  We deal with zero-delay source coding of a vector Gaussian autoregressive
(AR) source subject to an average mean squared error (MSE) fidelity criterion.
Toward this end, we consider the nonanticipative rate distortion function
(NRDF) which is a lower bound to the causal and zero-delay rate distortion
function (RDF). We use the realization scheme with feedback proposed in [1] to
model the corresponding optimal "test-channel" of the NRDF, when considering
vector Gaussian AR(1) sources subject to an average MSE distortion. We give
conditions on the vector Gaussian AR(1) source to ensure asymptotic
stationarity of the realization scheme (bounded performance). Then, we encode
the vector innovations due to Kalman filtering via lattice quantization with
subtractive dither and memoryless entropy coding. This coding scheme provides a
tight upper bound to the zero-delay Gaussian RDF. We extend this result to
vector Gaussian AR sources of any finite order. Further, we show that for
infinite dimensional vector Gaussian AR sources of any finite order, the NRDF
coincides with the zero-delay RDF. Our theoretical framework is corroborated
with a simulation example.
