With the increasing demand for faster communication systems, soon data rates in the terabit regime (100 Gbit/s and beyond) are required, which yields new challenges for the design of analog-to-digital converters (ADCs) since high bandwidths imply high sampling rates. For sampling rates larger than 300MHz, which we now achieve with 5G, the ADC power consumption per conversion step scales quadratically with the sampling rate. Thus, ADCs become a major energy consumption bottleneck. To circumvent this problem, we consider digital receivers based on 1-bit quantization and oversampling. We motivate this concept by a brief comparison of the energy efficiency of a recently proposed system employing 1-bit quantization and oversampling to the conventional approach using high resolution quantization and Nyquist rate sampling. Our numerical results show that the energy efficiency can be improved significantly by employing 1-bit quantization and oversampling at the receiver at the cost of increased bandwidth. The main part of this work is concerned with the synchronization of fully digital receivers using 1-bit quantization and oversampling. As a first step, we derive performance bounds for phase, timing, and frequency estimation in order to gain a deeper insight into the impact of 1-bit quantization and oversampling. We identify uniform phase and sample dithering as crucial to combat the non-linear behavior introduced by 1-bit quantization. This dithering can be implemented by sampling at an irrational intermediate frequency and with an oversampling factor with respect to the symbol rate that is irrational, respectively. Since oversampling results in noise correlation, a closed form expression of the likelihood function is not available. To enable an analytical treatment we thus study a system model with white noise by adapting the receive filter bandwidth to the sampling rate. Considering the aforementioned dithering, we obtain very tight closed form lower bounds on the Cramér-Rao lower bound (CRLB) in the large sample regime. We show that with uniform phase and sample dithering, all large sample properties of the CRLB of the unquantized receiver are preserved under 1-bit quantization, except for a signal-to-noise ratio (SNR) dependent performance loss that can be decreased by oversampling. For the more realistic colored noise case, we discuss a numerically computable upper bound of the CRLB and show that the properties of the CRLB for white noise still hold for colored noise except that the performance loss due to 1-bit quantization is reduced. Assuming a neglectable frequency offset, we use the least squares objective function to derive a typical digital matched filter receiver with a data-and timing-aided phase estimator and a timing estimator that is based on square time recovery. We show that both estimators are consistent under very general assumptions, e.g., arbitrary colored noise and stationary ergodic transmit symbols. Performance evaluations are done via simulations and are compared against the numerically computable upper bound of the CRLB. For low SNR the estimators perform well but for high SNR they converge to an error floor. The performance loss of the phase estimator due to decision-directed operation or estimated timing information is marginal. In summary, we have derived practical solutions for the design of fully digital receivers using 1-bit quantization and oversampling and presented a mathematical analysis of the proposed receiver structure. This is an important step towards enabling energy efficient future wireless communication systems with data rates of 100 Gbit/s and beyond.

Fedosov has described a geometro-algebraic method to construct in a canonical way a deformation of the Poisson algebra associated with a finite-dimensional symplectic manifold (\\\"phase space\\\"). His algorithm gives a non-commutative, but associative, product (a so-called \\\"star-product\\\") between smooth phase space functions parameterized by Planck\\\''s constant ℏ, which is treated as a deformation parameter. In the limit as ℏ goes to zero, the star product commutator goes to ℏ times the Poisson bracket, so in this sense his method provides a quantization of the algebra of classical observables. In this work, we develop a generalization of Fedosov\\\''s method which applies to the infinite-dimensional symplectic \\\"manifolds\\\" that occur in Lagrangian field theories. We show that the procedure remains mathematically well-defined, and we explain the relationship of this method to more standard perturbative quantization schemes in quantum field theory.

Die AdS/CFT-Dualität zwischen N=4 SYM und dem AdS_5 × S^5 Superstring zeigt Quanten-Integrabilität im planaren Limes und erlaubte die Konstruktion mächtiger Methoden, welche das Spektrale Problem zu lösen scheinen. Unser Verständnis der direkten Quantisierung des AdS_5 × S^5 Superstrings ist jedoch weiterhin unbefriedigend und besonders das Spektrum kurzer Stringzustände war bisher nur in führender Ordnung in starker ''t Hooft-Kopplung bekannt. In dieser Arbeit untersuchen wir verschiedene Methoden der perturbativen Quantisierung kurzer Strings über die führende Ordnung hinaus, wodurch wir uns auch einen besseres Verständnis der vorhandenen Quanten-Symmetrien erhoffen. Wir fokusieren auf die niedrigst angeregten Stringzustände, dual zum Konishi-Supermultiplet, und begutachten kritisch eine angeblichen Berechnung der Konishi anomalen Skalendimension im Pure-Spinor-Superstring-Formalismus. Als nächstes betrachten wir den bosonischen AdS_5 × S^5 String in statischer Eichung und konstruieren eine sog. Einzelmoden-Stringlösung, eine Veralgemeinerung des pulsierenden Strings durch unbeschränkte Nullmoden. Diese ist klassisch integrabel und quanteninvariant unter den Isometrien SO(2,4) × SO(6). Mögliche Korrekturen der vernachlässigten Supersymmetrie werden heuristisch berücksichtigt, wodurch die ersten Quantenkorrekturen der Konishi anomale Skalendimension reproduzieren werden. Wir implementieren statische Eichung für den AdS_5 × S^5 Superstring und finden elegante Ausdrücke für die Lagrangedichte und Superladungen. Unter Beschränkung auf das Superteilchen finden wir auf zwei unterschiedliche Arten kanonische Koordinaten in quadratischer Ordnung in Fermionen. Schließlich betrachten wir eine weitere Quantisierungsmethode: Da der Einzelmoden-String die SO(2,4) × SO(6)-Bahn des pulsierenden Strings ist, wenden wir Bahn-Methoden-Quantisierung auf das Teilchen und Spinning Strings in bosonischem AdS_3 × S^3 an und erhalten konsistente Ergebnisse für die Spektra. / The initial AdS/CFT duality pair, the duality between N=4 SYM and the AdS_5 × S^5 superstring, appears to enjoy quantum integrability in the planar limit, which allowed to devise powerful methods ostensibly solving the spectral problem. However, quantization of the AdS_5 × S^5 superstring from first principles is still an open question and especially the spectrum of short string states has previously been derived only at leading order in large ''t Hooft coupling. In this thesis we investigate possible routes to quantize short string states perturbatively beyond the leading order, where equally our aim is to gain better appreciation of the quantum symmetries at play. A prominent role is played by the lowest excited string states, dual to the Konishi supermultiplet, and we start by reviewing critically an asserted derivation of the Konishi anomalous dimension in the setup of pure spinor string theory. Next, we constrain ourselves to bosonic AdS_5 × S^5 String in static gauge, where we construct a so-called single-mode string solution, a generalization of the pulsating string allowing for unconstrained zero-modes. This solution shows classical integrability and invariance under the isometries SO(2,4) × SO(6) at the quantum level. Arguing heuristically about the effects of supersymmetry, we indeed recover the first non-trivial quantum correction to the Konishi anomalous dimension. We continue by implementing static gauge for the full AdS_5 × S^5 superstring and find elegant expressions for the Lagrangian density and the supercharges. We then constrain our interest to the superparticle and, using two different methods, find canonical coordinates at quadratic order in fermions. We conclude by exploring another quantization scheme: As the single-mode string is nothing but the SO(2,4) × SO(6) orbit of the pulsating string, we apply orbit method quantization to the particle and spinning string solutions in bosonic AdS_3 × S^3 yielding consistent results for the spectra.

1-bit analog-to-digital conversion is very attractive for low-complexity communications receivers. A major drawback is, however, the small spectral efficiency when sampling at symbol rate. This can be improved through oversampling by exploiting the signal distortion caused by the transmission channel. This paper analyzes the achievable data rate of band-limited communications channels that are subject to additive noise and inter-symbol-interference with 1-bit quantization and oversampling at the receiver. It is shown that not only the channel noise but also the inter-symbol-interference can be exploited to benefit from oversampling.

To meet the exponentially increasing traffic data driven by the rapidly growing mobile subscriptions, both industry and academia are exploring the potential of a new genera- tion (5G) of wireless technologies. An important 5G goal is to achieve high data rate. Small cells with spectrum sharing and multiple-input multiple-output (MIMO) techniques are one of the most promising 5G technologies, since it enables to increase the aggregate data rate by improving the spectral efficiency, nodes density and transmission bandwidth, respectively. However, the increased interference in the densified networks will in return limit the achievable rate performance if not properly managed. The considered setup can be modeled as MIMO interference networks, which can be classified into the K-user MIMO interference channel (IC) and the K-cell MIMO interfering broadcast channel/multiple access channel (MIMO-IBC/IMAC) according to the number of mobile stations (MSs) simultaneously served by each base station (BS). The thesis considers two physical layer (PHY) resource allocation problems that deal with the interference for both models: 1) Pareto boundary computation for the achiev- able rate region in a K-user single-stream MIMO IC and 2) grouping-based interference alignment (GIA) with optimized IA-Cell assignment in a MIMO-IMAC under limited feedback. In each problem, the thesis seeks to provide a deeper understanding of the system and novel mathematical results, along with supporting numerical examples. Some of the main contributions can be summarized as follows. It is an open problem to compute the Pareto boundary of the achievable rate region for a K-user single-stream MIMO IC. The K-user single-stream MIMO IC models multiple transmitter-receiver pairs which operate over the same spectrum simultaneously. Each transmitter and each receiver is equipped with multiple antennas, and a single desired data stream is communicated in each transmitter-receiver link. The individual achievable rates of the K users form a K-dimensional achievable rate region. To find efficient operating points in the achievable rate region, the Pareto boundary computation problem, which can be formulated as a multi-objective optimization problem, needs to be solved. The thesis transforms the multi-objective optimization problem to two single-objective optimization problems–single constraint rate maximization problem and alternating rate profile optimization problem, based on the formulations of the ε-constraint optimization and the weighted Chebyshev optimization, respectively. The thesis proposes two alternating optimization algorithms to solve both single-objective optimization problems. The convergence of both algorithms is guaranteed. Also, a heuristic initialization scheme is provided for each algorithm to achieve a high-quality solution. By varying the weights in each single-objective optimization problem, numerical results show that both algorithms provide an inner bound very close to the Pareto boundary. Furthermore, the thesis also computes some key points exactly on the Pareto boundary in closed-form. A framework for interference alignment (IA) under limited feedback is proposed for a MIMO-IMAC. The MIMO-IMAC well matches the uplink scenario in cellular system, where multiple cells share their spectrum and operate simultaneously. In each cell, a BS receives the desired signals from multiple MSs within its own cell and each BS and each MS is equipped with multi-antenna. By allowing the inter-cell coordination, the thesis develops a distributed IA framework under limited feedback from three aspects: the GIA, the IA-Cell assignment and dynamic feedback bit allocation (DBA), respec- tively. Firstly, the thesis provides a complete study along with some new improvements of the GIA, which enables to compute the exact IA precoders in closed-form, based on local channel state information at the receiver (CSIR). Secondly, the concept of IA-Cell assignment is introduced and its effect on the achievable rate and degrees of freedom (DoF) performance is analyzed. Two distributed matching approaches and one centralized assignment approach are proposed to find a good IA-Cell assignment in three scenrios with different backhaul overhead. Thirdly, under limited feedback, the thesis derives an upper bound of the residual interference to noise ratio (RINR), formulates and solves a corresponding DBA problem. Finally, numerical results show that the proposed GIA with optimized IA-Cell assignment and the DBA greatly outperforms the traditional GIA algorithm.

To meet the exponentially increasing traffic data driven by the rapidly growing mobile subscriptions, both industry and academia are exploring the potential of a new genera- tion (5G) of wireless technologies. An important 5G goal is to achieve high data rate. Small cells with spectrum sharing and multiple-input multiple-output (MIMO) techniques are one of the most promising 5G technologies, since it enables to increase the aggregate data rate by improving the spectral efficiency, nodes density and transmission bandwidth, respectively. However, the increased interference in the densified networks will in return limit the achievable rate performance if not properly managed. The considered setup can be modeled as MIMO interference networks, which can be classified into the K-user MIMO interference channel (IC) and the K-cell MIMO interfering broadcast channel/multiple access channel (MIMO-IBC/IMAC) according to the number of mobile stations (MSs) simultaneously served by each base station (BS). The thesis considers two physical layer (PHY) resource allocation problems that deal with the interference for both models: 1) Pareto boundary computation for the achiev- able rate region in a K-user single-stream MIMO IC and 2) grouping-based interference alignment (GIA) with optimized IA-Cell assignment in a MIMO-IMAC under limited feedback. In each problem, the thesis seeks to provide a deeper understanding of the system and novel mathematical results, along with supporting numerical examples. Some of the main contributions can be summarized as follows. It is an open problem to compute the Pareto boundary of the achievable rate region for a K-user single-stream MIMO IC. The K-user single-stream MIMO IC models multiple transmitter-receiver pairs which operate over the same spectrum simultaneously. Each transmitter and each receiver is equipped with multiple antennas, and a single desired data stream is communicated in each transmitter-receiver link. The individual achievable rates of the K users form a K-dimensional achievable rate region. To find efficient operating points in the achievable rate region, the Pareto boundary computation problem, which can be formulated as a multi-objective optimization problem, needs to be solved. The thesis transforms the multi-objective optimization problem to two single-objective optimization problems–single constraint rate maximization problem and alternating rate profile optimization problem, based on the formulations of the ε-constraint optimization and the weighted Chebyshev optimization, respectively. The thesis proposes two alternating optimization algorithms to solve both single-objective optimization problems. The convergence of both algorithms is guaranteed. Also, a heuristic initialization scheme is provided for each algorithm to achieve a high-quality solution. By varying the weights in each single-objective optimization problem, numerical results show that both algorithms provide an inner bound very close to the Pareto boundary. Furthermore, the thesis also computes some key points exactly on the Pareto boundary in closed-form. A framework for interference alignment (IA) under limited feedback is proposed for a MIMO-IMAC. The MIMO-IMAC well matches the uplink scenario in cellular system, where multiple cells share their spectrum and operate simultaneously. In each cell, a BS receives the desired signals from multiple MSs within its own cell and each BS and each MS is equipped with multi-antenna. By allowing the inter-cell coordination, the thesis develops a distributed IA framework under limited feedback from three aspects: the GIA, the IA-Cell assignment and dynamic feedback bit allocation (DBA), respec- tively. Firstly, the thesis provides a complete study along with some new improvements of the GIA, which enables to compute the exact IA precoders in closed-form, based on local channel state information at the receiver (CSIR). Secondly, the concept of IA-Cell assignment is introduced and its effect on the achievable rate and degrees of freedom (DoF) performance is analyzed. Two distributed matching approaches and one centralized assignment approach are proposed to find a good IA-Cell assignment in three scenrios with different backhaul overhead. Thirdly, under limited feedback, the thesis derives an upper bound of the residual interference to noise ratio (RINR), formulates and solves a corresponding DBA problem. Finally, numerical results show that the proposed GIA with optimized IA-Cell assignment and the DBA greatly outperforms the traditional GIA algorithm.

Communications receivers that rely on 1-bit analogto-digital conversion are advantageous in terms of hardware complexity and power dissipation. Performance limitations due to the 1-bit quantization can be tackled with oversampling. This paper considers the oversampling gain from an information-theoretic perspective by analyzing the channel capacity with 1-bit quantization and oversampling at the receiver for the particular case of AWGN channels. This includes a numerical computation of the capacity and optimal transmit symbol constellations, as well as the derivation of closed-form expressions for large oversampling ratios and for high signal-to-noise ratios of the channel.

1-bit analog-to-digital conversion is very attractive for low-complexity communications receivers. A major drawback is, however, the small spectral efficiency when sampling at symbol rate. This can be improved through oversampling by exploiting the signal distortion caused by the transmission channel. This paper analyzes the achievable data rate of band-limited communications channels that are subject to additive noise and inter-symbol-interference with 1-bit quantization and oversampling at the receiver. It is shown that not only the channel noise but also the inter-symbol-interference can be exploited to benefit from oversampling.

In dieser Arbeit werden Algorithmen für die Berechnung elektronischer Korrelations- und Anregungsenergien mittels der Coupled-Cluster Methode auf adaptiven Gittern entwickelt und implementiert. Die jeweiligen Funktionen und Operatoren werden adaptiv durch Multiskalenanalyse dargestellt, was eine Basissatz unabängige Beschreibung mit kontrollierter numerischer Genauigkeit ermöglicht. Gleichungen für die Coupled-Cluster Methode werden in einem verallgemeinerten Rahmen, unabhängig von virtuellen Orbitalen und globalen Basissätzen, neu formuliert. Hierzu werden die amplitudengewichteten Anregungen in virtuelle Orbitale ersetzt durch Anregungen in n-Elektronenfunktionen, welche durch Gleichungen im n-Elektronen Ortsraum bestimmt sind. Die erhaltenen Gleichungen können, analog zur Basissatz abh¨angigen Form, mit leicht angepasster Interpretation diagrammatisch dargestellt werden. Aufgrund des singulären Coulomb Potentials werden die Arbeitsgleichungen mit einem explizit korrelierten Ansatz regularisiert. Coupled-Cluster singles mit genäherten doubles (CC2) und ähnliche Modelle werden, für geschlossenschalige Systeme und in regularisierter Form, in die MADNESS Bibliothek (eine allgemeine Bibliothek zur Darstellung von Funktionen und Operatoren mittels Multiskalenanalyse) implementiert. Mit der vorgestellten Methode können elektronische CC2 Paarkorrelationsenergien und Anregungsenergien mit bestimmter numerischer Genauigkeit unabhängig von globalen Basissätzen berechnet werden, was anhand von kleinen Molekülen verifiziert wird / In this work algorithms for the computation of electronic correlation and excitation energies with the Coupled-Cluster method on adaptive grids are developed and implemented. The corresponding functions and operators are adaptively represented with multiresolution analysis allowing a basis-set independent description with controlled numerical accuracy. Equations for the coupled-cluster model are reformulated in a generalized framework independent of virtual orbitals and global basis-sets. For this, the amplitude weighted excitations into virtuals are replaced by excitations into n-electron functions which are determined by projected equations in the n-electron position space. The resulting equations can be represented diagrammatically analogous to basis-set dependent approaches with slightly adjusted rules of interpretation. Due to the singular Coulomb potential, the working equations are regularized with an explicitly correlated ansatz. Coupled-cluster singles with approximate doubles (CC2) and similar models are implemented for closed-shell systems and in regularized form into the MADNESS library (a general library for the representation of functions and operators with multiresolution analysis). With the presented approach electronic CC2 pair-correlation energies and excitation energies can be computed with definite numerical accuracy and without dependence on global basis sets, which is verified on small molecules.