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“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.