  This paper investigates the problem of zero-delay joint source-channel coding
of a vector Gauss-Markov source over a multiple-input multiple-output (MIMO)
additive white Gaussian noise (AWGN) channel with feedback. In contrast to the
classical problem of causal estimation using noisy observations, we examine a
system where the source can be encoded before transmission. An encoder,
equipped with feedback of past channel outputs, observes the source state and
encodes the information in a causal manner as inputs to the channel while
adhering to a power constraint. The objective of the code is to estimate the
source state with minimum mean square error at the infinite horizon. This work
shows a fundamental theorem for two scenarios: for the transmission of an
unstable vector Gauss-Markov source over either a multiple-input single-output
(MISO) or a single-input multiple-output (SIMO) AWGN channel, finite estimation
error is achievable if and only if the sum of logs of the unstable eigenvalues
of the state gain matrix is less than the Shannon channel capacity. We prove
these results by showing an optimal linear innovations encoder that can be
applied to sources and channels of any dimension and analyzing it together with
the corresponding Kalman filter decoder.
