In this article, the concept and definitions of the Fractional Order Moment (FOM) and Fractional Order Cumulant (FOC) are proposed, which is based on the fractional derivative of the fractional order Moment-generating function and the fractional order Cumulant-generating function of stochastic processes. The moment and cumulant are defined on an expanded set from positive integer to the whole positive real. This development not only provides a new technology for signal processing, also complements the existing theory in the field. The properties of the FOC have been derived, and their uniformity and particularity with the High Order Cumulant are compared and commented. In addition, the transformation between the FOM and the FOC are derived and discussed in detail. As one of the applications of the new concept to the α and Gaussian processes, a new method of suppressing α and Gaussian noise is proposed. Furthermore, a FOC-based parameter estimation algorithm is developed for the non-minimum phase ARMA processes in α and/or Gaussian noise. Simulation examples are used to demonstrate the effectiveness of the proposed parameter estimation algorithm.

This paper studies the effects of cross-terms in the local polynomial periodograms (LPP) by defining and examining the cross local polynomial periodogram (CLPP) and the time-frequency correlative coefficients (TFCC). Our observations show that the cross-terms of signal components can be ignored if they do not intersect and the window length used in the LPP is sufficient. In addition, the TFCC is useful for the analysis of time-frequency coherency between LFM signals and gives us a parametric measure on the amount of the cross-terms in the LPP.