<p> The rapid increasing bandwidth requirement of communication systems demands powerful numerical simulation tools for optics fiber. The computational efficient, memory saving and stable are of the most important characteristics for any simulation tools used for long-haul and broadband optics fiber. An optimized split-step digital filtering method is developed in this paper. The concept of Fourier integral and Fourier series are used in extracting a FIR filter which is used to fit the original transfer function. A further optimization process which employs windowing technique to improve computation efficiency had also been done. Compared with split-step frequency method, our method improves the computation efficiency. Only simple shifts and multiplications are needed in our method. This optimized digital filtering method differs from the former digital filtering method in a sense that the filter length of the FIR filter we extracted is reduced to a very small number. The computation time can be saved as much as 96% than before. This method can also be used to solve coupled nonlinear Schrodinger equation which governs polarization mode dispersion effect in fibers. A new simulation scheme for PMD is proposed to save computation time. The propagation results shows good accordance to those already published results. </p> / Thesis / Master of Applied Science (MASc)

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Ultrasound has been widely used in medical diagnostic imaging to image soft tissues. Compared with other methods, ultrasound is superior with no ionizing-radiation, easy portability, low cost, and the capability to provide elasticity information. Conventional ultrasound images provide distorted image information when the ultrasound beam is not normal to the bone structures. In this thesis, we present two imaging algorithms: reverse time migration (RTM) and split-step Fourier migration (SSFM), to image long bones using ultrasound. The methods are tested using simulated data sets. The reconstructed images show accurate cortical thickness measurement and provide the correct fracture dip. The images also clearly illustrate the healing process of a 1-mm wide crack with different in-filled tissue velocities simulating fracture healing. Two in-vitro examples using fractured bones are also presented. The study has showed that the migration methods have great potential to quantify bone fractures and monitor the fracture healing process.

La simulation numérique de la propagation des ondes électromagnétiques sur de longues distances et au-dessus de terrain a ces dernières années reçu une attention particulière du fait de son fort impact sur les systèmes radar et de télécommunications. Habituellement, il est considéré une modélisation bidimensionnelle pour traiter ces problématiques, cependant par une telle approche il est impossible de considérer les effets transverses au plan vertical passant par l'émetteur et le récepteur ainsi que la dépolarisation de l'onde. Pour pallier ces problèmes, une approche tridimensionnelle doit obligatoirement être considérée.<br /><br />La méthode de modélisation proposée est basée sur l'Equation Parabolique 3D (EP3D). Deux résolutions de celle-ci ont été considérées : nommées Split-Step Fourier (SSF) et Différences Finies (DF). La résolution SSF est basée sur une décomposition en un spectre angulaire d'ondes planes par l'intermédiaire d'une transformée de Fourier. La résolution de l'EP3D par DF développée utilise quant à elle un algorithme dit de Crank-Nicholson. Afin d'optimiser le temps de calcul et l'espace mémoire nécessaire, la méthode des directions alternées a été appliquée pour résoudre cette équation de propagation. Toutes deux ont été couplées avec la condition aux limites de Léontovich pour pouvoir prendre en compte le relief 3D.<br /><br />Ces deux méthodes ont été implémentées et validées sur différents cas tests canoniques. On a ainsi pu constater la capacité de ces méthodes à modéliser les phénomènes de réflexion, diffraction et réfraction. Celles-ci ont ensuite été appliquées au-dessus de scènes tridimensionnelles réalistes. Ces applications ont permis de comparer les deux méthodes développées ainsi que de mettre en relief les effets 3D dus au terrain et souligné les avantages d'une résolution tridimensionnelle.

Stochastic differential equations(SDEs) play an important role in many branches of engineering and science including economics, finance, chemistry, biology, mechanics etc. SDEs (with m-dimensional Wiener process) arising in many applications do not have explicit solutions, which implies the development of effective numerical methods for such systems. For SDEs, one can classify the numerical methods into three classes: fully implicit methods, semi-implicit methods and explicit methods. In order to solve SDEs, the computation of Newton iteration is necessary for the implicit and semi-implicit methods whereas for the explicit methods we do not need such computation. In this thesis the common theme is to construct explicit numerical methods with strong order 1.0 and 1.5 for solving Itˆo SDEs. The five-stage Milstein(FSM)methods, split-step forward Milstein(SSFM)methods and M-stage split-step strong Taylor(M-SSST) methods are constructed for solving SDEs. The FSM, SSFM and M-SSST methods are fully explicit methods. It is proved that the FSM and SSFM methods are convergent with strong order 1.0, and M-SSST methods are convergent with strong order 1.5.Stiffness is a very important issue for the numerical treatment of SDEs, similar to the case of deterministic ordinary differential equations. Stochastic stiffness can lead someone to use smaller step-size for the numerical simulation of the SDEs. However, such issues can be handled using numerical methods with better stability properties. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the FSM and SSFM methods are larger than the Milstein and three-stage Milstein methods. The M-SSST methods possess large mean square stability region as compared to the order 1.5 strong Itˆo-Taylor method. SDE systems simulated with the FSM, SSFM and M-SSST methods show the computational efficiency of the methods. In this work, we also consider the problem of computing numerical solutions for stochastic delay differential equations(SDDEs) of Itˆo form with a constant lag in the argument. The fully explicit methods, the predictor-corrector Euler(PCE)methods, are constructed for solving SDDEs. It is proved that the PCE methods are convergent with strong order γ = ½ in the mean-square sense. The conditions under which the PCE methods are MS-stable and GMS-stable are less restrictive as compared to the conditions for the Euler method.

This study explores the application of GPU-based algorithms in radiowave propagation modeling, specifically through the scope of solving parabolic wave equations. Radiowave propagation models are crucial in the field of wireless communications, where they help predict how radio waves travel through different environments, which is vital for planning and optimization. The research specifically examines the implementation of two numerical methods: the Split Step Method and the Finite Difference Method. Both methods are adapted to utilize the parallel processing capabilities of modern GPUs, harnessing a parallel computing framework known as CUDA to achieve considerable speed enhancements compared to traditional CPU-based methods.Our findings reveal that the Split Step method generally achieves higher speedup factors, especially in scenarios involving large system sizes and high-frequency simulations, making it particularly effective for expansive and complex models. In contrast, the Finite Difference Method shows more consistent speedup across various domain sizes and frequencies, suggesting its robustness across a diverse range of simulation conditions. Both methods maintained high accuracy levels, with differences in computed norms remaining low when comparing GPU implementations against their CPU counterparts.

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La propagation à longue distance est un problème majeur dans les télécommunications, la navigation et la surveillance. L'objectif de cette thèse est de développer une méthode rapide pour simuler la propagation des ondes dans une atmosphère en 2D et 3D. Dans ce travail, deux contributions principales vers cet objectif sont obtenues. Tout d'abord, des méthodes auto-cohérentes,c'est-à-dire basées sur une théorie discrète de l'électromagnétisme, sont développées en 2D et 3D. Ensuite, une méthode rapide 2D basée sur les ondelettes est proposée. Pour simuler la propagation d'ondes électromagnétiques dans une atmosphère 2D, la méthode split-step Fourier (SSF) est largement utilisée. Le calcul est effectué itérativement en distances en tenant compte d'une réfractivité variable, du relief et des caractéristiques du sol. À chaque pas, le signal est transformé du domaine spatial au domaine spectral. La méthode des écrans de phase est appliquée pour modéliser les effets de réfraction. D'autre part, pour modéliser un sol impédant, la transformée mixte de Fourier discrète (SSF-DMFT) est utilisée. Le concept de la théorie électromagnétique auto-cohérente implique que l'utilisation d'équations de Maxwell discrètes pour la simulation numérique évite les solutions parasites. Dans la méthode couramment utilisée SSF-DMFT, la transformée spectrale est basée sur la condition aux limites d'impédance discrète, alors que le propagateur provient de l'équation continue. Pour pallier cette incohérence, une méthode auto-cohérente est proposée, notée la DSSF-DMFT. La formulation est dérivée des équations discrètes pour obtenir l'auto-cohérence. Des tests numériques montrent que SSF-DMFT présente des oscillations parasites dans certaines conditions de simulation, tandis que DSSF-DMFT reste précis. En effet, l'auto-cohérence empêche certaines instabilités numériques. Pour simuler la propagation dans des environnements en 3D, les méthodes précédentes doivent être étendues en 3D. Tout d'abord, la 3D-SSF est présentée comme une extension naturelle de la SSF. Ensuite, la 3D-DSSF est dérivée d'équations discrètes. Pour considérer un sol impédant, la 3D-DSSF-DMFT est développée conduisant à de nouvelles expressions pour les propagateurs. Ces méthodes sont testées dans plusieurs configurations incluant un profil de réfractivité extrait de mesures. Les résultats montrent une grande précision et une capacité à prendre en compte les effets latéraux. Cependant, pour la propagation dans de grand domaines, les ressources nécessaires en temps et en mémoire deviennent la préoccupation principale. Pour alléger la charge de calcul, une méthode split-step en ondelettes (SSW) est proposée en 2D comme une méthode alternative à SSF. Elle est basée sur la transformée rapide en ondelettes dont la complexité est faible et qui permet de compresser les champs. La propagation est réalisée à partir d'une combinaison linéaire d'ondelettes propagées individuellement. La compression est appliquée pour augmenter l'efficacité. Afin de considérer la réflexion sur le sol, une nouvelle méthode de source image locale dédiée à la propagation des ondelettes est proposée. Les tests numériques montrent que la SSW a une efficacité de calcul plus élevée que la SSF tout en gardant une bonne précision. / The long-range propagation of electromagnetic waves is a major issue in telecommunication, navigation, and surveillance. The objective of this Ph.D. thesis is to develop fast and accurate modeling methods for the tropospheric propagation in 2D and 3D. In this work, two main contributions towards this objective are achieved. Firstly, self-consistent methods, i.e. based on the discrete electromagnetic theory, are developed in 2D and 3D. Secondly, a fast wavelet-based 2D method is proposed. For simulating the electromagnetic wave propagation in a 2D atmosphere, the split-step Fourier method (SSF) is widely used. The computation is performed marching on in distances taking into account a variable refractivity, an irregular relief, and the electric characteristics of the ground. At each step, the signal is transformed from the spatial to the spectral domains. The phase screens method is applied to model refraction. Besides, to model an impedance ground, the discrete mixed Fourier transform (SSF-DMFT) is used. The concept of the self-consistent electromagnetic theory implies that the use of discrete Maxwell equations for numerical simulations does not lead to spurious solutions. In the widely used SSF-DMFT, the spectral transform is based on the discrete impedance boundary condition, while the propagator is derived from the continuous equation. To overcome this inconsistency, a discrete formulation of SSF-DMFT is proposed, denoted as DSSF-DMFT. The spectral transform and propagator are both derived from the discrete equations to achieve self-consistency. Numerical tests show that SSF-DMFT has spurious oscillations in certain simulation conditions, whereas DSSF-DMFT remains accurate. Indeed, the self-consistency prevents from numerical instabilities. To simulate the propagation in 3D environments, the previous methods are extended to 3D. First, 3D-SSF is presented as a natural extension of SSF. Then, 3D-DSSF is derived from discrete equations. To consider an impedance ground, 3D-DSSF-DMFT is developed leading to new expressions for the propagators. These methods are tested for several configurations, including a refractivity profile extracted from measurements. Results show that they have a high accuracy. They notably consider lateral effects. However, for the propagation in a large computation domain, time and memory occupations become the main concern. To improve the computation burden, a split-step wavelet method (SSW) is proposed in 2D as an alternative to SSF. It is based on the fast wavelet transform, which complexity is weak and which allows for data compression. The propagation is performed by means of a linear combination of wavelets that are individually propagated. Data compression is applied to increase the efficiency. A new local image source method dedicated to wavelet propagation is proposed to consider the ground reflection. Numerical tests show that this method has a higher computational efficiency than SSF while keeping a good accuracy.

This document describes radar propagation modelling using split step parabolic wave equation (PWE) method. A computer program using Fourier split-step (FSS) marching technique is developed for predicting the electromagnetic wave propagation in troposphere. The program allows specification of frequency, polarization, antenna radiation pattern, antenna altitude, elevation angle and terrain profile. Both staircase terrain modelling and conformal mapping are used to model the irregular terrain. Mixed Fourier transform is used to implement the impedance boundary conditions. The conditions and the limits of different approximations are stated. The propagation code, RPPT (Radar Propagation Prediction Tool) is developed in Matlab 6.0 with a user friendly GUI. Different PWE methods can be selected in RPPT for different applications. The results are presented as one-way propagation factor and path loss in decibels versus range.Comparisons are made between different PWE techniques and other propagation models to demonstrate the ability and accuracy of the present model to accommodate various situations. It is assumed that the reader is familiar with the tropospheric propagation.

Radars are used for range estimation of distant objects. They operate on the principle of sending electromagnetic pulses that are reflected off a target. This leads to the propagation of electromagnetic waves over large distances. As the waves propagate, they are affected by several aspects that decrease the performance of the radar system. In this master thesis, we derive a mathematical model that describes electromagnetic propagation in the troposphere. The model developed is based on a parabolic equation and uses the split-step Fourier method for its numerical solution. Using the model, we estimate the influence of a varying, complex, refractive index of the atmosphere, different lossy materials in the ground, terrain, and oceans. The terrain is described using a piecewise linear shift map method. The modelling of the ocean is done using a novel model which is a combination of terrain for large swells and Miller surface roughness for smaller waves, both based on a Pierson-Moskowitz sea spectrum. The model is validated and found to agree very well, with results found in the literature.