ESTACIONARIDAD EN LAS REALIZACIONES DE LA FUNCIÓN DE TASA DISTORSIÓN CAUSAL PARA FUENTES ESTACIONARIAS UNILATERALES

Abstract

This paper derives novel results on the characterization of the the causal informationrate-distortion function (IRDF) Ritc (D) for arbitrarily-distributed one-sided stationaryk-th order Markov source x11 = x(1); x(2); : : :. It is first shown that Gorbunov & Pinsker'sresults on the stationarity of the realizations to the causal IRDF (stated for two-sidedstationary sources) do not apply to the commonly used family of asymptotic averagesingle-letter (AASL) distortion criteria. Moreover, we show that, in general, a reconstructionsequence cannot be both jointly stationary with a one-sided stationary sourcesequence and causally related to it. This implies that, in general, the causal IRDF forone-sided stationary sources cannot be realized by a stationary distribution. However, weprove that for an arbitrarily distributed one-sided stationary source and a large class ofdistortion criteria (including AASL), the search for Ritc (D) can be restricted to distributionswhich yield the output sequence y11 jointly stationary with the source after k samples.Finally, we improve the definition of the stationary causal IRDF Ritc (D) previouslyintroduced by Derpich and stergaard for two-sided Markovian stationary sources andshow that Ritc (D) for a two-sided source : : : ; x(-1); x(0); x(1); : : : equals Ritc (D) for theassociated one-sided source x(1); x(2); : : :. This implies that, for the Gaussian quadraticcase, the practical zero-delay encoder-decoder pairs proposed by Derpich & stergaardfor approaching Ritc (D) achieve an operational data rate which exceeds Ritc (D) by lessthan 1 + 0;5 log2(2pi e =12) ' 1;254 bits per sample.