14 research outputs found
Tensor-Based Methods for Blind Spatial Signature Estimation in Multidimensional Sensor Arrays
The estimation of spatial signatures and spatial frequencies is crucial for several practical applications such as radar, sonar, and wireless communications. In this paper, we propose two generalized iterative estimation algorithms to the case in which a multidimensional (R-D) sensor array is used at the receiver. The first tensor-based algorithm is an R-D blind spatial signature estimator that operates in scenarios where the source’s covariance matrix is nondiagonal and unknown. The second tensor-based algorithm is formulated for the case in which the sources are uncorrelated and exploits the dual-symmetry of the covariance tensor. Additionally, a new tensor-based formulation is proposed for an L-shaped array configuration. Simulation results show that our proposed schemes outperform the state-of-the-art matrix-based and tensor-based techniques
Multipath mitigation in time-delay estimation via tensorbased techniques for antenna array-based gnss receivers
Dissertação (mestrado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2017.Clientes de Sistemas Globais de Navegação por Satélites, do inglês Global Navigation Satellite System (GNSS), dependem da estimação do atraso para estimar a posição do usuário. [1] Isto é feito fazendo a correlação do sinal recebido com sequências-réplicas para separar o sinal de cada satélite e estimar o atraso. Como componentes de multipercurso são cópias atrasadas do sinal original, estes alteram a função de correlação cruzada, assim gerando erros na estimação de atraso. Nesta dissertação estudamos um algoritmo estado-da-arte em mitigação de multipercursos para estimação de atraso baseado no autofiltro da decomposição em valores singular de alta ordem, do inglês Higher-Order Singular Value Decomposition (HOSVD), de posto unitário, [2] e propomos dois esquemas tensoriais para mitigação de multipercurso e estimação de atraso, para qual o esquema baseado em HOSVD é usado para comparação. O primeiro esquema tensorial é um método em três etapas que aplica estimação da direção de chegada, do inglês Direction of Arrival (DoA), e fatorização Khatri-Rao, do inglês Khatri-Rao factorization (KRF), para separar o código de cada componente incidente de forma fechada. O segundo esquema calcula uma matrix de covariância multimodo como aproximação do desdobramento Hermitian duplamente simétrico [3] com qual, alternando entre a solução do problema ortogonal de Procrustes, do inglês Orthogonal Procrustes Problem (OPP), [4] e fatorização Khatri-Rao de mínimos quadrados, do inglês Least Squares Khatri-Rao Factorization (LSKRF), [5] se estima iterativamente as matrizes-fator do canal, que são então usadas para separar o código de cada componente incidente. Ambos esquemas geram resultados melhores que o estado-da-arte baseado no autofiltro de alta ordem.Global Navigation Satellite System (GNSS) clients rely on time-delay estimation to estimate a user’s position. [1] This is done by correlating the incoming signal with replica sequences to separate each satellite and perform time-delay estimation. Since multipath components are delayed copies of the original signal, this affects the cross-correlation function, thus impacting time-delay estimation. 1 In this thesis, we study a state-of-the-art approach for multipath mitigation time-delay estimation algorithm based on the rank-one Higher-Order Singular Value Decomposition (HOSVD) eigenfilter, [2] and propose two tensorbased schemes for multipath mitigation and time-delay estimation, for which the HOSVD-based scheme is a basis of comparison. The first scheme is a three step tensor-based approach applying direction of arrival (DoA) estimation and Khatri-Rao factorization (KRF) to separate the code for each impinging component in a closed fashion. The second approach calculates a multimode covariance matrix as an approximation of the dualsymmetric Hermitian unfolding [3] with which, by alternating between a solution to the orthogonal Procrustes problem (OPP) [4] and least squares Khatri-Rao factorization, [5] iteratively estimates the channel factor matrices which are then used to separate the code of each impinging component. Both our schemes outperforms the HOSVD-based eigenfilter state-of-the-art solution
Time-delay estimation under non-clustered and clustered scenarios for GNSS signals
Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2021.Aplicações que empregam sistemas globais de navegação por satélite, do inglês Global
Navigation Satellite Systems (GNSS) para prover posicionamento acurado estão sujeitos a
degradação drástica não só por intereferências eletromagnéticas, como também componentes
de multipercurso causados por reflexões e refrações no ambiente. Aplicações de segurança
crítica como veículos autonômos e aviação civil, e aplicações de risco crítico como gestão de
pesca, pedágio automático, e agricultura de precisão dependem de posicionamento acurado
sob cenários complicados. Tipicamente quanto mais agrupamento ocorre entre o componente de linha de visada, do inglês line-of-sight (LOS) e componentes de multipercurso ou
não-linha de visada, do inglês non-line-of-sight (NLOS), menos acurada é a estimação da
posição. Abordagens tensorials estado da arte para receptores GNSS baseado em arranjos
de antenas utilizam processamento tensorial de sinais para separar o componente LOS dos
componentes NLOS, assim mitigando os efeitos destes, utilizando decomposição em valores singulares multilinear, do inglês multilinear singular value decomposition (MLSVD)
para gerar um autofiltro de order superior, do inglês higher-order eigenfilter (HOE) com
pré-processamento por média frente-costas, do inglês forward-backward averaging (FBA),
e suavização espacial expandida, do inglês expanded spatial smoothing (ESPS), estimação
de direção de chegada, do inglês direction of arrival (DoA) e fatorização Khatri-Rao, do
inglês Khatri-Rao factorization (KRF), estimação de Procrustes e fatorização Khatri-Rao
(ProKRaft), e o sistema semi-algébrico de decomposição poliádica canônica por diagonalização matricial simultânea, do inglês semi-algebraic framework for approximate canonical
polyadic decomposition via simultaneous matrix diagonalization (SECSI), respectivamente.
Propomos duas abordagens de processamento para estimação de atraso, do inglês time-delay
estimation (TDE). A primeira é a abordagem em lotes utilizando dados de vários períodos
do sinal. Usando estimação em lotes propomos duas abordagens algébricas para TDE, em
que diagonalizaçao é efetivada por decomposição generalizada em autovalores, do inglês
generalized eigenvalue decomposition (GEVD), das primeiras duas fatias frontais do tensor núcleo do tensor de dados, estimado por MLSVD. Esta primeira abordagem, como os
métodos citados, na quais simulações foram feitas com 1 componente LOS e 1 componente
NLOS, assim os dados observados tem posto cheio em todos seus modos, não faz suposições
sobre o posto do tensor de dados. A segunda abordagem supõe cenários nos quais mais de
1 componente NLOS está presente e são agregados (clustered em inglês), assim vários vetores de uma das matrizes-fator que formam o tensor de dados são altamente correlacionaiii
dos, resultando num tensor de dados que é de posto deficiente em pelo menos um modo.
Os esquemas algébricos baseados em tensores propostos utilizam a decomposição poliádica
canônica por decomposição generalizada em autovalores, do inglês canonical polyadic decomposition via generalized eigenvalue decomposition (CPD-GEVD), e a decomposição em
termos de posto-(Lr, Lr, 1) por decomposição generalizada em autovalores, do inglês decomposition in multilinear rank-(Lr, Lr, 1) terms via generalized eigenvalue decomposition
((Lr, Lr, 1)-GEVD) para melhorar a TDE do componente LOS sob cenários desafiadores. A
segunda é a abordagem de processamento adaptativo de amostras individuais utilizando rastreamento de subespaço a cada período de código, epoch em inglês. Usando processamento
adaptativo propomos duas abordagem, uma aplicando FBA expandido (EFBA) e ESPS ao
dados e estimando um HOE, e outra usando usa estimação paramétrica para estimar a DoA.
Estendendo o modelo para um arranjo retangular uniforme, do inglês uniform rectangular
array (URA), o fluxo de dados são tensores de terceira ordem. Para este modelo propomos
três abordagens para TDE baseado em HOE, CPD-GEVD, e ESPRIT tensorial, respectivamente e empregando uma estratégia de truncamento sequencial para reduzir a quantidade de
operações necessárias para cada modo do tensorCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).Applications employing Global Navigation Satellite Systems (GNSS) to provide accurate positioning are subject to drastic degradation not only due to electromagnetic interference, but also due to multipath components caused by reflections and refractions in the
environment. Safety-critical applications such as autonomous vehicles and civil aviation,
and liability-critical applications such as fisheries management, automatic tolling, and precision agriculture depend on accurate positioning under such demanding scenarios. Typically,
the more clustering occurs between the line-of-sight (LOS) component and multipath or
non-line-of-sight (NLOS) components, the more inaccurate is the estimation of the positioning. State-of-the-art tensor based approaches for antenna array-based GNSS receivers apply
tensor-based signal processing to separate the LOS components from NLOS components,
thus mitigating the effects of the latter, using the multilinear singular value decomposition
(MLSVD) to generate a higher-order eigenfilter (HOE) with forward-backward averaging
(FBA) and expanded spatial smoothing (ESPS) preprocessing, direction of arrival (DoA) estimation and Khatri-Rao factorization (KRF), Procrustes estimation and Khatri-Rao factorization (ProKRaft), and the semi-algebraic framework for approximate canonical polyadic
decomposition via simultaneous matrix diagonalization (SECSI), respectively. These approaches use filtering, parameter estimation and filtering, iterative algebraic factor matrix
estimation and filtering, and algebraic factor matrix estimation, respectively. We propose
two processing approaches to time-delay estimation (TDE). The first is batch processing
taking data from several signal periods. Using batch processing we propose two algebraic
approaches to TDE, in which diagonalization is achieved using the generalized eigenvalue
decomposition (GEVD) of the first two frontal slices of the measurement tensor’s core tensor,
estimated via MLSVD. The former approach, like the cited methods, in which simulations
were performed with 1 LOS component and 1 NLOS component, and thus the measured data
has full-rank tensor in all its modes, makes no assumption about the rank of the measurement tensor. The latter approach assumes scenarios in which more than 1 NLOS component
is present and these are clustered, thus several vectors of one of the factor matrices which
forms the tensor data are highly correlated, resulting in a rank-deficient measurement tensor
in at least one mode. These proposed algebraic tensor-based schemes utilize the canonical
polyadic decomposition via generalized eigenvalue decomposition (CPD-GEVD) and the decomposition in multilinear rank-(Lr, Lr, 1) terms via generalized eigenvalue decomposition
((Lr, Lr, 1)-GEVD) in order to improve the TDE of the LOS component in challenging scev
narios. The second approach is adaptive processing of individual samples utilizing subspace
tracking to iteratively estimate the subspace at each epoch. Using adaptive processing we
propose two approaches, one applying FBA and ESPS to the data and estimating a higherorder eigenfilter, and the other using a parametric approach using DoA estimation. By extending the data model for an uniform rectangular array, we have a data stream of third-order
tensors. For this model we propose three approaches to TDE based on HOE, CPD-GEVD,
and standard tensor ESPRIT, respectively and employing a sequential truncation strategy to
reduce the amount of operations necessary for each tensor mode
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
Advanced Algebraic Concepts for Efficient Multi-Channel Signal Processing
Unsere moderne Gesellschaft ist Zeuge eines fundamentalen Wandels in der Art und Weise
wie wir mit Technologie interagieren. Geräte werden zunehmend intelligenter - sie verfügen
über mehr und mehr Rechenleistung und häufiger über eigene Kommunikationsschnittstellen.
Das beginnt bei einfachen Haushaltsgeräten und reicht über Transportmittel bis zu großen
überregionalen Systemen wie etwa dem Stromnetz. Die Erfassung, die Verarbeitung und der
Austausch digitaler Informationen gewinnt daher immer mehr an Bedeutung. Die Tatsache,
dass ein wachsender Anteil der Geräte heutzutage mobil und deshalb batteriebetrieben ist,
begründet den Anspruch, digitale Signalverarbeitungsalgorithmen besonders effizient zu gestalten.
Dies kommt auch dem Wunsch nach einer Echtzeitverarbeitung der großen anfallenden
Datenmengen zugute.
Die vorliegende Arbeit demonstriert Methoden zum Finden effizienter algebraischer Lösungen
für eine Vielzahl von Anwendungen mehrkanaliger digitaler Signalverarbeitung. Solche Ansätze
liefern nicht immer unbedingt die bestmögliche Lösung, kommen dieser jedoch häufig recht
nahe und sind gleichzeitig bedeutend einfacher zu beschreiben und umzusetzen. Die einfache
Beschreibungsform ermöglicht eine tiefgehende Analyse ihrer Leistungsfähigkeit, was für den
Entwurf eines robusten und zuverlässigen Systems unabdingbar ist. Die Tatsache, dass sie nur
gebräuchliche algebraische Hilfsmittel benötigen, erlaubt ihre direkte und zügige Umsetzung
und den Test unter realen Bedingungen.
Diese Grundidee wird anhand von drei verschiedenen Anwendungsgebieten demonstriert.
Zunächst wird ein semi-algebraisches Framework zur Berechnung der kanonisch polyadischen
(CP) Zerlegung mehrdimensionaler Signale vorgestellt. Dabei handelt es sich um ein sehr
grundlegendes Werkzeug der multilinearen Algebra mit einem breiten Anwendungsspektrum
von Mobilkommunikation über Chemie bis zur Bildverarbeitung. Verglichen mit existierenden
iterativen Lösungsverfahren bietet das neue Framework die Möglichkeit, den Rechenaufwand
und damit die Güte der erzielten Lösung zu steuern. Es ist außerdem weniger anfällig gegen eine
schlechte Konditionierung der Ausgangsdaten. Das zweite Gebiet, das in der Arbeit besprochen
wird, ist die unterraumbasierte hochauflösende Parameterschätzung für mehrdimensionale Signale,
mit Anwendungsgebieten im RADAR, der Modellierung von Wellenausbreitung, oder
bildgebenden Verfahren in der Medizin. Es wird gezeigt, dass sich derartige mehrdimensionale
Signale mit Tensoren darstellen lassen. Dies erlaubt eine natürlichere Beschreibung und eine
bessere Ausnutzung ihrer Struktur als das mit Matrizen möglich ist. Basierend auf dieser Idee
entwickeln wir eine tensor-basierte Schätzung des Signalraums, welche genutzt werden kann
um beliebige existierende Matrix-basierte Verfahren zu verbessern. Dies wird im Anschluss
exemplarisch am Beispiel der ESPRIT-artigen Verfahren gezeigt, für die verbesserte Versionen
vorgeschlagen werden, die die mehrdimensionale Struktur der Daten (Tensor-ESPRIT),
nichzirkuläre Quellsymbole (NC ESPRIT), sowie beides gleichzeitig (NC Tensor-ESPRIT) ausnutzen.
Um die endgültige Schätzgenauigkeit objektiv einschätzen zu können wird dann ein
Framework für die analytische Beschreibung der Leistungsfähigkeit beliebiger ESPRIT-artiger
Algorithmen diskutiert. Verglichen mit existierenden analytischen Ausdrücken ist unser Ansatz
allgemeiner, da keine Annahmen über die statistische Verteilung von Nutzsignal und
Rauschen benötigt werden und die Anzahl der zur Verfügung stehenden Schnappschüsse beliebig
klein sein kann. Dies führt auf vereinfachte Ausdrücke für den mittleren quadratischen
Schätzfehler, die Schlussfolgerungen über die Effizienz der Verfahren unter verschiedenen Bedingungen
zulassen. Das dritte Anwendungsgebiet ist der bidirektionale Datenaustausch mit
Hilfe von Relay-Stationen. Insbesondere liegt hier der Fokus auf Zwei-Wege-Relaying mit Hilfe
von Amplify-and-Forward-Relays mit mehreren Antennen, da dieser Ansatz ein besonders gutes
Kosten-Nutzen-Verhältnis verspricht. Es wird gezeigt, dass sich die nötige Kanalkenntnis
mit einem einfachen algebraischen Tensor-basierten Schätzverfahren gewinnen lässt. Außerdem
werden Verfahren zum Finden einer günstigen Relay-Verstärkungs-Strategie diskutiert. Bestehende
Ansätze basieren entweder auf komplexen numerischen Optimierungsverfahren oder auf
Ad-Hoc-Ansätzen die keine zufriedenstellende Bitfehlerrate oder Summenrate liefern. Deshalb
schlagen wir algebraische Ansätze zum Finden der Relayverstärkungsmatrix vor, die von relevanten
Systemmetriken inspiriert sind und doch einfach zu berechnen sind. Wir zeigen das
algebraische ANOMAX-Verfahren zum Erreichen einer niedrigen Bitfehlerrate und seine Modifikation
RR-ANOMAX zum Erreichen einer hohen Summenrate. Für den Spezialfall, in dem
die Endgeräte nur eine Antenne verwenden, leiten wir eine semi-algebraische Lösung zum
Finden der Summenraten-optimalen Strategie (RAGES) her. Anhand von numerischen Simulationen
wird die Leistungsfähigkeit dieser Verfahren bezüglich Bitfehlerrate und erreichbarer
Datenrate bewertet und ihre Effektivität gezeigt.Modern society is undergoing a fundamental change in the way we interact with technology.
More and more devices are becoming "smart" by gaining advanced computation capabilities
and communication interfaces, from household appliances over transportation systems to large-scale
networks like the power grid. Recording, processing, and exchanging digital information
is thus becoming increasingly important. As a growing share of devices is nowadays mobile
and hence battery-powered, a particular interest in efficient digital signal processing techniques
emerges.
This thesis contributes to this goal by demonstrating methods for finding efficient algebraic
solutions to various applications of multi-channel digital signal processing. These may not
always result in the best possible system performance. However, they often come close while
being significantly simpler to describe and to implement. The simpler description facilitates a
thorough analysis of their performance which is crucial to design robust and reliable systems.
The fact that they rely on standard algebraic methods only allows their rapid implementation
and test under real-world conditions.
We demonstrate this concept in three different application areas. First, we present a semi-algebraic
framework to compute the Canonical Polyadic (CP) decompositions of multidimensional
signals, a very fundamental tool in multilinear algebra with applications ranging from
chemistry over communications to image compression. Compared to state-of-the art iterative
solutions, our framework offers a flexible control of the complexity-accuracy trade-off and
is less sensitive to badly conditioned data. The second application area is multidimensional
subspace-based high-resolution parameter estimation with applications in RADAR, wave propagation
modeling, or biomedical imaging. We demonstrate that multidimensional signals can
be represented by tensors, providing a convenient description and allowing to exploit the
multidimensional structure in a better way than using matrices only. Based on this idea,
we introduce the tensor-based subspace estimate which can be applied to enhance existing
matrix-based parameter estimation schemes significantly. We demonstrate the enhancements
by choosing the family of ESPRIT-type algorithms as an example and introducing enhanced
versions that exploit the multidimensional structure (Tensor-ESPRIT), non-circular source
amplitudes (NC ESPRIT), and both jointly (NC Tensor-ESPRIT). To objectively judge the
resulting estimation accuracy, we derive a framework for the analytical performance assessment
of arbitrary ESPRIT-type algorithms by virtue of an asymptotical first order perturbation
expansion. Our results are more general than existing analytical results since we do not need
any assumptions about the distribution of the desired signal and the noise and we do not
require the number of samples to be large. At the end, we obtain simplified expressions for the
mean square estimation error that provide insights into efficiency of the methods under various
conditions. The third application area is bidirectional relay-assisted communications. Due to
its particularly low complexity and its efficient use of the radio resources we choose two-way
relaying with a MIMO amplify and forward relay. We demonstrate that the required channel
knowledge can be obtained by a simple algebraic tensor-based channel estimation scheme. We
also discuss the design of the relay amplification matrix in such a setting. Existing approaches
are either based on complicated numerical optimization procedures or on ad-hoc solutions
that to not perform well in terms of the bit error rate or the sum-rate. Therefore, we propose
algebraic solutions that are inspired by these performance metrics and therefore perform well
while being easy to compute. For the MIMO case, we introduce the algebraic norm maximizing
(ANOMAX) scheme, which achieves a very low bit error rate, and its extension Rank-Restored
ANOMAX (RR-ANOMAX) that achieves a sum-rate close to an upper bound. Moreover, for
the special case of single antenna terminals we derive the semi-algebraic RAGES scheme which
finds the sum-rate optimal relay amplification matrix based on generalized eigenvectors. Numerical
simulations evaluate the resulting system performance in terms of bit error rate and
system sum rate which demonstrates the effectiveness of the proposed algebraic solutions
Synchronization Problems in Computer Vision
The goal of \u201csynchronization\u201d is to infer the unknown states of a network of nodes, where only the ratio (or difference) between pairs of states can be measured. Typically, states are represented by elements of a group, such as the Symmetric Group or the Special Euclidean Group. The former can represent local labels of a set of features, which refer to the multi-view matching application, whereas the latter can represent camera reference frames, in which case we are in the context of structure from motion, or local coordinates where 3D points are represented, in which case we are dealing with multiple point-set registration. A related problem is that of \u201cbearing-based network localization\u201d where each node is located at a fixed (unknown) position in 3-space and pairs of nodes can measure the direction of the line joining their locations. In this thesis we are interested in global techniques where all the measures are considered at once, as opposed to incremental approaches that grow a solution by adding pieces iteratively
Theory and Algorithms for Reliable Multimodal Data Analysis, Machine Learning, and Signal Processing
Modern engineering systems collect large volumes of data measurements across diverse sensing modalities. These measurements can naturally be arranged in higher-order arrays of scalars which are commonly referred to as tensors. Tucker decomposition (TD) is a standard method for tensor analysis with applications in diverse fields of science and engineering. Despite its success, TD exhibits severe sensitivity against outliers —i.e., heavily corrupted entries that appear sporadically in modern datasets. We study L1-norm TD (L1-TD), a reformulation of TD that promotes robustness. For 3-way tensors, we show, for the first time, that L1-TD admits an exact solution via combinatorial optimization and present algorithms for its solution. We propose two novel algorithmic frameworks for approximating the exact solution to L1-TD, for general N-way tensors. We propose a novel algorithm for dynamic L1-TD —i.e., efficient and joint analysis of streaming tensors. Principal-Component Analysis (PCA) (a special case of TD) is also outlier responsive. We consider Lp-quasinorm PCA (Lp-PCA) for