Calculus of variations on time scales and applications to economics

We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.

Multi-standard reconfigurable wireless transmitter architectures

The continuous demand for highly efficient wireless transmitter systems has triggered an increased interest in switching mode techniques to handle the required power amplification. The RF carrier amplitude-burst transmitter, i.e. a wireless transmitter chain where a phase-modulated carrier is modulated in amplitude in an on-off mode, according to some prescribed envelope-to-time conversion, such as pulse-width or sigma-delta modulation, constitutes a promising architecture capable of efficiently transmitting signals of highly demanding complex modulation schemes. However, the tested practical implementations present results that are way behind the theoretically advanced promises (perfect linearity and efficiency). My original contribution to knowledge presented in this thesis is the first thorough study and model of the power efficiency and linearity characteristics that can be actually achieved with this architecture. The analysis starts with a brief revision of the theoretical idealized behavior of these switched-mode amplifier systems, followed by the study of the many sources of impairments that appear when the real system is implemented. In particular, a special attention is paid to the dynamic load modulation caused by the often ignored interaction between the narrowband signal reconstruction filter and the usual single-ended switched-mode power amplifier, which, among many other performance impairments, forces a two transistor implementation. The performance of this architecture is clearly explained based on the presented theory, which is supported by simulations and corresponding measured results of a fully working implementation. The drawn conclusions allow the development of a set of design rules for future improvements, one of which is proposed and verified in this thesis. It suggests a significant modification to this traditional architecture, where now the phase modulated carrier is always on – and thus allowing a single transistor implementation – and the amplitude is impressed into the carrier phase according to a bi-phase code.