  In this paper, we study the zero-delay source-channel coding problem, and
specifically the problem of obtaining the vector transformations that optimally
map between the m-dimensional source space and the k-dimensional channel space,
under a given transmission power constraint and for the mean square error
distortion. We first study the functional properties of this problem and show
that the objective is concave in the source and noise densities and convex in
the density of the input to the channel. We then derive the necessary
conditions for optimality of the encoder and decoder mappings. A well known
result in information theory pertains to the linearity of optimal encoding and
decoding mappings in the scalar Gaussian source and channel setting, at all
channel signal-to-noise ratios (CSNRs). In this paper, we study this result
more generally, beyond the Gaussian source and channel, and derive the
necessary and sufficient condition for linearity of optimal mappings, given a
noise (or source) distribution, and a specified power constraint. We also prove
that the Gaussian source-channel pair is unique in the sense that it is the
only source-channel pair for which the optimal mappings are linear at more than
one CSNR values. Moreover, we show the asymptotic linearity of optimal mappings
for low CSNR if the channel is Gaussian regardless of the source and, at the
other extreme, for high CSNR if the source is Gaussian, regardless of the
channel. Our numerical results show strict improvement over prior methods. The
numerical approach is extended to the scenario of source-channel coding with
decoder side information. The resulting encoding mappings are shown to be
continuous relatives of, and in fact subsume as special case, the Wyner-Ziv
mappings encountered in digital distributed source coding systems.
