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.

Environmental and biological characteristics of lagoons on the southwest coast of Ireland

Coastal lagoons are defined as shallow coastal water bodies partially separated from the adjacent sea by a restrictive barrier. Coastal lagoons are protected under Annex I of the European Habitats Directive (92/43/EEC). Lagoons are also considered to be “transitional water bodies” and are therefore included in the “register of protected areas” under the Water Framework Directive (2000/60/EC). Consequently, EU member states are required to establish monitoring plans and to regularly report on lagoon condition and conservation status. Irish lagoons are considered relatively rare and unusual because of their North Atlantic, macrotidal location on high energy coastlines and have received little attention. This work aimed to assess the physicochemical and ecological status of three lagoons, Cuskinny, Farranamanagh and Toormore, on the southwest coast of Ireland. Baseline salinity, nutrient and biological conditions were determined in order to provide reference conditions to detect perturbations, and to inform future maintenance of ecosystem health. Accumulation of organic matter is an increasing pressure in coastal lagoon habitats worldwide, often compounding existing eutrophication problems. This research also aimed to investigate the in situ decomposition process in a lagoon habitat together with exploring the associated invertebrate assemblages. Re-classification of the lagoons, under the guidelines of the Venice system for the classifications of marine waters according to salinity, was completed by taking spatial and temporal changes in salinity regimes into consideration. Based on the results of this study, Cuskinny, Farranamanagh and Toormore lagoons are now classified as mesohaline (5 ppt – 18 ppt), oligohaline (0.5 ppt – 5 ppt) and polyhaline (18 ppt – 30 ppt), respectively. Varying vertical, longitudinal and transverse salinity patterns were observed in the three lagoons. Strong correlations between salinity and cumulative rainfall highlighted the important role of precipitation in controlling the lagoon environment. Maximum effect of precipitation on the salinity of the lagoon was observed between four and fourteen days later depending on catchment area geology, indicating the uniqueness of each lagoon system. Seasonal nutrient patterns were evident in the lagoons. Nutrient concentrations were found to be reflective of the catchment area and the magnitude of the freshwater inflow. Assessment based on the Redfield molar ratio indicated a trend towards phosphorus, rather than nitrogen, limitation in Irish lagoons. Investigation of the decomposition process in Cuskinny Lagoon revealed that greatest biomass loss occurred in the winter season. Lowest biomass loss occurred in spring, possibly due to the high density of invertebrates feeding on the thick microbial layer rather than the decomposing litter. It has been reported that the decomposition of plant biomass is highest in the preferential distribution area of the plant species; however, no similar trend was observed in this study with the most active zones of decomposition varying spatially throughout the seasons. Macroinvertebrate analysis revealed low species diversity but high abundance, indicating the dominance of a small number of species. Invertebrate assemblages within the lagoon varied significantly from communities in the adjacent freshwater or marine environments. Although carried out in coastal lagoons on the southwest coast of Ireland, it is envisaged that the overall findings of this study have relevance throughout the entire island of Ireland and possibly to many North Atlantic coastal lagoon ecosystems elsewhere.