Signal Detection in Fractional Gaussian Noise

Fractional Gaussian noise provides a useful model for phenomena exhibiting strong long-term dependence, such as 1/f noise in oscillators or interference on some communication channels. The problem of detecting signals in the presence of additive fractional Gaussian noise is considered. To facilitate the study of this problem, several results related to the reproducing kernel Hilbert space of fractional Brownian motion are presented. In particular, this reproducing kernel Hilbert space is completely characterized, and an alternative characterization for the restriction of this class of functions to a compact interval [0, T] is given. Infinite- and finite-interval whitening filters for fractional Brownian motion are also developed. Application of these results to the signal detection problem yields necessary and sufficient conditions for a deterministic or stochastic signal to produce a nonsingular shift when embedded in additive fractional Gaussian noise. Finally, a formula for the likelihood ratio corresponding to any deterministic nonsingular shift is developed.