Wen-Liang Hwang
We show that non-orthogonal wavelets can characterize the fractional Brownian motion (fBm) that is in white noise. We demonstrate the point that discriminating the parameter of a fBm from that of noise is equivalent to discriminating the composite singularity formed by superimposing a peak singularity upon 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 an $\frac{1}{f}$ process. Comparative studies of our methods with those of Wornell and Oppenheim are shown in numerical simulations.