On optimal linear estimation of signals with general spectral distribution (Corresp.)

It is standard practice to assume that second-order stationary signal and noise processes involved in a linear estimation procedure have spectral densities. A heuristic justification may be based on the reasoning that the part of the signal having singular spectral distribution can be precisely determined, and the part of the noise having singular spectral distribution can be completely eliminated. Rigorous phrasing and proof of this claim are given here for the most general case, i.e., where the singular parts of the spectral distributions may contain continuous components, in which situation the intuitive picture is obscure. The discussion includes both discrete-time and continuous-time processes, and the multivariable case is also treated.