Fonollosa and Nikias proposed a class of multidimensional distributions to evaluate the evolution of discrete-time signals' higher order moment spectra. However, to satisfy some important property called marginal condition, their distribution is defined even for a time/frequency for which the signal and its higher order moment spectrum are not defined. The main objective of this paper is to present an alternative definition of such multidimensional distributions, named generalized discrete higher order moment spectra (GDHOMS), which overcome the drawback. The proposed GDHOMS are completely characterized in terms of multivariate functions called kernels. The second objective of this paper is to clarify the standard requirements for the kernels to realize the desirable properties of the GDHOMS, similar to those of the Cohen's generalized class of joint time-frequency distributions. In addition, we propose a systematic kernel design of the GDHOMS based on the hybrid steepest descent method. The proposed design realizes a kernel that satisfies given standard requirements as good as possible.
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