Instantaneous spectral moments

Density functions find application in many fields of science, math and engineering. In many cases, the density can be sufficiently characterized by some of its moments, particularly the mean, variance, skew and kurtosis. In signal analysis, densities of interest are the instantaneous power of the signal, the spectral density, and for signals with time-varying spectral content, the joint time–frequency density. As with densities in general, these signal densities may also be characterized by their low-order moments. For the case of the joint density, the moments are conditional moments, e.g., the mean frequency at a particular time. The first- and second-conditional moments of a time–frequency density have been well-studied in the past two decades; the third, fourth and higher conditional moments have not. In this paper, we propose candidates, utilizing an operator-theoretic approach, for the instantaneous spectral moments of a signal, in terms of its amplitude and phase. From these instantaneous spectral moments, we obtain expressions for the instantaneous spectral mean, variance, skew and kurtosis. We also address the question of designing kernels in the Cohen class of time–frequency distributions to obtain distributions with these moments.