The propagation of waves in solids, especially when characterized by dispersion, remains a topic of profound interest in the field of signal processing. Dispersion represents a phenomenon where wave speed becomes a function of frequency and results in multiple oscillatory modes. Such signals find application in structural healthmonitoring for identifying potential damage sensitive features in complex materials. Consequently, it becomes important to find matched time-frequency representations for characterizing the properties of the multiple frequency-dependent modes of propagation in dispersive material. Various time-frequency representations have been used for dispersive signal analysis. However, some of them suffered from poor timefrequency localization or were designed to match only specific dispersion modes with known characteristics, or could not reconstruct individual dispersive modes. This thesis proposes a new time-frequency representation, the nonlinear synchrosqueezing transform (NSST) that is designed to offer high localization to signals with nonlinear time-frequency group delay signatures. The NSST follows the technique used by reassignment and synchrosqueezing methods to reassign time-frequency points of the short-time Fourier transform and wavelet transform to specific localized regions in the time-frequency plane. As the NSST is designed to match signals with third order polynomial phase functions in the frequency domain, we derive matched group delay estimators for the time-frequency point reassignment. This leads to a highly localized representation for nonlinear time-frequency characteristics that also allow for the reconstruction of individual dispersive modes from multicomponent signals. For the reconstruction process, we propose a novel unsupervised learning approach that does not require prior information on the variation or number of modes in the signal. We also propose a Bayesian group delay mode merging approach for reconstructing modes that overlap in time and frequency. In addition to using simulated signals, we demonstrate the performance of the new NSST, together with mode extraction, using real experimental data of ultrasonic guided waves propagating through a composite plate.

Dynamical decoupling (DD) is a promising approach to mitigate the detrimental effects that interactions with the environment have on a quantum system. In DD, the finite-dimensional system is rotated about specified axes using strong and fast controls that eliminate system-environment interactions and protect the system fromdecoherence. In this thesis, the framework of DD is theoretically studied, and later it discusses how this framework can be implemented on an infinite-dimensional system that amplifies system components rather than suppressing them through quadrature squeezing operations. It begins by studying the impact of system-environment interactions on a quantum system, and then it analyzes how DD suppresses these interactions. The conditions for protecting a finite-dimensional system through DD are reviewed, and a numerical analysis of the DD conditions for simple systems is conducted. Using bang-bang controls, a framework for decoupling decoherence-inducing components from a general finite-dimensional system is studied. Later, following an overview of schemes that amplify the strength of a quantum signal through reversible squeezing, a theoretical study of Hamiltonian Amplification (HA) for quantum harmonic oscillators is presented. By implementing the DD framework with squeezing operations, HA achieves speed-up in the dynamics of quantum harmonic oscillators, which translates into the strengthening of interactions between harmonic oscillators. Finally, the application of HA in amplifying the third-order nonlinearity in a Kerr medium is proposed to obtain a speed-up in the implementation of controlled phase gates for optical quantum computations. Numerically simulated results show that large amplification in nonlinearity is feasible with sufficient squeezing resources, completing the set of universal quantum gates in optical quantum computing.