Optimal constrained representation and filtering of signals

A random signal is observed in independent random noise. We are addressing the problem of finding the optimum signal estimate that is constrained to lie in a given linear subspace. The optimality is defined in the sense of weighted mean square error. In the second step, we find the optimum linear subspace of given dimensionality. It is shown to be the linear manifold spanned by the eigenvectors of the simultaneous diagonalization of the signal and noise covariance, that correspond to the largest eigenvalues. The result is valid for both discrete and continuous time. For large observation time and stationary signals, it is shown that the constrained optimal estimate is determined by the two spectral densities and a weighted Fourier Transform of the noise observations. The above result applies to both discrete time and continuous time signals.The Wiener filter emerges as a special case of the constrained filtering estimate, when the linear subspace is enlarged to coincide with the total measurement space.