Modern power systems are at risk of largely reducing the inertia of generation assets and prone to experience extreme dynamics. The consequence is that, during electromechanical transients triggered by large contingencies, transmission of electrical power may take place in a wide spectrum well beyond the single fundamental component. Traditional modeling approaches rely on the phasor representation derived from the Fourier Transform (FT) of the signal under analysis. During large transients, though, FT-based analysis may fail to accurately identify the fundamental component parameters, in terms of amplitude, frequency and phase. Taking inspiration from the theory on analytic signals, this paper proposes a different approach to model signals of power systems electromechanical transients based on the Hilbert transform (HT). We compare FT- and HT-based approaches during representative operating conditions, i.e., amplitude modulations, frequency ramps and step changes, in synthetic and real-world datasets. We further validate the approaches using a contingency analysis on the IEEE 39-bus.