Dans cette thèse, l'holographie numérique dans l'axe (DIH) est la principale méthode optique utilisée pour analyser des inclusions dans une gouttelette. L'holographie numérique dans l'axe est utilisée pour caractériser des inclusions du point de vue de leur taille, leur position 3D et leur trajectoire à l'intérieur de la gouttelette. Comme les particules sont situées à l'intérieur d'une gouttelette, le front d'onde incident sur l'inclusion est modifié avant qu'il l'illumine. Le défi de ce travail est double : premièrement de prendre en compte la forme de la gouttelette dans le modèle d'holographie et deuxièmement d'étendre l'analyse aux inclusions transparentes (type objet de phase). Pour décrire l'hologramme enregistré par le capteur CCD, l'intégrale d'Huygens-Fresnel et le formalisme des matrices ABCD ont été utilisés. Dans ce modèle, nous introduisons les polynômes de Zernike pour décrire la fonction de transmission d'une particule. Pour l'analyse des hologrammes, l'outil mathématique de la transformation de Fourier fractionnaire 2D (2D-FRFT) est utilisé pour restituer l'image des inclusions et dans ce cas une mesure la taille de l'inclusion et de sa position 3D sont réalisées. Les trajectoires des inclusions dans la goutte est possible avec un long temps de pose de l'obturateur du capteur CCD. Nous avons également proposé un nouveau modèle pour décrire des objets de phases quelconque et des particules opaques. Pour ce nouveau modèle, les mêmes procédés ont été utilisés. Dans le cas d'inclusions filiformes à l'intérieur d'une géométrie cylindrique comme un canal, une méthode de simulation d'imagerie interférométrique multi-coeurs est proposée. Dans ce cas, une somme de distributions de Dirac, localisées le long d'une droite, introduite dans l'intégrale de Fresnel généralisée (c'est-à-dire le formalisme des matrices ABCD et l'intégrale de Fresnel) permet d'obtenir un bon degré de similitude entre l'expérience et la simulation. / In this thesis, the digital in-line holography (DIH) is the main optical method used to analyze inclusions in a droplet. The digital in-line holography is used to characterize the inclusions in terms of of their size, their 3D position, and their trajectories inside the droplet. Since the particles are located within a droplet, the incident wavefront is changed before it illuminates the inclusions. The challenge of this work has two points : first to take into account the shape of the droplet in the holographic model and secondly to extend the analysis to the transparent inclusions (phase object). To describe the hologram recorded by the CCD sensor, the Huygens-Fresnel integral and the ABCD matrix formalism were used. In this model, we introduce the Zernike polynomials to describe the transmission function of a particle. For the analysis of holograms, the2D fractional Fourier transformation (2D-FRFT) is used to reconstruct the image of inclusions and in this case the size and their 3D position of the inclusions are performed.The trajectories of the inclusions in the drop are possible tracked with a long exposure shutter speed of the CCD. We also proposed a new simulation to describe objects of any phases and opaque particles. For this simulation, the same methods of reconstruction were used. In the case of micro-channel inclusions inside a cylindrical geometry such as a pipe, the interferometric imaging of multi-core pipe is proposed. In this case, summation of Dirac delta distribution, located along a line, introduced into the generalized Fresnel integral allows us to get a good agreement between the experiment and the simulation.

Mild traumatic brain injury (mTBI) is a common and often inconspicuous wound that is frequently associated with chronic low-grade symptoms and cognitive dysfunction. Previous evidence suggests that daily blue wavelength light therapy may be effective at reducing fatigue and improving sleep in patients recovering from mTBI. However, the effects of light therapy on recovering brain structure remain unexplored. In this study, we analyzed white matter diffusion properties, including generalized fractional anisotropy, and the quantity of water diffusion in isotropic (i.e., isotropic diffusion) and anisotropic fashion (i.e., quantitative anisotropy, QA) for fibers crossing 11 brain areas known to be significantly affected following mTBI. Specifically, we investigated how 6 weeks of daily morning blue light exposure therapy (compared to an amber-light placebo condition) impacted changes in white matter diffusion in individuals with mTBI. We observed a significant impact of the blue light treatment (relative to the placebo) on the amount of water diffusion (QA) for multiple brain areas, including the corpus callosum, anterior corona radiata, and thalamus. Moreover, many of these changes were associated with improvements in sleep latency and delayed memory. These findings suggest that blue wavelength light exposure may serve as one of the potential non-pharmacological treatments for facilitating structural and functional recovery following mTBI; they also support the use of QA as a reliable neuro-biomarker for mTBI therapies.

This thesis is split in two different topics. Firstly, we study anomalous transport from kinetic models. Secondly, we consider the equations coming from weak wave turbulence theory and we study them via mean-field limits of finite stochastic particle systems. $\textbf{Anomalous transport from kinetic models.}$ The goal is to understand how fractional diffusion arises from kinetic equations. We explain how fractional diffusion corresponds to anomalous transport and its relation to the classical diffusion equation. In previous works it has been seen that particles systems undergoing free transport and scattering with the media can give rise to fractional phenomena in two cases: firstly, if in the dynamics of the particles there is a heavy-tail equilibrium distribution; and secondly, if the scattering rate is degenerate for small velocities. We use these known results in the literature to study the emergence of fractional phenomena for some particular kinetic equations. Firstly, we study BGK-type equations conserving not only mass (as in previous results), but also momentum and energy. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density making use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momentum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavy-tailed equilibria. Secondly, we will study diffusion phenomena arising from transport of energy in an anharmonic chain. More precisely, we will consider the so-called FPU-$\beta$ chain, which is a very simple model for a one-dimensional crystal in which atoms are coupled to their nearest neighbours by a harmonic potential, weakly perturbed by a nonlinear quartic potential. The starting point of our mathematical analysis is a kinetic equation; lattice vibrations, responsible for heat transport, are modelled by an interacting gas of phonons whose evolution is described by the Boltzmann Phonon Equation. Our main result is the derivation of an anomalous diffusion equation for the temperature. $\textbf{Weak wave turbulence theory and mean-field limits for stochastic particle systems.}$ The isotropic 4-wave kinetic equation is considered in its weak formulation using model homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting. We also consider finite stochastic particle systems undergoing instantaneous coagulation-fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit).

The thesis is concerned with capacity and coverage enhancement of OFDMA heterogeneous cellular systems with a specific focus on fractional frequency reuse (FFR), femtocells and amplify-and-forward (AF) relay systems. The main aim of the thesis is to develop new mathematical analysis for the spectral efficiency and outage probability of multi-cells multi-tier systems in diverse traffic, interference and fading scenarios. In the first part of the thesis, a new unified mathematical framework for performance analysis of FFR and soft frequency reuse (SFR) schemes is developed. This leads to new exact expressions of FFR and SFR area spectral efficiency in downlink and uplink scenarios which account for a mixture of frequency reuse factors in a homogeneous cellular system. The mathematical framework is extended to include modelling and performance analysis of FFR systems with elastic data traffic. Further analysis is carried out in relation to the performance of FFR and/or SFR schemes, in terms of energy efficiency and base station cooperation. The new proposed analytical framework can lead to a better understanding and computationally efficient performance analysis of next generation heterogeneous cellular systems. Next generation cellular systems are characterized by an increase in the spatial node density to improve the spectral efficiency and coverage, especially for users at home and at the cell edges. In this regard, relays and femtocells play a major role. Therefore, relays and femocells are the focus of the second part of the thesis. Firstly, we present a new and unified spectral efficiency analysis in dual-hop fixed-gain AF relay systems over generalised interferences models. The generalised interference models are either based on the Nakagami-m fading with arbitrary distance or on spatial Poisson Point Process in case of randomly deployed heterogeneous interferers. The models have been considered separately in the open literature due to the complexity of the mathematical analysis. Secondly, the outage probability is utilised to deduce the femtocell exclusion region for FFR system and a new static resource allocation scheme is proposed for femtocells which improve the capacity. The work presented in the thesis has resulted in the publication of seven scientific papers in prestigious IEEE journals and conferences.

O objetivo deste trabalho é realizar uma análise sobre a cinética de nêutrons utilizando um modelo fracionário, recente na literatura. Este novo modelo envolve uma nova relação constitutiva entre a corrente de nêutrons e o fluxo escalar de nêutrons, em que uma das suposições realizada para a relação constitutiva clássica (lei de Fick) não é feita: a densidade de corrente não mais é irrelevante se comparada `a taxa de colisão. Ao considerar esta hipótese, este novo modelo aqui apresentado se torna mais abrangente que o modelo fickiano clássico. Para manter o caráter geral do modelo é considerado um operador diferencial de ordem fracionaria em sua dedução, o que acresce ao modelo dois novos parâmetros: o tempo de relaxação e a ordem da derivada fracionária. Considerando este modelo, é realizada a dedução das equações de cinética de nêutrons, com ênfase na cinética pontual de nêutrons. A equação fracionária de cinética pontual de nêutrons é resolvida analiticamente através do método da decomposição, a fim de estudar o novo modelo minimizando os erros numéricos. Os resultados obtidos através desta metodologia são comparados com os resultados clássicos em diversos casos, com a finalidade de analisar a influência dos parâmetros fracionários no modelo. / The objective of this work is to make an analysis about the neutron kinetics using a recent model on the literature. This new model involves a new constitutive relation between the neutron current and the neutron scalar flux, in which one of the simplifications made to find the classical constitutive relation (Fick’s law) is not considered: the current density no longer is irrelevant as compared to the collision rate. By considering this hypothesis, this new model here presented becomes more general than the classic fickian model. In order to keep the general property of the model, a differential operator with fractional order is considered in its deduction, which adds two new parameters to the model: the relaxation time and the fractional derivative order. Considering this model, the neutron kinetics equations are deduced, with emphasis on the neutron point kinetics. The neutron point kinetic equation is analytically solved by the Adomian’s decomposition method, in order to study the new model minimizing numerical errors. The results obtained by this methodology are compared to the classic results in several cases, in order to analyse the fractional parameters influences to the model.

Surface temperature control of a thin plate is investigated. Temperature is controlled on one side of the plate using the other side temperature measurements. This is a decades-old problem, reactivated more recently by the awareness that this is a fractional-order problem that justifies the investigation of the use of fractional order calculus. The approach is based on a transfer function obtained from the one-dimensional heat conduction equation solution that results in a fractional-order s-domain representation. Both the inverse problem approach and the fractional controller approach are studied here to control the surface temperature, the first one using inverse problem plus a Proportional only controller, and the second one using only the fractional controller. The direct problem defined as the ratio of the output to the input, while the inverse problem defined as the ratio of the input to the output. Both transfer functions are obtained, and the resulting fractional-order transfer functions were approximated using Taylor expansion and Zero-Pole expansion. The finite number of terms transfer functions were used to form an open-loop control scheme and a closed-loop control scheme. Simulation studies were done for both control schemes and experiments were carried out for closed-loop control schemes. For the fractional controller approach, the fractional controller was designed and used in a closed-loop scheme. Simulations were done for fractional-order-integral, fractional-order-derivative and fractional-integral-derivative controller designs. The experimental study focussed on the fractional-order-integral-derivative controller design. The Fractional-order controller results are compared to integer-order controller’s results. The advantages of using fractional order controllers were evaluated. Both Zero-Pole and Taylor expansions are used to approximate the plant transfer functions and both expansions results are compared. The results show that the use of fractional order controller performs better, in particular concerning the overshoot.

Multi-tier heterogeneous networks have become an essential constituent for next generation cellular networks. Meanwhile, energy efficiency (EE) has been considered a critical design criterion along with the traditional spectral efficiency (SE) metric. In this context, we study power and spectrum allocation for the recently proposed two-tier network architecture known as phantom cellular networks. The optimization framework includes both EE and SE. First, we consider sparsely deployed cells experiencing negligible interference and assume perfect channel state information (CSI). For this setting, we propose an algorithm that finds the SE and EE resource allocation strategies. Then, we compare the performance of both design strategies versus number of users, and phantom cells share of the total available resource units (RUs). We aim to investigate the effect of some system parameters to achieve improved SE performance at a non-significant loss in EE performance, or vice versa. It is found that increasing phantom cells share of RUs decreases the SE performance loss due to EE optimization when compared with the optimized SE performance. Second, we consider the densely deployed phantom cellular networks and model the EE optimization problem having into consideration the inevitable interference and imperfect channel estimation. To this end, we propose three resource allocation strategies aiming at optimizing the EE performance metric of this network. Furthermore, we investigate the effect of changing some of the system parameters on the performance of the proposed strategies, such as phantom cells share of RUs, number of deployed phantom cells within a macro cell coverage, number of pilots and the maximum power available for transmission by the phantom cells BSs. It is found that increasing the number of pilots deteriorates the EE performance of the whole setup, while increasing maximum power available for phantom cells transmissions reduces the EE of the whole setup in a less severe way than increasing the number of pilots. It is found also that increasing phantom cells share increases the EE metric in the dense deployment case. Thus, it is always useful to allocate most of the network RUs to the phantom cells tier.

Der Beitrag stellt Zweischicht-Bruchlochwicklungen mit einem Wickelschritt von zwei Nuten als einen m¨oglichen Kompromiss vor, welcher weniger ohmsche Verluste als bei klassischen verteilten Wicklungen und weniger Ummagnetisierungsverluste als bei Zahnspulenwicklungen ermöglicht. Er diskutiert Entwurf und Eigenschaften solcher Wicklungen f¨ur drei- und sechssträngige Synchronmaschinen anhand ausgewählter Beispiele. / This paper presents two layer fractional slot windings with a winding step of two slots as a possible compromise which enables less ohmic loss than classic distributed windings would have and less hysteresis and eddy loss than fractional slot concentrated windings would have. It discusses properties and design of such windings for three- and six-phase synchronous machines based on selected examples.

This paper concentrates on the description of fractional order LTI SISO systems using generalized Laguerre functions. There are properties of generalized Laguerre functions described in the paper, and an orthogonal base of these functions is shown. Next the concept of fractional derivatives is explained. The last part of this paper deals with the representation of fractional order LTI SISO systems using generalized Laguerre functions. Several examples were solved to demonstrate the benefits of using these functions for the representation of LTI SISO systems.

Wireless communication systems are based on frequency synthesizers that generate carrier signals, which are used to transmit information. Frequency synthesizers use voltage controlled oscillators (VCO) to produce the required frequencies within a specified period of time. In the process of generating frequency, the VCO and other electronic components such as amplifiers produce some unwanted short-term frequency variations, which cause frequency instability within the frequency of interest known as phase noise (PN). PN has a negative impact on the performance of the overall wireless communication system. A literature study conducted on this research reveals that the existing PN cancellation techniques have some limitations and drawbacks that require further attention. A new PN correction technique based on the combination of least mean square (LMS) adaptive filtering and single-loop single-bit Sigma Delta (SD) modulator is proposed. The new design is also based on the Cascaded Resonator Feedback (CRFB) architecture. The noise transfer function (NTF) of the architecture was formulated in way that made it possible to stabilize the frequency fluctuations within the in-band (frequency of interest) by locating its poles and zeros within the unit circle. The new design was simulated and tested on a commercially available software tool called Agilent Advanced Design System (ADS). Simulation results show that the new technique achieves better results when compared with existing techniques as it achieves a 104 dB signal-to-noise (SNR), which is an improvement of 9 dB when compared with the existing technique accessed from the latest publications. The new design also achieves a clean signal with minimal spurious tones within the inband with a phase noise level of -141 dBc/Hz (lower phase noise level by 28 dBc/Hz) when compared with the existing techniques. / Thesis (MEng)--University of Pretoria, 2017. / Electrical, Electronic and Computer Engineering / MEng / Unrestricted