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.

An investigation into reactivity and selectivity in cycloadditions of 1,3-dipoles with formamidines

The objective of this research was to investigate the synthesis of nitrile oxides and to study their reactivity in 1,3-dipolar cycloadditions with formamidines. Chapter one looks at the literature surrounding the 1,3-dipolar cycloaddition reaction. It explores the generation of 1,3-dipoles (mainly nitrile oxides) and dipolarophiles (predominantly amidines). It discusses the potential synthetic uses of the 1,3-dipolar cycloadducts. It examines both and inter- and intra-molecular cycloaddition reactions. It recognises the use of the 1,3-dipolar cycloadditions as a successful method in building natural products and oxadiazolines. The decomposition of oxadiazolines as a route to nitriles is also outlined in this chapter. Chapter two discusses the results of this research candidate. The preparation of nitrile oxide precursors - hydroximoyl halides - is outlined at first. The generation of nitrile oxides is then demonstrated, followed by the preparation of furoxans. Methods for preparing the reference materials (nitriles and ureas), which result from decomposition of oxadiazolines, then follow. The preparation of series of Δ2-1,2,4- oxadiazolines via the 1,3-dipolar cycloaddition reaction is illustrated in this chapter. The selectivity of the addition of nitrile oxides to dipolarophiles was tested by competition reactions, which are also described in this chapter. NMR techniques were used in the study of the kinetics of the 1,3-dipolar cycloadditions used for the preparation of a series of Δ2-1,2,4-oxadiazolines, which is addressed in this chapter. Chapter three charts the experimental procedures followed to gain results which are discussed in chapter two. It also outlines all analytical data produced during the course of this research.