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-242

“Signal Modeling with Extended Self-Similar Processes”

by Lance M. Kaplan and C.-C. Jay Kuo

September 1993

The fractional Brownian motion (fBm) model has proven to be valuable in modeling many natural processes because of its self-similarity character. However, the model is characterized by one single parameter which cannot distinguish between short and long term correlation effects. This work investigates the idea of generalizing self-similarity to create extended self-similar (ESS) processes for which fBm processes are a subset. Properties of ESS processes are discussed and examples are provided. Additionally, an ESS increment model parameterized by variables controlling short and long term correlation effects is examined. We derive a theorem about the variance progression of the output coefficients of the Haar transform applied to the ESS increments and justify the ``whitening" effect of the Haar transform applied to decaying stationary processes. These results lead to a fast parameter estimation algorithm for ESS processes. We demonstrate the performance of this parameter estimation algorithm with numerical simulations.

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