Alpha stable distribution P order spectrum with its frequency domain characteristics analysis

Alpha stable distribution has not the usual power spectral density. The current fractional lower order spectrum definitions, such as alpha spectrum, covariation spectrum, fractional lower order covariance spectrum and geometric power spectrum, etc., all use the fractional lower order transform so that there are two order statistics for transformed signals. But the fractional lower order transform will cause the changes of frequency spectrum. For this purpose, the first order spectrum of alpha stable distribution is defined in this paper, it is simple in calculation and does not change the frequency domain characteristics, and can be extended to p order spectrum. The frequency domain characteristics of fractional lower order transform are analyzed, introduced the frequency domain of the highest order of convergence of the p order spectrum and related characteristics. Simulation results show the correctness of the theory.