This study focuses on the characterization of nonstationary, weakly nonlinear, weakly damped mechanical systems, subjected to an unknown ambient excitation modeled as white Gaussian noise. A theoretical analysis using a novel approach of a one degree of freedom system, whose dynamic response is investigated in time domain as a Markov process, allows simple relations to be established on the expectation and standard deviation of its amplitude and instantaneous frequency. This theoretical study of the system behavior is coupled with the use of the Continuous Wavelet Transform (CWT) ridge, which gives the amplitude and instantaneous frequency of its vibrations. A discussion on the tuning of the mother wavelet function for the analysis of these signals is presented. A simple procedure based on the use of the CWT, characterizing the system's nonlinear, nonstationary behavior, is then proposed. Finally, numerical simulations are carried out, and the proposed method is implemented. Results show good agreement with those of the theoretical study.