Minimum Entropy Rate Simplification of Stochastic Processes: Supplemental Material

A.1 Gaussian MERS Solution We first consider the purely nondeterministic case; the result is easily extended to arbitrary stationary and ergodic Gaussian processes using the Wold decomposition. Let X and X̃ be two discrete-time stationary and ergodic purely nondeterministic univariate Gaussian processes, with spectral power density functions RX ( e ) and RX̃ ( e ) respectively. These are by necessity positive and defined on (−π, π]. The process X̃ is considered given, and we are trying to find a corresponding X which minimizes the differential entropy rate h∞ (X) under a differential KL-divergence rate constraint d∞(X || X̃) ≤ d and a constraint RX ( e ) ≥ rmin on the minimum power at any frequency. Following [1], the differential entropy rate of X can be calculated as