Interaction of additive and quantization noises in digital PLLs

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.

Learning to 'understand backwards' in time: children's temporal cognition and the primary history curriculum

This study examines children’s temporal ways of knowing and it highlights the centrality of temporal cognition in the development of children’s historical understanding. It explores how young children conceptualise time and it examines the provision for temporal cognition at the levels of the intended, enacted and received history curriculum in the Irish primary school context. Positioning temporality as a prerequisite second-order concept, the study recognises the essential role of both first-order and additional second-order concepts in historical understanding. While the former can be defined as the basic, substantive content to be taught, the latter refers to a number of additional key concepts that are deemed fundamental to children's capacity to make meaningful sense of history. The study argues for due recognition to be given to temporality, in the belief that both sets of knowledge, the content and skills, are required to develop historical thinking (Lévesque, 2011). The study addresses a number of key research questions, using a mixed methods research design, comprising an analysis of history textbooks, a survey among final year student teachers about their teaching of history, and school-based interviews with primary school children: What opportunities are available for children to develop temporal ways of knowing? How do student teachers experience being apprenticed into the available culture for teaching history and understanding temporality at primary level? What insights do the cognitive-developmental and sociocultural perspectives on learning provide for understanding the dynamics of children’s temporal ways of knowing? The study argues that the skill of developing a deeper understanding of time is a key prerequisite in connecting with, and constructing, understandings and frameworks of the past. The study advances a view of temporality as complex, multi-faceted and developmental. The findings have a potential contribution to make in influencing policy and pedagogy in establishing an elaborated and well-defined curriculum framework for developing temporal cognition at both national and international levels.