Statistical properties of first-order bang-bang Pll with nonzero loop delay

A method to solve the stationary state probability is presented for the first-order bang-bang phase-locked loop (BBPLL) with nonzero loop delay. This is based on a delayed Markov chain model and a state How diagram for tracing the state history due to the loop delay. As a result, an eigenequation is obtained, and its closed form solutions are derived for some cases. After obtaining the state probability, statistical characteristics such as mean gain of the binary phase detector and timing error variance are calculated and demonstrated.

Hardware reduction in digital delta-sigma modulators via error masking - part I: MASH DDSM

Two classes of techniques have been developed to whiten the quantization noise in digital delta-sigma modulators (DDSMs): deterministic and stochastic. In this two-part paper, a design methodology for reduced-complexity DDSMs is presented. The design methodology is based on error masking. Rules for selecting the word lengths of the stages in multistage architectures are presented. We show that the hardware requirement can be reduced by up to 20% compared with a conventional design, without sacrificing performance. Simulation and experimental results confirm theoretical predictions. Part I addresses MultistAge noise SHaping (MASH) DDSMs; Part II focuses on single-quantizer DDSMs..

Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

A corpus-driven error analysis of the oral and written production of Leaving Certificate Spanish learners

The Leaving Certificate (LC) is the national, standardised state examination in Ireland necessary for entry to third level education – this presents a massive, raw corpus of data with the potential to yield invaluable insight into the phenomena of learner interlanguage. With samples of official LC Spanish examination data, this project has compiled a digitised corpus of learner Spanish comprised of the written and oral production of 100 candidates. This corpus was then analysed using a specific investigative corpus technique, Computer-aided Error Analysis (CEA, Dagneaux et al, 1998). CEA is a powerful apparatus in that it greatly facilitates the quantification and analysis of a large learner corpus in digital format. The corpus was both compiled and analysed with the use of UAM Corpus Tool (O’Donnell 2013). This Tool allows for the recording of candidate-specific variables such as grade, examination level, task type and gender, therefore allowing for critical analysis of the corpus as one unit, as separate written and oral sub corpora and also of performance per task, level and gender. This is an interdisciplinary work combining aspects of Applied Linguistics, Learner Corpus Research and Foreign Language (FL) Learning. Beginning with a review of the context of FL learning in Ireland and Europe, I go on to discuss the disciplinary context and theoretical framework for this work and outline the methodology applied. I then perform detailed quantitative and qualitative analyses before going on to combine all research findings outlining principal conclusions. This investigation does not make a priori assumptions about the data set, the LC Spanish examination, the context of FLs or of any aspect of learner competence. It undertakes to provide the linguistic research community and the domain of Spanish language learning and pedagogy in Ireland with an empirical, descriptive profile of real learner performance, characterising learner difficulty.