  Most existing works on optimal filtering of linear time-invariant (LTI)
stochastic systems with arbitrary unknown inputs assume perfect knowledge of
the covariances of the noises in the filter design. This is impractical and
raises the question of whether and under what conditions one can identify the
process and measurement noise covariances (denoted as $Q$ and $R$,
respectively) of systems with unknown inputs. This paper considers the
identifiability of $Q$/$R$ using the correlation-based measurement difference
approach. More specifically, we establish (i) necessary conditions under which
$Q$ and $R$ can be uniquely jointly identified; (ii) necessary and sufficient
conditions under which $Q$ can be uniquely identified, when $R$ is known; (iii)
necessary conditions under which $R$ can be uniquely identified, when $Q$ is
known. It will also be shown that for achieving the results mentioned above,
the measurement difference approach requires some decoupling conditions for
constructing a stationary time series, which are proved to be sufficient for
the well-known strong detectability requirements established by Hautus.
