  The problem of joint source-channel coding is considered for a stationary
remote (noisy) Gaussian source and a Gaussian channel. The encoder and decoder
are assumed to be causal and their combined operations are subject to a delay
constraint. It is shown that, under the mean-square error distortion metric, an
optimal encoder-decoder pair from the linear and time-invariant (LTI) class can
be found by minimization of a convex functional and a spectral factorization.
The functional to be minimized is the sum of the well-known cost in a
corresponding Wiener filter problem and a new term, which is induced by the
channel noise and whose coefficient is the inverse of the channel's
signal-to-noise ratio. This result is shown to also hold in the case of
vector-valued signals, assuming parallel additive white Gaussian noise
channels. It is also shown that optimal LTI encoders and decoders generally
require infinite memory, which implies that approximations are necessary. A
numerical example is provided, which compares the performance to the lower
bound provided by rate-distortion theory.
