Synthesis of accurate fractional Gaussian noise by filtering

This paper describes a method for generating long sample paths of accurate fractional Gaussian noise (fGn), the increment process of fractional Brownian motion (fBm). The method is based on a Wold decomposition in which fGn is expressed as the output of a finite impulse response filter with discrete white Gaussian noise as input. The form of the ideal filter is derived analytically in the continuous-time case. For the finite-length discrete-time case, an iterative projection algorithm incorporating a Newton-Raphson step is described for computing the coefficients of a length-N filter in a time approximately proportional to N. Fast convolution of discrete white Gaussian noise with the computed filter impulse response yields arbitrarily long sequences which exactly match the correlation structure of fGn over a finite range of lags. For values of the Hurst parameter H smaller than a critical value H-cr (it) approximate to 0.85, and large N, the finite-length autocorrelation sequence of fGn is positive definite and this range of lags can be as large as the filter autocorrelation length. When H > H-cr (it), the finite-length autocorrelation sequence of fGn is no longer positive definite and a modification is made to allow a Wold decomposition. The generated sequences then exactly match the autocorrelation structure of fGn over a more restricted range of lags which becomes smaller as H approaches unity.