Instantaneous Frequency Estimation via Spectrogram Projection and P-Spline Approximation

Frequency Modulated (FM) signals are usually characterized by two or more components having different waveforms and time-dependent frequencies. The estimation of the instantaneous frequency (IF) represents a very challenging topic when there are two or more interfering components. In this case, available approaches, like the reassignment-based ones, fail in correspondence to the interference region, where IFs of each individual mode are not far enough to be considered separable. In this paper, a novel approach for IF estimation based on spectrogram analysis is proposed. Firstly, it is proved that the projection of signal spectrogram on the temporal axis is itself an amplitude and frequency modulated signal. Its IF depends on the instantaneous frequencies of the individual signal components and the functional dependence can be explicitely given. Then, under proper assumptions, a regression method for the individual IFs estimation is provided using a few selected 'good' points lying in the interference region. The regression method benefits from a penalized-spline (P-spline) model where a second-order discrete penalty term is also used. Experimental results show that the proposed model is very effective to estimate the individual IFs of multicomponent FM signals in the interference region, making it a suitable method for correcting reassignment estimation where the separability condition is not satisfied.