Optimality conditions are derived for reduced-order sampled-data estimation satisfying an observation constraint that involves a prespecified, possibly unstable, subspace of the system dynamics. It is shown that a generalized hold device possessing a time-varying exponential intersample profile is essential in order to account for the unstable dynamics. In particular, it is shown that with a continuous-time quadratic performance measure a zero-order-hold device may result in an unbounded (infinite) cost. An additional feature of the problem is the utilization of an averaging A/D device within the continuous-time/discrete-time conversion which results in averaged measurements depending upon delayed samples of the state.