Investigation and optimum design of the generalized second-order phase-locked loop

In this dissertation, a second-order phase-locked loop (PLL) in which the loop filter contains complex zeros is investigated in both its linear and nonlinear modes of operation; prior designs used a filter containing a simple zero on the negative real axis. This generalized second-order PLL had been heretofore essentially unexplored. The basic characteristics of the generalized second-order PLL operating in the linear mode including the open and closed-loop responses and the corresponding root locus were generated and compared against those of the conventional second-order PLL. As in the conventional case, the generalized second-order PLL is found to be unconditionally stable. The closed-loop response of the generalized second-order PLL indicates a noise bandwidth which is theoretically infinite, thus making the predetection filter critical to the performance of this PLL. Such is not the case in the conventional PLL. A method is presented for achieving an optimum design for the generalized second-order PLL for a number of useful modulation types including a single-channel FM speech signal, FDM-FM and FDM-PM. This optimum design is in terms of threshold performance and theoretically predicts that superior performance is possible over the conventional second-order PLL. Using the Continuous System Modeling Program (CSMP), a nonlinear model of the generalized second-order PLL was simulated for the test-tone modulation case, both in the absence of noise and with the signal corrupted by bandpass additive Gaussian noise. In addition, preliminary simulation results were obtained for the case of a single-channel FM speech signal. Using simulation techniques, a measure of the mean-square phase error at threshold for the generalized second-order PLL was obtained. This parameter is useful in the optimum design procedure. Additional insight into the operation of the generalized second-order PLL was obtained through investigation of its acquisition and tracking behavior. This was accomplished using phase plane techniques to study the nonlinear differential equation which governs the loop operation. The results indicate that two distinct types of behavior are theoretically possible depending upon the loop parameters. In one case the behavior is not unlike that of the conventional second-order PLL. In the second case, however, additional singularities are introduced into the phase plane and the behavior is seen to change markedly

On the Schr\"odinger equation with potential in modulation spaces

This work deals with Schr\"odinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations. We shall give a representation of such evolution as the composition of a metaplectic operator and a pseudodifferential operator having symbol in certain classes of modulation spaces. About propagation of singularities, we use a new notion of wave front set, which allows the expression of optimal results of propagation in our context. To support this claim, many comparisons with the existing literature are performed in this work.Comment: 25 page

Generalized Metaplectic Operators and the Schr\"odinger Equation with a Potential in the Sj\"ostrand Class

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential opertators in a Sj\"ostrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the one-parameter group generated by a Hamiltonian operator with a potential in the Sj\"ostrand class consists of generalized metaplectic operators. As a consequence, the Schr\"odinger equation preserves the phase-space concentration, as measured by modulation space norms.Comment: 23 page

Photonic quasicrystals for general purpose nonlinear optical frequency conversion

We present a general method for the design of 2-dimensional nonlinear photonic quasicrystals that can be utilized for the simultaneous phase-matching of arbitrary optical frequency-conversion processes. The proposed scheme--based on the generalized dual-grid method that is used for constructing tiling models of quasicrystals--gives complete design flexibility, removing any constraints imposed by previous approaches. As an example we demonstrate the design of a color fan--a nonlinear photonic quasicrystal whose input is a single wave at frequency $\omega$ and whose output consists of the second, third, and fourth harmonics of $\omega$, each in a different spatial direction