  Input estimation is a signal processing technique associated with
deconvolution of measured signals after filtering through a known dynamic
system. Kitanidis and others extended this to the simultaneous estimation of
the input signal and the state of the intervening system. This is normally
posed as a special least-squares estimation problem with unbiasedness. The
approach has application in signal analysis and in control. Despite the
connection to optimal estimation, the standard algorithms are not necessarily
stable, leading to a number of recent papers which present sufficient
conditions for stability. In this paper we complete these stability results in
two ways in the time-invariant case: for the square case, where the number of
measurements equals the number of unknown inputs, we establish exactly the
location of the algorithm poles; for the non-square case, we show that the best
sufficient conditions are also necessary. We then draw on our previous results
interpreting these algorithms, when stable, as singular Kalman filters to
advocate a direct, guaranteed stable implementation via Kalman filtering. This
has the advantage of clarity and flexibility in addition to stability. En
route, we decipher the existing algorithms in terms of system inversion and
successive singular filtering. The stability results are extended to the
time-varying case directly to recover the earlier sufficient conditions for
stability via the Riccati difference equation.
