Linear Prediction of Discrete-Time $1/f$ Processes

In this letter, the linear predictability of discrete-time stationary stochastic processes with 1/|f|^α-shaped power spectral density (PSD) is considered. In particular, the spectral flatness measure (SFM)-which yields a lower bound for the normalized mean-squared-error (NMSE) of any linear one-step-ahead (OSA) predictor-is obtained analytically as a function of α ∈ [0, 1]. By comparing the SFM bound to the NMSE of the p -tap linear minimum-mean-square error (LMMSE) predictor, it is shown that close to optimal NMSE performance may be achieved for relatively moderate values of p. The performance of the LMMSE predictor for the discrete-time fractional Gaussian noise (DFGN), which may be viewed as the conventional discrete-time counterpart of continuous-time processes with 1/|f|^α-shaped PSD, shows that the DFGN is more easily predicted than the discrete-time processes considered herein.