Fast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises

In this letter, it is shown that fast and exact fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) signals can be synthesized by the circulant embedding method (CEM). CEM consists in embedding the N/spl times/N covariance matrix of the stationary fGn process in a larger 2M/spl times/2M circulant matrix such that M /spl ges/N-1. CEM is exact, since second-order statistics of the generated data are those of the Gaussian fGn. CEM is fast, since the optimal case M=N-1 can be reached. Fast and exact fBm sequences can be easily recovered from fGn ones.