K\"ahler information manifolds of signal processing filters in weighted Hardy spaces

Abstract

We extend the framework of K\"ahler information manifolds for complex-valued signal processing filters by introducing weighted Hardy spaces and smooth transformations of transfer functions. We demonstrate that the Riemannian geometry induced from weighted Hardy norms for the smooth transformations of its transfer function is a K\"ahler manifold. In this setting, the K\"ahler potential of the linear system geometry corresponds to the squared weighted Hardy norm of the composite transfer function. With the inherent structure of K\"ahler manifolds, geometric quantities on the manifold of linear systems in weighted Hardy spaces can be computed more efficiently and elegantly. Moreover, this generalized framework unifies a variety of well-known information manifolds within the structure of K\"ahler information manifolds for signal filters. Several illustrative examples from time series models are provided, wherein the metric tensor, Levi-Civita connection, and K\"ahler potentials are explicitly expressed in terms of polylogarithmic functions of the poles and zeros of transfer functions parameterized by weight vectors.