Fractal estimation from noisy data via discrete fractional Gaussian noise (DFGN) and the Haar basis

We show that the application of the discrete wavelet transform (DWT) using the Haar basis to the increments of fractional Brownian motion (f Bm), also known as discrete fractional Gaussian noise (DFGN), yields coefficients which are weakly correlated and have a variance that is exponentially related to scale. Similar results were derived by Flandrin, Tewfik, and Kim for a continuous-time fBm going through a continuous wavelet transform (CWT). The new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by Wornell and Oppenheim. The performance of the new algorithm is compared with that of Wornell and Oppenheim’s algorithm in numerical simulation.