The USC Andrew and Erna Viterbi School of Engineering USC Signal and Image Processing Institute USC Ming Hsieh Department of Electrical Engineering University of Southern California

Technical Report USC-SIPI-212

“Fractal Estimation from Noisy Measurements via Discrete Fractional Gaussian Noise (DFGN) and the Haar Basis”

by Lance Kaplan and C.-C. Jay Kuo

July 1992

In this research we first show that when the increments of sampled fractional Brownian motion (fBm), also known as discrete fractional Gaussian noise (DFGN), is set equal to the finest scale wavelet approximation coefficients and when the Haar basis is selected, the discrete wavelet transform (DWT) coefficients 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), and the theoretical results justify a fractal estimation algorithm recently proposed by Wornell and Oppenheim. However, since discrete samples of fBm data are passed through the DWT in practice, Wornell and Oppenheim's algorithm may not yield an accurate estimate. Thus, motivated by our theoretical result, a new fractal estimation algorithm is proposed by handling the increments of the sampled fBm rather than the sampled fBm itself. The performance of the new algorithm is compared with that of Wornell and Oppenheim's algorithm in numerical simulation.

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