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.

The political philosophy of Miki Kiyoshi: A close reading of the philosophical foundations of cooperative communitarianism

The thesis is a historical and philological study of the mature political theory of Miki Kiyoshi (1897-1945) focused on Philosophical Foundations of Cooperative Communitarianism (1939), a full translation of which is included. As the name suggests, it was a methodological and normative communitarianism, which critically built on liberalism, Marxism and Confucianism to realise a regional political community. Some of Miki’s Western readers have wrongly considered him a fascist ideologue, while he has been considered a humanist Marxist in Japan. A closer reading cannot support either view. The thesis argues that the Anglophone study of Japanese philosophy is a degenerating research programme ripe for revolution in the sense of returning full circle to an original point. That means returning to the texts, reading them contextually and philologically, in principle as early modern European political theory is read by intellectual historians, such as the representatives of Cambridge School history of political thought. The resulting reading builds critically on the Japanese scholarship and relates it to contemporary Western and postcolonial political theory and the East Asian tradition, particularly neo-Confucianism. The thesis argues for a Cambridge School perspective radicalised by the critical addendum of geo-cultural context, supplemented by Geertzian intercultural hermeneutics and a Saidian ‘return to philology’. As against those who have seen radical reorientations in Miki’s political thought, the thesis finds gradual progression and continuity between his neo-Kantian, existentialist, Marxian anthropology, Hegelian and finally communitarian phases. The theoretical underpinnings are his philosophical anthropology, a structurationist social theory of praxis, and a critique of liberalism, Marxism, nationalism and idealism emphasising concrete as opposed to abstract theory and the need to build on existing cultural traditions to modernise rather than westernise East Asia. This post-Western fusion was imagined to be the beginning of a true and pluralistic universalism.

From the demise of theory to its resurgence. A retrospective on the postmodern turn to rethink the 'desire for narrative' and the sublime untimeliness of emancipation

Cultural Marxist Theory, commonly known as theory, enjoyed a moment of extraordinary success in the 1970s, when the works of leading post-war French philosophers were published in English. After relocating to Anglophone academia, however, theory disavowed its original concerns and lost its ambition to understand the world as a whole, becoming the play of heterogeneities associated with postcolonialism, multiculturalism and identity politics, commonly referred to as postmodern theory. This turn, which took place during a period that seemed to have spelt the death of Marxism, the 1990s, induced many of its supporters to engage in an ongoing funeral wake designating the merits of theory and dreaming its resurgence. According to them, had theory been resurrected in historical circumstances completely different from those which had led to its rise, it would have never reacquired the significance that had originally connoted it. This thesis demonstrates how theory has survived its demise and entirely regained its prominence in our socio-political context marked by the effects of the latest crisis of capitalism and by the global threat of terrorisms rooted in messianic eschatologies. In its current form theory does no longer need to show allegiance to certain intellectual stances or political groupings in order to produce important reformulations of the projects it once gave life to. Though less overtly radical and epistemologically bounded, theory remains a necessary form of enquiry justified by the political commitment which originated it in the first place. Its voice continues to speak to us about justice ‘where it is not yet, not yet there, where it is no longer’ (Derrida, 1993, XVIII).