  The effects of quantization and coding on the estimation quality of
Gauss-Markov processes are considered, with a special attention to the
Ornstein-Uhlenbeck process. Samples are acquired from the process, quantized,
and then encoded for transmission using either infinite incremental redundancy
(IIR) or fixed redundancy (FR) coding schemes. A fixed processing time is
consumed at the receiver for decoding and sending feedback to the transmitter.
Decoded messages are used to construct a minimum mean square error (MMSE)
estimate of the process as a function of time. This is shown to be an
increasing functional of the age-of-information (AoI), defined as the time
elapsed since the sampling time pertaining to the latest successfully decoded
message. Such functional depends on the quantization bits, codewords lengths
and receiver processing time. The goal, for each coding scheme, is to optimize
sampling times such that the long-term average MMSE is minimized. This is then
characterized in the setting of general increasing functionals of AoI, not
necessarily corresponding to MMSE, which may be of independent interest in
other contexts.
  We first show that the optimal sampling policy for IIR is such that a new
sample is generated only if the AoI exceeds a certain threshold, while for FR
it is such that a new sample is delivered just-in-time as the receiver finishes
processing the previous one. Enhanced transmissions schemes are then developed
in order to exploit the processing times to make new data available at the
receiver sooner. For both IIR and FR, it is shown that there exists an optimal
number of quantization bits that balances AoI and quantization errors, and
hence minimizes the MMSE. It is also shown that for longer receiver processing
times, the relatively simpler FR scheme outperforms IIR.
