The increasing demand for mobile data has placed significant strain on radio access networks (RANs), leading to a continuous need for increased network capacity. In keeping with that, a significant advancement in modern RANs is the ability to utilize several receivers and transmitters, to allow for beamforming. One way to increase the capacity of the network is therefore to optimize the resource allocation by preprocessing the transmitted signals, which involves several costly matrix multiplications (MMs). The aim of the project was to investigate the potential of accelerating Ericsson's RAN simulations by using AI based approximate matrix multiplication (AMM) algorithms. The main focus was on the multiply additionless (MADDNESS) algorithm, a product quantization technique that has achieved speedups of up to 100 times compared to exact MM, and 10 times faster than previous AMM methods. A complex matrix handling version of MADDNESS was implemented in Java and Python respectively, and its speed and accuracy were evaluated against Ericsson's current MM implementation. The proposed implementation did not beat the benchmark with respect to speed, instead resulting in a 4-10 times slowdown in runtime. However, this may largely be due to the fact that the used languages do not allow for complete control over memory resource allocation. As such, the implementations at hand do not incorporate all the crucial features of the algorithm. Particularly, the handicapped version does not fully leverage the vectorization potential, which is one of the key contributors to the speed of the algorithm. Consequently, further improvements are necessary before employing the techniques in an end-to-end implementation. / Den växande efterfrågan på mobildata har ökat belastningen på dagens radionätverk (RAN) och har medfört ett behov av att utvidga dess kapacitet. En betydande innovation inom RAN är beamforming, vilket är förmågan att fokusera digitala signaler mot mottagaren och på så vis öka singalstyrkan. En metod för att öka kapaciteten i ett nätverk är att optimera både kvaliteten av och resursallokeringen mellan nätverkets digitala kanaler, vilket medför tidskrävande matrismultiplikationer. Syftet med denna studie var att utforska om AI-baserade approximativa matrismultiplikationsalgoritmer har potentialen att accelerera Ericssons digitala tvilling-simuleringar. Studien fokuserade i huvudsak på produktkvantiseringsalgoritmen MADDNESS som påvisat potentialen att accelerera exakta matrismultiplikationer med en faktor 100, samt en faktor 10 snabbare än jämförbara approximativa metoder. En modifierad version av MADDNESS, som behandlar komplexa matriser, implementerades i Java samt Python, varefter precisionen och hastigheten utvärderades. Den föreslagna implementationen resulterade i en försämring med avseende på hastigheten med en faktor 4-10 jämfört med Ericssons nuvarande algoritmer. Den föreslagna implementationen saknar effektiv minnesallokering och misslyckas följaktligen att till fullo ta tillvara på vektoriseringspotentialen i MADDNESS. Detta indikerar att det är nödvändigt för ytterligare förbättringar innan algoritmen är användbar i den givna simuleringsmiljön.

Indiana University-Purdue University Indianapolis (IUPUI) / I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.

This thesis presents two methods that enable high internal quantizer resolution in delta-sigma analog-to-digital converters. Increasing the quantizer resolution in a delta-sigma modulator can increase SNR, improve stability and reduce integrator power consumption. However, each added bit of quantizer resolution also causes an exponential increase in the power dissipation, required area and complexity of the dynamic element matching (DEM) circuit required to attenuate digital-to-analog converter (DAC) mismatch errors. One way to overcome these drawbacks is to segment the feedback signal, creating a "coarse" signal and a "fine" signal. This reduces the DEM circuit complexity, power dissipation, and size. However, it also creates additional problems. The negative consequences of segmentation are presented, along with two potential solutions: one that uses calibration to cancel mismatch between the "coarse" DAC and the "fine" DAC, and another that frequency-shapes this mismatch error. Mathematical analysis and behavioral simulation results are presented. A potential circuit design for the frequency-shaping method is presented in detail. Circuit simulations for one of the proposed implementations show that the delay through the digital path is under 7 ns, thus permitting a 50 MHz clock frequency for the overall ADC.

In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schrödinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.

Sustainably realizing applications of the future with high performance demands requires that energy efficiency becomes a central design criterion for the entire system. For example, the power consumption of the analog-to-digital converter (ADC) can become a major factor when transmitting at large bandwidths and carrier frequencies, e.g., for ultra-short range high data rate communication. The consumed energy per conversion step increases with the sampling rate such that high resolution ADCs become unfeasible in the sub-THz regime at the very high sampling rates required. This makes signaling schemes adapted to 1-bit quantizers a promising alternative. We therefore quantify the performance of bandlimited 1-bit quantized wireless communication channels using techniques like oversampling and faster-than-Nyquist (FTN) signaling to compensate for the loss of achievable rate. As a limiting case, we provide bounds on the mutual information rate of the hard bandlimited 1-bit quantized continuous-time – i.e., infinitely oversampled – additive white Gaussian noise channel in the mid-to-high signal-to-noise ratio (SNR) regime. We derive analytic expressions using runlength encoded input signals. For real signals the maximum value of the lower bound on the spectral efficiency in the high-SNR limit was found to be approximately 1.63 bit/s/Hz. Since in practical scenarios the oversampling ratio remains finite, we derive bounds on the achievable rate of the bandlimited oversampled discrete-time channel. These bounds match the results of the continuous-time channel remarkably well. We observe spectral efficiencies up to 1.53 bit/s/Hz in the high-SNR limit given hard bandlimitation. When excess bandwidth is tolerable, spectral efficiencies above 2 bit/s/Hz per domain are achievable w.r.t. the 95 %-power containment bandwidth. Applying the obtained bounds to a bandlimited oversampled 1-bit quantized multiple-input multiple-output channel, we show the benefits when using appropriate power allocation schemes. As a constant envelope modulation scheme, continuous phase modulation is considered in order to relieve linearity requirements on the power amplifier. Noise-free performance limits are investigated for phase shift keying (PSK) and continuous phase frequency shift keying (CPFSK) using higher-order modulation alphabets and intermediate frequencies. Adapted waveforms are designed that can be described as FTN-CPFSK. With the same spectral efficiency in the high-SNR limit as PSK and CPFSK, these waveforms provide a significantly improved bit error rate (BER) performance. The gain in SNR required for achieving a certain BER can be up to 20 dB. / Die nachhaltige Realisierung von zukünftigen Übertragungssystemen mit hohen Leistungsanforderungen erfordert, dass die Energieeffizienz zu einem zentralen Designkriterium für das gesamte System wird. Zum Beispiel kann die Leistungsaufnahme des Analog-Digital-Wandlers (ADC) zu einem wichtigen Faktor bei der Übertragung mit großen Bandbreiten und Trägerfrequenzen werden, z. B. für die Kommunikation mit hohen Datenraten über sehr kurze Entfernungen. Die verbrauchte Energie des ADCs steigt mit der Abtastrate, so dass hochauflösende ADCs im Sub-THz-Bereich bei den erforderlichen sehr hohen Abtastraten schwer einsetzbar sind. Dies macht Signalisierungsschemata, die an 1-Bit-Quantisierer angepasst sind, zu einer vielversprechenden Alternative. Wir quantifizieren daher die Leistungsfähigkeit von bandbegrenzten 1-Bit-quantisierten drahtlosen Kommunikationssystemen, wobei Techniken wie Oversampling und Faster-than-Nyquist (FTN) Signalisierung eingesetzt werden, um den durch Quantisierung verursachten Verlust der erreichbaren Rate auszugleichen. Wir geben Grenzen für die Transinformationsrate des Extremfalls eines strikt bandbegrenzten 1-Bit quantisierten zeitkontinuierlichen – d.h. unendlich überabgetasteten – Kanals mit additivem weißen Gauß’schen Rauschen bei mittlerem bis hohem Signal-Rausch-Verhältnis (SNR) an. Wir leiten analytische Ausdrücke basierend auf lauflängencodierten Eingangssignalen ab. Für reelle Signale ist der maximale Wert der unteren Grenze der spektralen Effizienz im Hoch-SNR-Bereich etwa 1,63 Bit/s/Hz. Da die Überabtastrate in praktischen Szenarien endlich bleibt, geben wir Grenzen für die erreichbare Rate eines bandbegrenzten, überabgetasteten zeitdiskreten Kanals an. Diese Grenzen stimmen mit den Ergebnissen des zeitkontinuierlichen Kanals bemerkenswert gut überein. Im Hoch-SNR-Bereich sind spektrale Effizienzen bis zu 1,53 Bit/s/Hz bei strikter Bandbegrenzung möglich. Wenn Energieanteile außerhalb des Frequenzbandes tolerierbar sind, können spektrale Effizienzen über 2 Bit/s/Hz pro Domäne – bezogen auf die Bandbreite, die 95 % der Energie enthält – erreichbar sein. Durch die Anwendung der erhaltenen Grenzen auf einen bandbegrenzten überabgetasteten 1-Bit quantisierten Multiple-Input Multiple-Output-Kanal zeigen wir Vorteile durch die Verwendung geeigneter Leistungsverteilungsschemata. Als Modulationsverfahren mit konstanter Hüllkurve betrachten wir kontinuierliche Phasenmodulation, um die Anforderungen an die Linearität des Leistungsverstärkers zu verringern. Beschränkungen für die erreichbare Datenrate bei rauschfreier Übertragung auf Zwischenfrequenzen mit Modulationsalphabeten höherer Ordnung werden für Phase-shift keying (PSK) and Continuous-phase frequency-shift keying (CPFSK) untersucht. Weiterhin werden angepasste Signalformen entworfen, die als FTN-CPFSK beschrieben werden können. Mit der gleichen spektralen Effizienz im Hoch-SNR-Bereich wie PSK und CPFSK bieten diese Signalformen eine deutlich verbesserte Bitfehlerrate (BER). Die Verringerung des erforderlichen SNRs zur Erreichung einer bestimmten BER kann bis zu 20 dB betragen.

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[pt] O emprego conjunto de Redes de Acesso por Rádio em Nuvem (CRANs) e sistemas de múltiplas entradas e múltiplas saídas (MIMO) de larga escala é uma solução chave para atender aos requisitos da quinta geração (5G) de redes sem fio. No entanto, alguns desafios ainda precisam ser superados como a redução do consumo de energia do sistema, a capacidade limitada dos links fronthaul e a redução dos custos de implantação e operação. Embora seja prejudicial para o desempenho do sistema, a quantização em baixa resolução é proposta como uma solução para estes desafios. Portanto, técnicas que melhoram o desempenho de sistemas quantizados grosseiramente são necessárias. Em sistemas móveis, os ADCs geralmente são precedidos por um controle de ganho automático (AGC). O AGC trabalha moldando a amplitude do sinal recebido dentro do intervalo do quantizador para usar eficientemente a resolução. A fim de solucionar esses problemas, esta dissertação apresenta uma otimização conjunta do AGC, que funciona nas cabeças de rádio remotas (RRHs), e um filtro de recepção linear de baixa resolução consciente (LRA) baseado no mínimo erro quadrático médio (MMSE), que funciona na unidade de nuvem (CU), para sistemas quantizados grosseiramente. Desenvolvemos receptores de cancelamento de interferência lineares e sucessivos (SIC) com base na proposta conjunta de AGC e LRA MMSE (AGC-LRA-MMSE). Uma análise da soma das taxas alcançáveis juntamente com um estudo de complexidade computacional também são realizadas. As simulações mostram que o projeto proposto fornece taxas de erro reduzidas e taxas alcançáveis mais altas do que as técnicas existentes. / [en] The joint employment of Cloud Radio Access Networks (C-RANs) and large-scale multiple-input multiple-output (MIMO) systems is a key solution to fulfill the requirements of the fifth generation (5G) of wireless networks. However, some challenges are still open to be overcome such as the high power consumption of large-scale MIMO systems, which employ a large number of analog-to-digital converters (ADCs), the capacity bottleneck of the fronthaul links and the system cost reduction. Although it often affects the system performance, the low-resolution quantization is a possible solution for these problems. Therefore, techniques that improve the performance of coarsely quantized systems are needed. In mobile applications, the ADCs are usually preceded by an automatic gain control (AGC). The AGC works shaping the received signal amplitude within the quantizer range to efficiently use the ADC resolution. Then, the optimization of an AGC is especially important. In order to present possible solutions for these issues, this thesis presents a joint optimization of the AGC, which works in the remote radio heads (RRHs), and a low-resolution aware (LRA) linear receive filter based on the minimum mean square error (MMSE), which works in the cloud unit (CU), for coarsely quantized large-scale MIMO with CRAN systems. We develop linear and successive interference cancellation (SIC) receivers based on the proposed joint AGC and LRA MMSE (AGCLRAMMSE) approach. An analysis of the achievable sum rates along with a computational complexity study is also carried out. Simulations show that the proposed design provides improved error rates and higher achievable rates than existing techniques.

Current trends on video technologies and services are demanding higher bit rates, highervideo resolutions and better video qualities. This issue results in the need of a new generationof video coding techniques to increase the quality and compression rates of previous standards.Since the release of HEVC, ITU-T VCEG and ISO/IEC MPEG have been studying the potentialneed for standardization of future video coding technologies with a compression capability thatsignificantly exceeds the ones from current standards. These new e↵orts of standardization andcompression enhancements are being implemented and evaluated over a software test modelknown under the name of Joint Exploration Model (JEM). One of the blocks being explored inJEM is an Adaptive Loop Filter (ALF) at the end of each frame’s processing flow. ALF aimsto minimize the error between original pixels and decoded pixels using Wiener-based adaptivefilter coefficients, reporting, in its JEM’s implementation, improvements of around a 1% in theBD MS-SSIM rate. A lot of e↵orts have been devoted on improving this block over the pastyears. However, current ALF implementations do not consider the potential use of adaptive QPalgorithms at the encoder. Adaptive QP algorithms enable the use of di↵erent quality levels forthe coding of di↵erent parts of a frame to enhance its subjective quality.In this thesis, we explore potential improvements over di↵erent dimensions of JEM’s AdaptiveLoop Filter block considering the potential use of adaptive QP algorithms. In the document, weexplore a great gamut of modification over ALF processing stages, being the ones with betterresults (i) a QP-aware implementation of ALF were the filter coefficients estimation, the internalRD-optimization and the CU-level flag decision process are optimized for the use of adaptiveQP, (ii) the optimization of ALF’s standard block activity classification stage through the useof CU-level information given by the di↵erent QPs used in a frame, and (iii) the optimizationof ALF’s standard block activity classification stage in B-frames through the application of acorrection weight on coded, i.e. not predicted, blocks of B-frames. These ALF modificationscombined obtained improvements of a 0.419% on average for the BD MS-SSIM rate in the lumachannel, showing each modification individual improvements of a 0.252%, 0.085% and 0.082%,respectively. Thus, we concluded the importance of optimizing ALF for the potential use ofadaptive-QP algorithms in the encoder, and the benefits of considering CU-level and frame-levelmetrics in ALF’s block classification stage. / Utvecklingen inom video-teknologi och service kräver högre bithastighet, högre videoupplösningoch bättre kvalitet. Problemet kräver en ny generation av kodning och tekniker för att ökakvaliteten och komprimeringsgraden utöver vad tidigare teknik kunnat prestera. Sedan lanseringenav HEVC har ITU-T VCEG och ISO/IEC MPEG studerat ett eventuellt behov av standardiseringav framtida video-kodings tekniker med kompressions kapacitet som vida överstigerdagens system. Dessa försök till standardisering och kompressionsframsteg har implementeratsoch utvärderats inom ramen för en mjukvara testmodell som kallas Joint Exploration Model(JEM). Ett av områdena som undersöks inom ramen för JEM är adaptiva loopfilter (ALF) somläggs till i slutet av varje bilds processflöde. ALF har som mål att minimera felet mellan originalpixel och avkodad pixel genom Wiener-baserade adaptiva filter-koefficienter. Mycket kraft harlagts på att förbättra detta område under de senaste åren. Men, nuvarande ALF-appliceringbeaktar inte potentialen av att använda adaptiva QP algoritmer i videokodaren. Adaptiva QPalgoritmer tillåter användningen av olika kvalitet på kodning av olika delar av bilden för attförbättra den subjektiva kvaliteten.I föreliggande uppsats kommer vi undersöka den potentiella förbättringen av JEM:s adaptivaloopfilter som kan uppnås genom att använda adaptiva QP algoritmer. I uppsatsen kommervi undersöka ett stort antal modifikationer i ALF:s process-stadier, för att ta reda på vilkenmodifikationer som har bäst resultat: (i) en QP-medveten implementering av ALF där filterkoefficientensuppskattning av den interna RD-optimeringen och CU-nivåns flaggbeslutsprocessär optimerade för användnngen av adaptiv QP, (ii) optimeringen av ALF:s standard blockaktivitets klassificerings stadie genom användning av CU-nivå-information producerad av deolika QP:n som används i en bild, och (iii) optimeringen av ALF:s standard block aktivitetsklassificerings stadier i B-bilders genom applicering av en korrektursvikt i tidigare kod, d.v.sej förutsedda, block av B-bilder. När dessa ALF modifikationer kombinerades förbättradesi genomsnitt BD MS-SSIM hastigheten i luma kanalen med 0.419%, där varje modifikationförbättrade med 0.252%, 0.085% och 0.082% var. Därigenom drog vi slutstatsen att det är viktigtatt optimera ALF för det potentiella användandet av adaptiva QP-algoritmer i kodaren, ochfördelarna av att beakta CU-nivåmätningar och bild-nivåmätningar i ALF:s block klassificeringsstadie.

After giving a brief background on the subject we introduce in section two the Phase-Integral Method of Fröman & Fröman in terms of the platform function of Yngve and Thidé. In section three we derive a different form of the radial Bohr-Sommerfeld condition in terms of the apsidal angle of the corresponding classical motion. Using the derived expression, we then show how easily one can calculate the exact energy eigenvalues of the hydrogen atom and the isotropic three-dimensional harmonic oscillator, we also derive an expression for higher order quantization condition. In section four we derive an expression for the angular frequencies of a restricted (0≤φ≤β) soap bubble and also give a proposition concerning the parameters l and m of the associated Legendre differential equation. / Vi använder Fröman & Frömans Fas-Integral Metod tillsammans med Yngve & Thidés plattformfunktion för att härleda kvantiseringsvilkoret för högre ordningar. I sektion tre skriver vi Bohr-Sommerfelds kvantiseringsvillkor på ett annorlunda sätt med hjälp av den så kallade apsidvinkeln (definierad i samma sektion) för motsvarande klassiska rörelse, vi visar också hur mycket detta underlättar beräkningar av energiegenvärden för väteatomen och den isotropa tredimensionella harmoniska oscillatorn. I sektion fyra tittar vi på en såpbubbla begränsad till området 0≤φ≤β för vilket vi härleder ett uttryck för dess (vinkel)egenfrekvenser. Här ger vi också en proposition angående parametrarna l och m tillhörande den associerade Legendreekvationen.