Convergence to the rate-distortion function for Gaussian sources

In this paper we derive an expression for the minimum-mean-square error achievable in encodingtsamples of a stationary correlated Gaussian source. It is assumed that the source output is not known exactly but is corrupted by correlated Gaussian noise. The expression is obtained in terms of the covariance matrices of the source and noise sequences. It is shown that ast rightarrow infty, the result agrees with a known asymptotic result, which is expressed in terms of the power spectra of the source and noise. The rate of convergence to the asymptotic results as a function of coding delay is investigated for the case where the source is first-order Markov and the noise is uncorrelated. WithDthe asymptotic minimum-mean-square error andD_tthe minimum-mean-square error achievable in transmittingtsamples, we findmid D_t - D mid leq O((t^{-1} log t) ^ {1/2})when we transmit the noisy source vectors over a noiseless channel andmid D_t - D mid leq O((t^{-1} log t)^ {1/3})when the channel is noisy.