IFESIS: Instantaneous frequencies estimation via subspace invariance properties of wavelet structures

• Complex Shifted Morlet Wavelets are proposed for the IF estimation.
• CSMW major advantages include robustness to noise and low computational cost.
• CSMW provide a means for estimating the number of the resulting harmonic components, using the non-zero singular values as an indicator.
• The proposed method can track efficiently either large or slight speed variations.
• Electric motor and rolling mill stand vibrations verify the efficiency of the CSMW approach.

According to the proposed method, a set of wavelet transforms of the signal is first obtained, using a structure of Complex Shifted Morlet Wavelets. No specific constraints are imposed on the center frequencies and the bandwidths of the individual wavelets, as well as on the number of wavelets used. In this way, a set of complex signals result in the time domain, equal to the number of the wavelets used. Then, the instantaneous frequencies of the signals are estimated by applying an appropriate subspace algorithm (as for e.g. ESPRIT), to the entire set of the resulting complex wavelet transforms, exploiting the corresponding subspace rotational invariance property of this set of complex signals. Since the method proposes the application of the subspace algorithm after the signal has been appropriately transformed by an appropriate wavelet structure, contrary to the classical subspace methods which are applied to the signal itself, the desired time–frequency features of the signal are enhanced, while simultaneously, the undesired frequency components, as well as the noise, are suppressed. In this way, the method combines the advantages of the Complex Shifted Morlet Wavelets with the advantages of subspace based approaches. Moreover, the method provides a means for estimating the number of the resulting harmonic components, using as a relevant indicator the number of the non-zero singular values of the corresponding singular value decomposition problem. It should be noted that the resulting singular values are also time dependent, providing valuable information on the possibly time variable dynamic structure of the signal.