Optimal estimation of an unknown deterministic signal vector using a time-invariant filter

The problem of estimating a deterministic signal vector
\undertilde{theta}
from
\undertilde{x} = \undertilde{theta} + \undertilde{n}
is considered using quadratic loss. It is assumed that the noise
\undertilde{n}
is weakly stationary, and that the vector size is large. These assumptions along with a time-invariant filter constraint allow the use of Fourier transforms and a filtering approach. It is noted that in the class of time-invariant data-independent filters, given spectral knowledge of the unknown deterministic signal vector
\undertilde{theta}
, the best performance is achieved by a form similar to the classical Wiener filter form. This provides the motivation for a simple empirical Wiener estimator, wherein the signal spectral information is estimated from the data. This estimator is shown to dominate the MLE at least in the case where the spectral signal-to-noise ratio is uniformly
\lsim
0.65.