Semi-classical signal analysis (SCSA) is a signal representation framework based on quantum mechanics principles and the inverse scattering transform. The signal of interest is decom- posed in a linear combination of the Schrodinger operator squared eigenfunctions, influenced by the semi-classical parameter. The framework has been utilized in several applications, in virtue of the adaptivity and localization of its components. In this thesis, we expand two direc- tions. From the theoretical perspective, up to date, the semi-classical parameter was selected in an error minimization context or a representation sparsity requirement. The framework is reinforced by providing the interval of this parameter, where a proper representation can be obtained. The lower bound is inspired by the semi-classical approximation and the sampling theorem, while the upper bound is based on the quantum perturbation theory. Such an interval defines the sampling theorem of the framework. Based on existing properties, we propose a non-uniform sampling of the semi-classical parameter, which can significantly increase the speed of convergence with minimal accuracy error. An immediate representation is also in- vestigated by providing an alternative convergence criterion drawn from signal features. Such criterion paves the way to a calculus-based parameter definition and extension to a filtering scenario. The semi-classical parameter exerts a strong influence on the SCSA components. Each component can be viewed as a soliton, a wave whose amplitude determines its width and velocity. In parallel, there exist arterial dynamics models where the solitons are solu- tions of the describing equations. We therefore propose that the soliton propagation velocity extracted from the algorithm is correlated with the pulse wave velocity, which is the blood pressure propagation velocity in the systolic phase. The velocity in the carotid-femoral seg- ment is considered the golden-standard to indicate cardiovascular risk. We therefore turn our attention to validate such a model and utilize it for arterial stiffness assessment. The model was validated based on an in-silico database fostering more than 3000 subjects. This SCSA-based model is proposed to be integrated into existing methods, where its calibration can yield single-point continuous velocity measurements.