  We study the problem of optimal estimation and control of linear systems
using quantized measurements, with a focus on applications over sensor
networks. We show that the state conditioned on a causal quantization of the
measurements can be expressed as the sum of a Gaussian random vector and a
certain truncated Gaussian vector. This structure bears close resemblance to
the full information Kalman filter and so allows us to effectively combine the
Kalman structure with a particle filter to recursively compute the state
estimate. We call the resulting filter the Kalman like particle filter (KLPF)
and observe that it delivers close to optimal performance using far fewer
particles than that of a particle filter directly applied to the original
problem. We show that the conditional state density follows a, so called,
generalized closed skew-normal (GCSN) distribution. We further show that for
such systems the classical separation property between control and estimation
holds and that the certainty equivalent control law is LQG optimal.
