We generalize K\"ahler information manifolds of complex-valued signal processing filters by introducing weighted Hardy spaces and smooth transformations of transfer functions. We prove that the Riemannian geometry of a linear filter induced from weighted Hardy norms for the smooth transformations of its transfer function is the K\"ahler manifold. Additionally, the K\"ahler potential of the linear system geometry corresponds to the square of the weighted Hardy norms of its composite transfer functions. Based on the properties of K\"ahler manifold, geometric objects on the manifolds from arbitrary weight vectors are computed in much simpler ways. Moreover, K\"ahler information manifolds of signal filters in weighted Hardy spaces incorporate various well-known information manifolds under the unified framework. We also cover several examples from time series models of which metric tensor, Levi-Civita connection, and K\"ahler potentials are represented with polylogarithm of poles and zeros from the transfer functions with the weight vectors are given as a family of exponential forms.

翻译：我们通过引入加权Hardy空间以及传递函数的光滑变换，推广了复值信号处理滤波器的Kähler信息流形。我们证明，由加权Hardy范数诱导的线性滤波器黎曼几何（针对其传递函数的光滑变换）即为Kähler流形。此外，该线性系统几何的Kähler势对应于其复合传递函数的加权Hardy范数的平方。基于Kähler流形的性质，从任意权向量出发的流形上的几何对象可通过更简便的方式计算得到。更重要的是，加权Hardy空间中信号滤波器的Kähler信息流形将多种已知信息流形纳入统一框架。我们还涵盖了来自时间序列模型的若干实例，其中度量张量、Levi-Civita联络以及Kähler势可用传递函数极点与零点的多重对数表示，且权向量族取指数形式。