Singularity power spectrum distribution

• We propose the definition of singularity power spectrum distribution (SPSD).
• The compatibility between SPSD and traditional PSD is proved.
• The approximation algorithm of SPSD is deduced.
• Simulation shows SPSD represents the new feature of real fractal time series.

Fractal and multifractal signal processing are hot topics in current studies. At present, the research mainly concentrates in fractal dimension, multifractal spectrum and time-singularity spectrum distribution, which are based on differentiability of fractal subsets, however fail to reflect the energy and power measurement of fractal singular subset. Therefore, singularity power spectrum distribution (SPSD) is proposed in this paper, based on the traditional power spectrum density function and multifractal spectrum. In theory, basic concept of SPSD is put forward based on the related singularity measure, the theoretical expression of SPSD is deduced, and the compatibility between SPSD and conventional power spectrum analysis is proved. In algorithm, the discrete approximation algorithm of SPSD for discrete fractal time series is introduced. Simulation based on the multifractal Brownian motion (mFBM) and actual sea clutter indicates that SPSD can reveal effectively the singular power distribution of sea clutter and mFBM, and identify fractal signals, even for those with approximate multifractal spectra or approximate power spectra. The SPSD will provide application prospect of multifractal signal processing, detection and distinguishing of multifractal time series.