Uniform FIR approximation of causal Wiener filters, with applications to causal coherence

Leveraging the relationship between Wiener filtering and the coherence function, a version of coherence is defined that captures the causal relationship between WSS processes. This causal coherence is interpreted in a modeling context and used to demonstrate what a frequency dependent measure for causality both can and can’t represent. To understand how well frequency dependent coherence spectra can be estimated with finite order approximations, the convergence of the FIR causal Wiener filters to the full IIR causal Wiener filter is investigated as filter length goes to infinity. The main results prove L convergence of the frequency responses for p = 1, 2,∞ under certain Hölder continuity conditions on the power spectra, as well as give asymptotic upper bounds for the convergence error.