Non-Linear, Non-Stationary spectral analysis methods! When and where?

Question

I have been reading about non-linear non-stationary signal analysis methods and it seems to do this type of analysis the go-to method is the Empirical Mode Decomposition (EMD), then Hilbert Transform (HT) to get instantaneous phase and frequency.

However I have been doing some analysis on a non-linear, non-stationary signal by splitting the time signal into intervals with a Hanning window and Fourier transforming each interval which I think might also be the idea behind the Short Time Fourier Transform (STFT), although I am not sure.

Can anyone tell me what is wrong with this method, and why the EMD / HT method is more advantageous? I have also seen wavelet analysis used in similar cases, is this something I should look into?

Answer 1

Non-linearity and non-stationarity are non-properties. Without more details, they do not say much about the methods that may perform well, and moreover the choice depends a lot on what you really do: analysis, feature extraction, enhancement, filtering, component separation, restoration? What follows are typical sets of tools you could use:

• Your moving-window-Fourier definitely has some STFT flavors, and there are many non-linear extensions: bilinear or quadratic time–frequency distributions (Wigner-Ville, Choi–Williams, Rihaczek)
• Non-linearity may come from changing the classical energy preservation into other norms, like robust DFTs (under an $\ell_1$ norm minimization), see for instance Robust $L$-estimation based forms of signal transforms and time-frequency representations, see also Using different algorithms/formulas to create spectrograms
• Some methods also alleviate the hypothesis that signal formation is linear: variance-stabilizing transforms, cepstral analysis or homomorphic signal processing (linear transforms applied in a $\log$ domain)
• Generalizations of linear approaches in polynomial or rational fraction modeling, with a recent interest in Rational Optimization for Nonlinear Reconstruction with Approximate $\ell_0$ Penalization.

All of the above can be combined somehow, like EMD at different scales in Intrinsic multi-scale analysis: a multi-variate empirical mode decomposition framework. EMD has many uses and some known limitations at the same time.