Long and short-term correlation properties of computer-generated fractional Gaussian noise

It is well known that fractional Gaussian noise (fGn) generated by synthesizing the fGn autocorrelation structure exhibits exact short and long-term dependence, as well as a good approximation to the theoretical power-law spectrum. In contrast, the spectral synthesis method (SSM), based on synthesizing the fGn power spectrum, generates time series that have the desired spectra, but not the correct dependence. This paper shows that the autocorrelation-based algorithm, called the fractional Gaussian process algorithm, produces a random, zero-frequency spectral component that accounts for the difference in behavior. Furthermore, any algorithm that does not produce the appropriate zero-frequency term cannot generate an fGn series with correct correlation properties. It is shown that the practice of using short segments of long SSM-generated time series does not fully compensate for the absence of the zero-frequency term.