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.

Forces of agglomeration and dispersal in rural business networks: a social network analysis

Two concepts in rural economic development policy have been the focus of much research and policy action: the identification and support of clusters or networks of firms and the availability and adoption by rural businesses of Information and Communication Technologies (ICT). From a theoretical viewpoint these policies are based on two contrasting models, with clustering seen as a process of economic agglomeration, and ICT-mediated communication as a means of facilitating economic dispersion. The study’s conceptual framework is based on four interrelated elements: location, interaction, knowledge, and advantage, together with the concept of networks which is employed as an operationally and theoretically unifying concept. The research questions are developed in four successive categories: Policy, Theory, Networks, and Method. The questions are approached using a study of two contrasting groups of rural small businesses in West Cork, Ireland: (a) Speciality Foods, and (b) firms in Digital Products and Services. The study combines Social Network Analysis (SNA) with Qualitative Thematic Analysis, using data collected from semi-structured interviews with 58 owners or managers of these businesses. Data comprise relational network data on the firms’ connections to suppliers, customers, allies and competitors, together with linked qualitative data on how the firms established connections, and how tacit and codified knowledge was sourced and utilised. The research finds that the key characteristics identified in the cluster literature are evident in the sample of Speciality Food businesses, in relation to flows of tacit knowledge, social embedding, and the development of forms of social capital. In particular the research identified the presence of two distinct forms of collective social capital in this network, termed “community” and “reputation”. By contrast the sample of Digital Products and Services businesses does not have the form of a cluster, but matches more closely to dispersive models, or “chain” structures. Much of the economic and social structure of this set of firms is best explained in terms of “project organisation”, and by the operation of an individual rather than collective form of “reputation”. The rural setting in which these firms are located has resulted in their being service-centric, and consequently they rely on ICT-mediated communication in order to exchange tacit knowledge “at a distance”. It is this factor, rather than inputs of codified knowledge, that most strongly influences their operation and their need for availability and adoption of high quality communication technologies. Thus the findings have applicability in relation to theory in Economic Geography and to policy and practice in Rural Development. In addition the research contributes to methodological questions in SNA, and to methodological questions about the combination or mixing of quantitative and qualitative methods.