Spectral decomposition of time-frequency distribution kernels

This paper addresses the general problem of approximating a given time-frequency distribution (TFD) in terms of other distributions with desired properties. It relates the approximation of two time-frequency distributions to their corresponding kernel approximation. It is shown that the singular-value decomposition (SVD) of the time-frequency (t-f) kernels allows the expression of the time-frequency distributions in terms of weighted sum of smoothed pseudo Wigner-Ville distributions or modified periodograms, which are the two basic nonparametric power distributions for stationary and nonstationary signals, respectively. The windows appearing in the decomposition take zero and/or negative values and, therefore, are different than the time and lag windows commonly employed by these two distributions. The centrosymmetry and the time-support properties of the kernels along with the fast decay of the singular values lead to computational savings and allow for an efficient reduced rank kernel approximations