Auto-Wigner–Ville distribution via non-adaptive and adaptive signal decomposition

In this work we derive general formulae for the first and the second conditional spectral moments, as well as the conditional spectral variance, according to the auto-Wigner–Ville distribution. We show that for any signal decomposition method, the first conditional spectral moment obtained via the auto-Wigner–Ville distribution is now exactly the weighted average instantaneous frequency of the decomposed signal and that it is also always real-valued. In addition, for many bilinear distributions there is no guarantee that the second conditional spectral moment (and therefore the conditional spectral variance) are always positive. We also show for any signal decomposition method based on any time–frequency elementary function with Gaussian envelopes and arbitrary polynomial phase, the above constraints are always satisfied. Although we use just two decomposition methods to verify our claim, Gabor and matching pursuit (MP), the derived expressions are true for any signal decomposition method. The auto-Wigner–Ville representation of the two different decomposition methods are examined together in addition to the original Wigner–Ville distribution with respect to: (i) concentration/resolution, (ii) noise reduction capabilities due to additive noise, and (iii) frequency and time resolvability.