Estimation of fractional Brownian motion embedded in a noisy environment using nonorthogonal wavelets

We show that nonorthogonal wavelets can characterize the fractional Brownian motion (fBm) that is in white noise. We demonstrate the point that discriminating the parameter of fBm from that of noise is equivalent to discriminating the composite singularity formed by superimposing a peak singularity on a Dirac singularity. We characterize the composite singularity by formalizing this problem as a nonlinear optimization problem. This yields our parameter estimation algorithm. For fractal signal estimation, Wiener filtering is explicitly formulated as a function of the signal and noise parameters and the wavelets. We show that the estimated signal is a 1/f process. Comparative studies through numerical simulations of our methods with those of Wornell and Oppenheim (1992) are presented