On the spectra of Riemann-Liouville fractional Brownian motion

We study the spectrum of the fractional Brownian motion of Riemann-Liouville type using two approaches, namely the double frequency spectral density and the Wigner-Ville spectrum. The bifrequency representation gives a complex-valued function which contains two distinctive terms. These terms can be identified as the diagonal and off-diagonal distribution of the spectrum in a frequency-frequency plane. The physical interpretation of these two terms is briefly discussed. A calculation of Wigner-Ville spectrum gives an alternative way of representing the spectrum of this nonstationary process in the time-frequency plane. Asymptotic approximation of the Wigner-Ville spectrum is obtained. We show that the large-time average spectrum of Riemann-Liouville fractional Brownian motion exhibits a power law in the high-frequency range.