Non-parametric power spectrum estimation for Riemann-Liouville fractional order signals

The power spectrum has served as a quick and fairly robust method to obtain access to the underlying dynamics of measured weakly stationary signals. As periodograms are noisy estimates of the underlying power spectrum, periodogram smoothing is performed in the frequency domain to obtain a more accurate description of the power spectrum. Frequency smoothing operates under the assumption that the power spectrum is continuously differentiable. Fractional order signals are becoming more important as they serve to describe important diffusion and dispersion effects encountered in mechanical, electrochemical and biomedical applications. Fractional time series of the Riemann-Liouville type have the property that the underlying power spectrum exists but it is not continuously differentiable. In this paper, we mathematically study these signals and propose a new frequency smoother to overcome the problem such that these signals can still be studied in a non-parametric way.