A variable-order fractional operator based synthesis method for multifractional Gaussian noise

Fractional Gaussian noise (fGn) with a constant Hurst parameter H can be used to more accurately characterize the long memory process than traditional short-range dependent stochastic processes, such as Markov, Poisson or ARMA processes. However, the ability of fGn is limited for modeling the stochastic processes with prescribed local forms. Therefore, the multifractional Gaussian noise (mGn) with local Hölder exponent which varies with a variable t (usually time), become more important both in theory and in practical applications. In this paper, by studying the relationship of white Gaussian noise (wGn), mGn and multifractional Brownian motion (mBm), a synthesis method which is based on variable-order fractional operators for synthesizing mGn is provided. Furthermore, the synthesis method of multifractional α-stable processes, the generalization of mGn, is proposed in the paper in order to more accurately characterize the processes with local scaling characteristic and heavy tailed distribution. Some synthetic examples of mGn and multifractional α-stable noises are provided in the paper.