Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.

Os requisitos de linearidade e ruído em aplicações multi-banda e multi-protocolo fazem que o projeto de misturadores RF seja uma tarefa muito desafiadora. Nesta dissertação dois misturadores com base na topologia célula de Gilbert são propostas. Linearidade e ruído foram as principais figuras de mérito consideradas para o misturadores propostos. Para aumento linearidade, foi utilizada uma técnica de cancelamento de harmônicas pós-distorção (PDHC). E, para redução de ruído, foi utilizado um circuito de redução dinâmica de corrente combinada com um filtro LC sintonizado na frequência do LO e cancelamento de ruído térmico. A análise por séries Volterra do estágio transcondutância do misturador proposto é reportada para mostrar a eficácia da técnica de cancelamento de harmônicos com pósdistorção. O circuito de linearização adicionado não aumenta o tamanho do misturador, nem degrada ganho de conversão, figura de ruído, ou consumo de potência. Simulações elétricas foram realizadas em nível de pós-layout para a primeira topologia e nível esquemático para a segunda topologia, usando processo CMOS de 0.13 mm da IBM. As melhorias em IIP2 e IIP3 são apresentadas em comparação com o misturador do tipo célula de Gilbert convencional. Para a primeira topologia, foi obtido um ganho de conversão de 10.2 dB com uma NF de 12 dB para o misturador projetado funcionando a 2 GHz, com uma frequência intermediária de 500 kHz. E um IIP2 e IIP3 de 55 dBm e 10.9 dBm, respectivamente, consumindo apenas 5.3 mW de uma fonte de 1.2 V. Para a segunda topologia, foram obtidos um ganho de conversão de [13.8 ~11] dB, um coeficiente de reflexão na entrada (S11) de [-18 ~-9.5] dB e um NF de [8.5 ~11] dB no intervalo de 1 a 6 GHz. Para as especificações de linearidade, um valor médio de IIP3 de 0 dBm foi alcançado para toda a faixa de frequência, consumindo 19.3 mW a partir de uma fonte de 1.2 V. Especificações adequadas para operação multi-banda e multi-protocolo. / The linearity and noise requirements in multi-band multi-standard applications make the design of RF CMOS mixers a very challenging task. In this dissertation two downconversion mixers based on the Gilbert-cell topology are proposed. Linearity and noise were the principal figures of merit for the proposed mixers. For linearity improvement, post distortion harmonic cancellation (PDHC) was employed. And, for noise reduction, dynamic current injection combined with an LC filter tuned at the LO frequency and thermal-noise cancellation were used. A Volterra series analysis of the transconductance stage is reported to show the effectiveness of the post-distortion harmonic cancellation technique. The added linearization circuitry does not increase the size of the mixer, nor does it degrade conversion gain, noise figure, or power consumption. Electrical simulations were performed on extracted layout level from the first topology and schematic level from the second topology. Using an IBM 0.13 mm CMOS process improvements on IIP3 and IIP2 in comparison to the conventional Gilbert-cell mixer are demonstrated. For the first topology, we achieved a conversion gain of 10.2 dB with a NF of 12 dB for the designed mixer working at 2 GHz, with a low-IF of 500 kHz and an IIP2 and IIP3 of 55 dBm and 10.9 dBm, respectively, while consuming only 5.3 mW from a 1.2 V supply. For the second topology, we achieved a conversion gain range of [13.8 ~11] dB, an input reflection coefficient (S11) of [-18 ~-9.5] dB and a NF of [8.5 ~11] dB in the frequency range of 1 to 6 GHz. For the linearity specs, an IIP3 of 0 dBm was achieved for the whole frequency range, while consuming 19.3 mW from a 1.2 V supply, making the second topology well suited for multi-band and multi-standard operation.

Os requisitos de linearidade e ruído em aplicações multi-banda e multi-protocolo fazem que o projeto de misturadores RF seja uma tarefa muito desafiadora. Nesta dissertação dois misturadores com base na topologia célula de Gilbert são propostas. Linearidade e ruído foram as principais figuras de mérito consideradas para o misturadores propostos. Para aumento linearidade, foi utilizada uma técnica de cancelamento de harmônicas pós-distorção (PDHC). E, para redução de ruído, foi utilizado um circuito de redução dinâmica de corrente combinada com um filtro LC sintonizado na frequência do LO e cancelamento de ruído térmico. A análise por séries Volterra do estágio transcondutância do misturador proposto é reportada para mostrar a eficácia da técnica de cancelamento de harmônicos com pósdistorção. O circuito de linearização adicionado não aumenta o tamanho do misturador, nem degrada ganho de conversão, figura de ruído, ou consumo de potência. Simulações elétricas foram realizadas em nível de pós-layout para a primeira topologia e nível esquemático para a segunda topologia, usando processo CMOS de 0.13 mm da IBM. As melhorias em IIP2 e IIP3 são apresentadas em comparação com o misturador do tipo célula de Gilbert convencional. Para a primeira topologia, foi obtido um ganho de conversão de 10.2 dB com uma NF de 12 dB para o misturador projetado funcionando a 2 GHz, com uma frequência intermediária de 500 kHz. E um IIP2 e IIP3 de 55 dBm e 10.9 dBm, respectivamente, consumindo apenas 5.3 mW de uma fonte de 1.2 V. Para a segunda topologia, foram obtidos um ganho de conversão de [13.8 ~11] dB, um coeficiente de reflexão na entrada (S11) de [-18 ~-9.5] dB e um NF de [8.5 ~11] dB no intervalo de 1 a 6 GHz. Para as especificações de linearidade, um valor médio de IIP3 de 0 dBm foi alcançado para toda a faixa de frequência, consumindo 19.3 mW a partir de uma fonte de 1.2 V. Especificações adequadas para operação multi-banda e multi-protocolo. / The linearity and noise requirements in multi-band multi-standard applications make the design of RF CMOS mixers a very challenging task. In this dissertation two downconversion mixers based on the Gilbert-cell topology are proposed. Linearity and noise were the principal figures of merit for the proposed mixers. For linearity improvement, post distortion harmonic cancellation (PDHC) was employed. And, for noise reduction, dynamic current injection combined with an LC filter tuned at the LO frequency and thermal-noise cancellation were used. A Volterra series analysis of the transconductance stage is reported to show the effectiveness of the post-distortion harmonic cancellation technique. The added linearization circuitry does not increase the size of the mixer, nor does it degrade conversion gain, noise figure, or power consumption. Electrical simulations were performed on extracted layout level from the first topology and schematic level from the second topology. Using an IBM 0.13 mm CMOS process improvements on IIP3 and IIP2 in comparison to the conventional Gilbert-cell mixer are demonstrated. For the first topology, we achieved a conversion gain of 10.2 dB with a NF of 12 dB for the designed mixer working at 2 GHz, with a low-IF of 500 kHz and an IIP2 and IIP3 of 55 dBm and 10.9 dBm, respectively, while consuming only 5.3 mW from a 1.2 V supply. For the second topology, we achieved a conversion gain range of [13.8 ~11] dB, an input reflection coefficient (S11) of [-18 ~-9.5] dB and a NF of [8.5 ~11] dB in the frequency range of 1 to 6 GHz. For the linearity specs, an IIP3 of 0 dBm was achieved for the whole frequency range, while consuming 19.3 mW from a 1.2 V supply, making the second topology well suited for multi-band and multi-standard operation.

abstract: In the field of infectious disease epidemiology, the assessment of model robustness outcomes plays a significant role in the identification, reformulation, and evaluation of preparedness strategies aimed at limiting the impact of catastrophic events (pandemics or the deliberate release of biological agents) or used in the management of disease prevention strategies, or employed in the identification and evaluation of control or mitigation measures. The research work in this dissertation focuses on: The comparison and assessment of the role of exponentially distributed waiting times versus the use of generalized non-exponential parametric distributed waiting times of infectious periods on the quantitative and qualitative outcomes generated by Susceptible-Infectious-Removed (SIR) models. Specifically, Gamma distributed infectious periods are considered in the three research projects developed following the applications found in (Bailey 1964, Anderson 1980, Wearing 2005, Feng 2007, Feng 2007, Yan 2008, lloyd 2009, Vergu 2010). i) The first project focuses on the influence of input model parameters, such as the transmission rate, mean and variance of Gamma distributed infectious periods, on disease prevalence, the peak epidemic size and its timing, final epidemic size, epidemic duration and basic reproduction number. Global uncertainty and sensitivity analyses are carried out using a deterministic Susceptible-Infectious-Recovered (SIR) model. The quantitative effect and qualitative relation between input model parameters and outcome variables are established using Latin Hypercube Sampling (LHS) and Partial rank correlation coefficient (PRCC) and Spearman rank correlation coefficient (RCC) sensitivity indices. We learnt that: For relatively low (R0 close to one) to high (mean of R0 equals 15) transmissibility, the variance of the Gamma distribution for the infectious period, input parameter of the deterministic age-of-infection SIR model, is key (statistically significant) on the predictability of the epidemiological variables such as the epidemic duration and the peak size and timing of the prevalence of infectious individuals and therefore, for the predictability these variables, it is preferable to utilize a nonlinear system of Volterra integral equations, rather than a nonlinear system of ordinary differential equations. The predictability of epidemiological variables such as the final epidemic size and the basic reproduction number are unaffected by (or independent of) the variance of the Gamma distribution for the infectious period and therefore for the choice on which type of nonlinear system for the description of the SIR model (VIE's or ODE's) is irrelevant. Although, for practical proposes, with the aim of lowering the complexity and number operations in the numerical methods, a nonlinear system of ordinary differential equations is preferred. The main contribution lies in the development of a model based decision-tool that helps determine when SIR models given in terms of Volterra integral equations are equivalent or better suited than SIR models that only consider exponentially distributed infectious periods. ii) The second project addresses the question of whether or not there is sufficient evidence to conclude that two empirical distributions for a single epidemiological outcome, one generated using a stochastic SIR model under exponentially distributed infectious periods and the other under the non-exponentially distributed infectious period, are statistically dissimilar. The stochastic formulations are modeled via a continuous time Markov chain model. The statistical hypothesis test is conducted using the non-parametric Kolmogorov-Smirnov test. We found evidence that shows that for low to moderate transmissibility, all empirical distribution pairs (generated from exponential and non-exponential distributions) for each of the epidemiological quantities considered are statistically dissimilar. The research in this project helps determine whether the weakening exponential distribution assumption must be considered in the estimation of probability of events defined from the empirical distribution of specific random variables. iii) The third project involves the assessment of the effect of exponentially distributed infectious periods on estimates of input parameter and the associated outcome variable predictions. Quantities unaffected by the use of exponentially distributed infectious period within low transmissibility scenarios include, the prevalence peak time, final epidemic size, epidemic duration and basic reproduction number and for high transmissibility scenarios only the prevalence peak time and final epidemic size. An application designed to determine from incidence data whether there is sufficient statistical evidence to conclude that the infectious period distribution should not be modeled by an exponential distribution is developed. A method for estimating explicitly specified non-exponential parametric probability density functions for the infectious period from epidemiological data is developed. The methodologies presented in this dissertation may be applicable to models where waiting times are used to model transitions between stages, a process that is common in the study of life-history dynamics of many ecological systems. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2014

Os requisitos de linearidade e ruído em aplicações multi-banda e multi-protocolo fazem que o projeto de misturadores RF seja uma tarefa muito desafiadora. Nesta dissertação dois misturadores com base na topologia célula de Gilbert são propostas. Linearidade e ruído foram as principais figuras de mérito consideradas para o misturadores propostos. Para aumento linearidade, foi utilizada uma técnica de cancelamento de harmônicas pós-distorção (PDHC). E, para redução de ruído, foi utilizado um circuito de redução dinâmica de corrente combinada com um filtro LC sintonizado na frequência do LO e cancelamento de ruído térmico. A análise por séries Volterra do estágio transcondutância do misturador proposto é reportada para mostrar a eficácia da técnica de cancelamento de harmônicos com pósdistorção. O circuito de linearização adicionado não aumenta o tamanho do misturador, nem degrada ganho de conversão, figura de ruído, ou consumo de potência. Simulações elétricas foram realizadas em nível de pós-layout para a primeira topologia e nível esquemático para a segunda topologia, usando processo CMOS de 0.13 mm da IBM. As melhorias em IIP2 e IIP3 são apresentadas em comparação com o misturador do tipo célula de Gilbert convencional. Para a primeira topologia, foi obtido um ganho de conversão de 10.2 dB com uma NF de 12 dB para o misturador projetado funcionando a 2 GHz, com uma frequência intermediária de 500 kHz. E um IIP2 e IIP3 de 55 dBm e 10.9 dBm, respectivamente, consumindo apenas 5.3 mW de uma fonte de 1.2 V. Para a segunda topologia, foram obtidos um ganho de conversão de [13.8 ~11] dB, um coeficiente de reflexão na entrada (S11) de [-18 ~-9.5] dB e um NF de [8.5 ~11] dB no intervalo de 1 a 6 GHz. Para as especificações de linearidade, um valor médio de IIP3 de 0 dBm foi alcançado para toda a faixa de frequência, consumindo 19.3 mW a partir de uma fonte de 1.2 V. Especificações adequadas para operação multi-banda e multi-protocolo. / The linearity and noise requirements in multi-band multi-standard applications make the design of RF CMOS mixers a very challenging task. In this dissertation two downconversion mixers based on the Gilbert-cell topology are proposed. Linearity and noise were the principal figures of merit for the proposed mixers. For linearity improvement, post distortion harmonic cancellation (PDHC) was employed. And, for noise reduction, dynamic current injection combined with an LC filter tuned at the LO frequency and thermal-noise cancellation were used. A Volterra series analysis of the transconductance stage is reported to show the effectiveness of the post-distortion harmonic cancellation technique. The added linearization circuitry does not increase the size of the mixer, nor does it degrade conversion gain, noise figure, or power consumption. Electrical simulations were performed on extracted layout level from the first topology and schematic level from the second topology. Using an IBM 0.13 mm CMOS process improvements on IIP3 and IIP2 in comparison to the conventional Gilbert-cell mixer are demonstrated. For the first topology, we achieved a conversion gain of 10.2 dB with a NF of 12 dB for the designed mixer working at 2 GHz, with a low-IF of 500 kHz and an IIP2 and IIP3 of 55 dBm and 10.9 dBm, respectively, while consuming only 5.3 mW from a 1.2 V supply. For the second topology, we achieved a conversion gain range of [13.8 ~11] dB, an input reflection coefficient (S11) of [-18 ~-9.5] dB and a NF of [8.5 ~11] dB in the frequency range of 1 to 6 GHz. For the linearity specs, an IIP3 of 0 dBm was achieved for the whole frequency range, while consuming 19.3 mW from a 1.2 V supply, making the second topology well suited for multi-band and multi-standard operation.

This thesis deals with the stochastic integral with respect to Gaussian processes, which can be expressed in the form Bt = t 0 K(t, s)dWs. Here W stands for a Brownian motion and K for a square integrable Volterra kernel. Such processes generalize fractional Brownian motion. Since these processes are not semimartin- gales, Itô calculus cannot be used and other methods must be employed to define the stochastic integral with respect to these proceses. Two ways are considered in this thesis. If both the integrand and the process B are regular enough, it is possible to define the integral in the pathwise sense as a generalization of Lebesgue-Stieltjes integral. The other method uses the methods of Malliavin cal- culus and defines the integral as an adjoint operator to the Malliavin derivative. As an application, the stochastic differential equation dSt = µStdt + σStdBt, which is used to model price of a stock, is solved. Implications of such a model are briefly discussed. 1

This thesis covers the development of a series of new methods and the application of adaptive filter theory which are combined to produce a generalised adaptive filter system which may be used to perform such tasks as pattern recognition. Firstly, the relevant background adaptive filter theory is discussed in Chapter 1 and methods and results which are important to the rest of the thesis are derived or referenced. Chapter 2 of this thesis covers the development of a new adaptive algorithm which is designed to give faster convergence than the LMS algorithm but unlike the Recursive Least Squares family of algorithms it does not require storage of a matrix with n2 elements, where n is the number of filter taps. In Chapter 3 a new extension of the LMS adaptive notch filter is derived and applied which gives an adaptive notch filter the ability to lock and track signals of varying pitch without sacrificing notch depth. This application of the LMS filter is of interest as it demonstrates a time varying filter solution to a stationary problem. The LMS filter is next extended to the multidimensional case which allows the application of LMS filters to image processing. The multidimensional filter is then applied to the problem of image registration and this new application of the LMS filter is shown to have significant advantages over current image registration methods. A consideration of the multidimensional LMS filter as a template matcher and pattern recogniser is given. In Chapter 5 a brief review of statistical pattern recognition is given, and in Chapter 6 a review of relevant connectionist models. In Chapter 7 the generalised adaptive filter is derived. This is an adaptive filter with the ability to model non-linear input-output relationships. The Volterra functional analysis of non-linear systems is given and this is combined with adaptive filter methods to give a generalised non-linear adaptive digital filter. This filter is then considered as a linear adaptive filter operating in a non-linearly extended vector space. This new filter is shown to have desirable properties as a pattern recognition system. The performance and properties of the new filter is compared with current connectionist models and results demonstrated in Chapter 8. In Chapter 9 further mathematical analysis of the networks leads to suggested methods to greatly reduce network complexity for a given problem by choosing suitable pattern classification indices and allowing it to define its own internal structure. In Chapter 10 robustness of the network to imperfections in its implementation is considered. Chapter 11 finishes the thesis with some conclusions and suggestions for future work.

<div>Radio Frequency Power Amplifiers (PA) are essential components of wireless systems and nonlinear in a permanent way. So, high efficiency and linearity at a time are imperative for power amplifiers. However, it is hard to obtain because high efficiency Power Amplifiers are nonlinear and linear Power Amplifiers have poor efficiency. To meet both linearity and efficiency, the linearization techniques such as Digital Predistortion (DPD) has arrested the most attention in industrial and academic sectors due to provide a compromising data between efficiency and linearity. This thesis proposed on digital predistortion techniques to control nonlinear distortion in radio frequency transmitters. </div><div>By using predistortion technique, both linearity and efficiency can obtain. In this thesis a new generic Saleh model for use in memoryless nonlinear power amplifier (PA) behavioral modelling is used. The results are obtained by simulations through MATLAB and experiments. We explore the baseband 13.56 MHz Power Amplifier input and output relationships and reveal that they apparent differently when the Power Amplifier shows long-term, short-term or memory less effects. We derive a SIMULINK based static DPD design depend on a memory polynomial. A polynomial improves both the non-linearity and memory effects in the Power Amplifier. As PA characteristics differs from time to time and operating conditions, we developed a model to calculate the effectiveness of DPD. We extended our static DPD design model into an adaptive DPD test bench using Indirect Learning Architecture (ILA) to implement adaptive DPD which composed of DPD subsystem and DPD coefficient calculation. By this technique, the output of PA achieves linear, amplitude and phase distortions are eliminated, and spectral regrowth is prevented. </div><div>The advanced linearity performance executed through the strategies and methods evolved on this thesis can allow a higher usage of the capability overall performance of existing and emerging exceptionally performance PAs, and therefore an anticipated to have an effect in future wireless communication systems. </div>

In the Thesis the problem of estimating an unknown parameter in a stochastic dif- ferential equation is studied. Linear equations with Volterra process as the source of noise are considered. Firstly, the properties of Volterra processes and the properties of stochastic integral with respect to a Volterra process are presented. Secondly, the prop- erties of the solution to the equation under consideration are discussed. This includes the existence of the strictly stationary solution, the properties of such solution and ergodic results. These results are then generalized to equations with a mixed noise. Ergodic results are used to derive strongly consistent estimators of the unknown parameter. 1

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