAFAC2 decomposition. Moreover, we exploit the novel contraction properties for element wise and slice-wise multiplications to model MIMO multi-carrier wireless communication systems. We show that this very general model can be used to derive the tensor model of the received signal for MIMO-OFDM (Multiple-Input Multiple-Output - Orthogonal Frequency Division Multiplexing), Khatri-Rao coded MIMO-OFDM, and randomly coded MIMO-OFDM systems. We propose the transmission techniques Khatri-Rao coding and random coding in order to impose an additional tensor structure of the transmit signal tensor that otherwise does not have a particular structure. Moreover, we show that this model can be extended to other multi-carrier techniques such as GFDM (Generalized Frequency Division Multiplexing). Utilizing these models at the receiver side, we design several types for receivers for these systems that outperform the traditional matrix based solutions in terms of the symbol error rate. In the last part of this thesis, we show the benefits of using tensor algebra in biomedical signal processing by jointly decomposing EEG (ElectroEncephaloGraphy) and MEG (MagnetoEncephaloGraphy) signals. EEG and MEG signals are usually acquired simultaneously, and they capture aspects of the same brain activity. Therefore, EEG and MEG signals can be decomposed using coupled tensor decompositions such as the coupled CP decomposition. We exploit the proposed coupled SECSI framework (one of the proposed extensions of the SECSI framework) for the computation of the coupled CP decomposition to first validate and analyze the photic driving effect. Moreover, we validate the effects of scull defects on the measurement EEG and MEG signals by means of a joint EEG-MEG decomposition using the coupled SECSI framework. Both applications show that we benefit from coupled tensor decompositions and the coupled SECSI framework is a very practical tool for the analysis of biomedical data.Zahlreiche messbare Signale in verschiedenen Bereichen der digitalen Signalverarbeitung, z.B. in der drahtlosen Kommunikation, im Mobilfunk, biomedizinischen Anwendungen, der Bild- oder akustischen Signalverarbeitung und dem maschinellen Lernen sind mehrdimensional. Tensoren erhalten die mehrdimensionale Struktur und stellen eine natürliche Darstellung dieser Signale/Daten dar. Darüber hinaus bieten Tensoren oft eine verbesserte Trennbarkeit von enthaltenen Signalkomponenten. Daher profitieren wir von der Verwendung der Tensor-Algebra in den oben genannten Anwendungen und vielen mehr. In dieser Arbeit stellen wir die Vorteile der Nutzung der Tensor-Algebra in zwei Bereichen der Signalverarbeitung vor: drahtlose MIMO (Multiple-Input Multiple-Output) Kommunikationssysteme und biomedizinische Signalverarbeitung. Darüber hinaus tragen wir zu theoretischen Aspekten der Tensor-Algebra bei, indem wir neue Eigenschaften und Berechnungsmethoden für die Tensor-Zerlegung ableiten. Oftmals verfügen wir lediglich über eine elementweise oder ebenenweise Beschreibung des Signalmodells, welche nicht die explizite Tensorstruktur zeigt. Daher ist die Ableitung aller Tensor-Unfoldings nicht offensichtlich, wodurch die multidimensionale Struktur dieser Modelle nicht trivial nutzbar ist. Wir schlagen eine alternative Darstellung der elementweisen Multiplikation oder der ebenenweisen Multiplikation auf der Grundlage des generalisierten Tensor-Kontraktionsoperators vor. Weiterhin nutzen wir diese neuartige Darstellung und deren Eigenschaften zur Ableitung der letztendlichen Tensor-Modelle. Es existieren eine Vielzahl von Tensor-Zerlegungen, die verschiedene Signalmodelle beschreiben, wie die HOSVD (Higher Order Singular Value Decomposition), CP/PARAFAC (Canonical Polyadic/ PARallel FACtors) Zerlegung, die BTD (Block Term Decomposition), die PARATUCK2-(PARAfac und TUCker2) und die PARAFAC2-Zerlegung (PARAllel FACtors2). Dabei ist die CP-Zerlegung am weitesten verbreitet und wird findet in zahlreichen Gebieten Anwendung. Daher ist die Entwicklung von Algorithmen zur effizienten Berechnung der CP-Zerlegung von besonderer Bedeutung. Das SECSI (Semi-Algebraic Framework for approximate CP decomposition via Simultaneaous matrix diagonalization) Framework ist ein effizientes und robustes Werkzeug zur Berechnung der approximierten Low-Rank CP-Zerlegung durch simultane Matrixdiagonalisierung. In dieser Arbeit stellen wir fünf Erweiterungen des SECSI-Frameworks vor, welche die Rechenkomplexität des ursprünglichen Frameworks reduzieren bzw. Einschränkungen für die Faktormatrizen einführen. Darüber hinaus werden die PARAFAC2- und die PARATUCK2-Zerlegung in der Regel mit einer ebenenweisen Notation beschrieben, die sich in Form der allgemeinen Tensor-Kontraktion, wie sie in dieser Arbeit vorgeschlagen wird, ausdrücken lässt. Wir nutzen diese neuartige Darstellung, um explizite Tensormodelle für diese beiden Zerlegungen abzuleiten. Darüber hinaus verwenden wir das PARAFAC2-Modell, um einen ALS-Algorithmus (Alternating Least-Squares) für die Berechnung der PARAFAC2-Zerlegungen abzuleiten. Weiterhin nutzen wir die neuartigen Kontraktionseigenschaften für elementweise und ebenenweise Multiplikationen, um MIMO Multi-Carrier-Mobilfunksysteme zu modellieren. Wir zeigen, dass dieses sehr allgemeine Modell verwendet werden kann, um das Tensor-Modell des empfangenen Signals für MIMO-OFDM- (Multiple- Input Multiple-Output - Orthogonal Frequency Division Multiplexing), Khatri-Rao codierte MIMO-OFDM- und zufällig codierte MIMO-OFDM-Systeme abzuleiten. Wir schlagen die Übertragungstechniken der Khatri-Rao-Kodierung und zufällige Kodierung vor, um eine zusätzliche Tensor-Struktur des Sendesignal-Tensors einzuführen, welcher gewöhnlich keine bestimmte Struktur aufweist. Darüber hinaus zeigen wir, dass dieses Modell auf andere Multi-Carrier-Techniken wie GFDM (Generalized Frequency Division Multiplexing) erweitert werden kann. Unter Verwendung dieser Modelle auf der Empfängerseite entwerfen wir verschiedene Typen von Empfängern für diese Systeme, die die traditionellen matrixbasierten Lösungen in Bezug auf die Symbolfehlerrate übertreffen. Im letzten Teil dieser Arbeit zeigen wir die Vorteile der Verwendung von Tensor-Algebra in der biomedizinischen Signalverarbeitung durch die gemeinsame Zerlegung von EEG-(ElectroEncephaloGraphy) und MEG- (MagnetoEncephaloGraphy) Signalen. Diese werden in der Regel gleichzeitig erfasst, wobei sie gemeinsame Aspekte derselben Gehirnaktivität beschreiben. Daher können EEG- und MEG-Signale mit gekoppelten Tensor-Zerlegungen wie der gekoppelten CP Zerlegung analysiert werden. Wir nutzen das vorgeschlagene gekoppelte SECSI-Framework (eine der vorgeschlagenen Erweiterungen des SECSI-Frameworks) für die Berechnung der gekoppelten CP Zerlegung, um zunächst den photic driving effect zu validieren und zu analysieren. Darüber hinaus validieren wir die Auswirkungen von Schädeldefekten auf die Messsignale von EEG und MEG durch eine gemeinsame EEG-MEG-Zerlegung mit dem gekoppelten SECSI-Framework. Beide Anwendungen zeigen, dass wir von gekoppelten Tensor-Zerlegungen profitieren, wobei die Methoden des gekoppelten SECSI-Frameworks erfolgreich zur Analyse biomedizinischer Daten genutzt werden können

Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis

The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under verymild and natural conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that match data properties, and to find more general latent components in the data than matrix-based methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker decomposition, HOSVD, tensor networks, Tensor Train

Widely Linear State Space Filtering of Improper Complex Signals

Complex signals are the backbone of many modern applications, such as power systems, communication systems, biomedical sciences and military technologies. However, standard complex valued signal processing approaches are suited to only a subset of complex signals known as proper, and are inadequate of the generality of complex signals, as they do not fully exploit the available information. This is mainly due to the inherent blindness of the algorithms to the complete second order statistics of the signals, or due to under-modelling of the underlying system. The aim of this thesis is to provide enhanced complex valued, state space based, signal processing solutions for the generality of complex signals and systems. This is achieved based on the recent advances in the so called augmented complex statistics and widely linear modelling, which have brought to light the limitations of conventional statistical complex signal processing approaches. Exploiting these developments, we propose a class of widely linear adaptive state space estimation techniques, which provide a unified framework and enhanced performance for the generality of complex signals, compared with conventional approaches. These include the linear and nonlinear Kalman and particle filters, whereby it is shown that catering for the complete second order information and system models leads to significant performance gains. The proposed techniques are also extended to the case of cooperative distributed estimation, where nodes in a network collaborate locally to estimate signals, under a framework that caters for general complex signals, as well as the cross-correlations between observation noises, unlike earlier solutions. The analysis of the algorithms are supported by numerous case studies, including frequency estimation in three phase power systems, DIFAR sonobuoy underwater target tracking, and real-world wind modeling and prediction.Open Acces