Mutually connected networks for clock signals distribution

Abstract

The development of distributed communication and computational systems with dynamical access requires precise synchronization among the nodes belonging to the established network and their study results in processes modeled by a high number of connected nodes, exchanging phase and frequency adjusted clock signals. Lü, Chen and Wu have obtained synchronization conditions for this type of network when they are composed of dynamical systems with linear coupling. These conditions are expressed by properties of the connectivity matrices associated to the network graphs. Concerning to clock signals distribution engineering, there are slightly different peculiarities from the cases described in the complex network synchronization literature. In the clock distribution networks, the nodes contain voltage-controlled oscillators (VCO) commanded by non-linear phase detectors. Additionally, the free-running frequencies of the VCOs are different and the final network synchronous state depends on the local gains and on the signal propagation times between the nodes. Therefore, as high connectivity wireless systems develop it is necessary to study and design clock distribution systems with complex topologies deriving existence and stability conditions for the network synchronous state, depending on the network connectivity that assumes a dynamical aspect with time varying graphs. Consequently, networks with phase-locked loops (PLLs) nodes with time varying weighting matrices generate an important problem to be studied.Existence and stability conditions for the synchronous state and lock-in and capture ranges expressions will be studied related to the corresponding graphs, the propagation times, and the node gains. Additionally, optimization algorithms for lock-in and capture ranges will be developed, trying to simplify the network designers tasks. Besides their engineering relevance, from the biological point of view, there are several models describing neural cells activity based on electrical circuits. So, the study of the several PLLs and VCOs network architectures and their properties, by using analytical tools and electrical circuits, gives useful results from the telecommunication and control engineering point of view and can be used to explain the oscillatory cortical activities emergence. Summarizing, the following questions will be studied: dynamical networks with PLL nodes, performance optimization for dynamical networks, and contribution of the models of VCO networks to Biology. (AU)