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.