  This paper investigates the distributionally robust filtering of signals
generated by state-space models driven by exogenous disturbances with noisy
observations in finite and infinite horizon scenarios. The exact joint
probability distribution of the disturbances and noise is unknown but assumed
to reside within a Wasserstein-2 ambiguity ball centered around a given nominal
distribution. We aim to derive a causal estimator that minimizes the worst-case
mean squared estimation error among all possible distributions within this
ambiguity set. We remove the iid restriction in prior works by permitting
arbitrarily time-correlated disturbances and noises. In the finite horizon
setting, we reduce this problem to a semi-definite program (SDP), with
computational complexity scaling with the time horizon. For infinite horizon
settings, we characterize the optimal estimator using Karush-Kuhn-Tucker (KKT)
conditions. Although the optimal estimator lacks a rational form, i.e., a
finite-dimensional state-space realization, it can be fully described by a
finite-dimensional parameter. {Leveraging this parametrization, we propose
efficient algorithms that compute the optimal estimator with arbitrary fidelity
in the frequency domain.} Moreover, given any finite degree, we provide an
efficient convex optimization algorithm that finds the finite-dimensional
state-space estimator that best approximates the optimal non-rational filter in
${\cal H}_\infty$ norm. This facilitates the practical implementation of the
infinite horizon filter without having to grapple with the ill-scaled SDP from
finite time. Finally, numerical simulations demonstrate the effectiveness of
our approach in practical scenarios.
