Mathematical Sciences
http://hdl.handle.net/10468/88
2015-06-11T09:29:30ZInteraction of additive and quantization noises in digital PLLs
http://hdl.handle.net/10468/1831
Interaction of additive and quantization noises in digital PLLs
Ó Tuama, Cillian
Phase-locked loops (PLLs) are a crucial component in modern communications systems. Comprising of a phase-detector, linear filter, and controllable oscillator, they are widely used in radio receivers to retrieve the information content from remote signals. As such, they are capable of signal demodulation, phase and carrier recovery, frequency synthesis, and clock synchronization. Continuous-time PLLs are a mature area of study, and have been covered in the literature since the early classical work by Viterbi [1] in the 1950s. With the rise of computing in recent decades, discrete-time digital PLLs (DPLLs) are a more recent discipline; most of the literature published dates from the 1990s onwards. Gardner [2] is a pioneer in this area. It is our aim in this work to address the difficulties encountered by Gardner [3] in his investigation of the DPLL output phase-jitter where additive noise to the input signal is combined with frequency quantization in the local oscillator. The model we use in our novel analysis of the system is also applicable to another of the cases looked at by Gardner, that is the DPLL with a delay element integrated in the loop. This gives us the opportunity to look at this system in more detail, our analysis providing some unique insights into the variance `dip' seen by Gardner in [3]. We initially provide background on the probability theory and stochastic processes. These branches of mathematics are the basis for the study of noisy analogue and digital PLLs. We give an overview of the classical analogue PLL theory as well as the background on both the digital PLL and circle map, referencing the model proposed by Teplinsky et al. [4, 5]. For our novel work, the case of the combined frequency quantization and noisy input from [3] is investigated first numerically, and then analytically as a Markov chain via its Chapman-Kolmogorov equation. The resulting delay equation for the steady-state jitter distribution is treated using two separate asymptotic analyses to obtain approximate solutions. It is shown how the variance obtained in each case matches well to the numerical results. Other properties of the output jitter, such as the mean, are also investigated. In this way, we arrive at a more complete understanding of the interaction between quantization and input noise in the first order DPLL than is possible using simulation alone. We also do an asymptotic analysis of a particular case of the noisy first-order DPLL with delay, previously investigated by Gardner [3]. We show a unique feature of the simulation results, namely the variance `dip' seen for certain levels of input noise, is explained by this analysis. Finally, we look at the second-order DPLL with additive noise, using numerical simulations to see the effects of low levels of noise on the limit cycles. We show how these effects are similar to those seen in the noise-free loop with non-zero initial conditions.
2014-01-01T00:00:00ZDefining metabolically healthy obesity: role of dietary and lifestyle factors
http://hdl.handle.net/10468/1474
Defining metabolically healthy obesity: role of dietary and lifestyle factors
Phillips, Catherine M.; Dillon, Christina; Harrington, Janas M.; McCarthy, Vera J. C.; Kearney, Patricia M.; Fitzgerald, Anthony P.; Perry, Ivan J.
Background: There is a current lack of consensus on defining metabolically healthy obesity (MHO). Limited data on dietary and lifestyle factors and MHO exist. The aim of this study is to compare the prevalence, dietary factors and lifestyle behaviours of metabolically healthy and unhealthy obese and non-obese subjects according to different metabolic health criteria. Method: Cross-sectional sample of 1,008 men and 1,039 women aged 45-74 years participated in the study. Participants were classified as obese (BMI ≥30kg/m2) and non-obese (BMI <30kg/m2). Metabolic health status was defined using five existing MH definitions based on a range of cardiometabolic abnormalities. Dietary composition and quality, food pyramid servings, physical activity, alcohol and smoking behaviours were examined. Results: The prevalence of MHO varied considerably between definitions (2.2% to 11.9%), was higher among females and generally increased with age. Agreement between MHO classifications was poor. Among the obese, prevalence of MH was 6.8% to 36.6%. Among the non-obese, prevalence of metabolically unhealthy subjects was 21.8% to 87%. Calorie intake, dietary macronutrient composition, physical activity, alcohol and smoking behaviours were similar between the metabolically healthy and unhealthy regardless of BMI. Greater compliance with food pyramid recommendations and higher dietary quality were positively associated with metabolic health in obese (OR 1.45-1.53 unadjusted model) and non-obese subjects (OR 1.37-1.39 unadjusted model), respectively. Physical activity was associated with MHO defined by insulin resistance (OR 1.87, 95% CI 1.19-2.92, p = 0.006).
2013-10-17T00:00:00ZAn investigation of post-primary students' images of mathematics
http://hdl.handle.net/10468/1096
An investigation of post-primary students' images of mathematics
Lane, Ciara Mary Frances
This research study investigates the image of mathematics held by 5th-year post-primary students in Ireland. For this study, “image of mathematics” is conceptualized as a mental representation or view of mathematics, presumably constructed as a result of past experiences, mediated through school, parents, peers or society. It is also understood to include attitudes, beliefs, emotions, self-concept and motivation in relation to mathematics. This study explores the image of mathematics held by a sample of 356 5th-year students studying ordinary level mathematics. Students were aged between 15 and 18 years. In addition, this study examines the factors influencing students‟ images of mathematics and the possible reasons for students choosing not to study higher level mathematics for the Leaving Certificate. The design for this study is chiefly explorative. A questionnaire survey was created containing both quantitative and qualitative methods to investigate the research interest. The quantitative aspect incorporated eight pre-established scales to examine students‟ attitudes, beliefs, emotions, self-concept and motivation regarding mathematics. The qualitative element explored students‟ past experiences of mathematics, their causal attributions for success or failure in mathematics and their influences in mathematics. The quantitative and qualitative data was analysed for all students and also for students grouped by gender, prior achievement, type of post-primary school attending, co-educational status of the post-primary school and the attendance of a Project Maths pilot school. Students‟ images of mathematics were seen to be strongly indicated by their attitudes (enjoyment and value), beliefs, motivation, self-concept and anxiety, with each of these elements strongly correlated with each other, particularly self-concept and anxiety. Students‟ current images of mathematics were found to be influenced by their past experiences of mathematics, by their mathematics teachers, parents and peers, and by their prior mathematical achievement. Gender differences occur for students in their images of mathematics, with males having more positive images of mathematics than females and this is most noticeable with regards to anxiety about mathematics. Mathematics anxiety was identified as a possible reason for the low number of students continuing with higher level mathematics for the Leaving Certificate. Some students also expressed low mathematical self-concept with regards to higher level mathematics specifically. Students with low prior achievement in mathematics tended to believe that mathematics requires a natural ability which they do not possess. Rote-learning was found to be common among many students in the sample. The most positive image of mathematics held by students was the “problem-solving image”, with resulting implications for the new Project Maths syllabus in post-primary education. Findings from this research study provide important insights into the image of mathematics held by the sample of Irish post-primary students and make an innovative contribution to mathematics education research. In particular, findings contribute to the current national interest in Ireland in post-primary mathematics education, highlighting issues regarding the low uptake of higher level mathematics for the Leaving Certificate and also making a preliminary comparison between students who took part in the piloting of Project Maths and students who were more recently introduced to the new syllabus. This research study also holds implications for mathematics teachers, parents and the mathematics education community in Ireland, with some suggestions made on improving students‟ images of mathematics.
2013-01-01T00:00:00ZDifferential and numerical models of hysteretic systems with stochastic and deterministic inputs
http://hdl.handle.net/10468/1138
Differential and numerical models of hysteretic systems with stochastic and deterministic inputs
McCarthy, Stephen Patrick
Many deterministic models with hysteresis have been developed in the areas of economics, finance, terrestrial hydrology and biology. These models lack any stochastic element which can often have a strong effect in these areas. In this work stochastically driven closed loop systems with hysteresis type memory are studied. This type of system is presented as a possible stochastic counterpart to deterministic models in the areas of economics, finance, terrestrial hydrology and biology. Some price dynamics models are presented as a motivation for the development of this type of model. Numerical schemes for solving this class of stochastic differential equation are developed in order to examine the prototype models presented. As a means of further testing the developed numerical schemes, numerical examination is made of the behaviour near equilibrium of coupled ordinary differential equations where the time derivative of the Preisach operator is included in one of the equations. A model of two phenotype bacteria is also presented. This model is examined to explore memory effects and related hysteresis effects in the area of biology. The memory effects found in this model are similar to that found in the non-ideal relay. This non-ideal relay type behaviour is used to model a colony of bacteria with multiple switching thresholds. This model contains a Preisach type memory with a variable Preisach weight function. Shown numerically for this multi-threshold model is a pattern formation for the distribution of the phenotypes among the available thresholds.
2013-01-01T00:00:00Z