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

• Defines a generalization of coherence that can capture causal relationships.
• Proves uniform convergence of the FIR causal Wiener filters under Hölder conditions.
• Proves uniform convergence of the PSDs of finite order AR approximations.
• Gives asymptotic upper bounds on the convergence errors.

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 cannot represent. To understand how well frequency dependent causal coherence spectra can be estimated with finite order approximations, the convergence of FIR causal Wiener filters to the full IIR causal Wiener filter is investigated as filter length goes to infinity. The main results prove Lp convergence of the frequency responses for p=1,2,∞p=1,2,∞ under certain Hölder continuity conditions on the power spectra, as well as give asymptotic upper bounds for the convergence error. As a corollary, under the same conditions, the uniform convergence of the power spectra of AR approximations is shown as model order goes to infinity.