  We characterize the rate-distortion function for zero-mean stationary
Gaussian sources under the MSE fidelity criterion and subject to the additional
constraint that the distortion is uncorrelated to the input. The solution is
given by two equations coupled through a single scalar parameter. This has a
structure similar to the well known water-filling solution obtained without the
uncorrelated distortion restriction. Our results fully characterize the unique
statistics of the optimal distortion. We also show that, for all positive
distortions, the minimum achievable rate subject to the uncorrelation
constraint is strictly larger than that given by the un-constrained
rate-distortion function. This gap increases with the distortion and tends to
infinity and zero, respectively, as the distortion tends to zero and infinity.
