Optimized Design of the Generalized Bilinear Transformation for Discretizing Analog Systems
For engineers designing digital systems from analog prototypes, this work offers a principled method to select the GBT parameter, though the contribution is incremental as it builds on existing transformation theory.
The paper provides a physical interpretation of the parameter α in the Generalized Bilinear Transformation (GBT) as a backward rectangular ratio and establishes its stable range as [0.5, 1]. An optimal design method for α is developed by minimizing normalized magnitude or phase error, validated through a low-pass filter design with strong agreement between theory and experiment.
A common approach to digital system design involves transforming a continuous-time (s-domain) transfer function into the discrete-time (z-domain) using methods such as Euler or Tustin. These transformations are shown to be specific cases of the Generalized Bilinear Transformation (GBT), characterized by a design parameter, $α$, whose physical interpretation and optimal selection remain inadequately explored. In this paper, we propose an alternative derivation of the GBT derived by employing a new hexagonal shape to approximate the enclosed area of the error function, and we define the parameter $α$ as a shape factor. We reveal, for the first time, the physical meaning of $α$ as the backward rectangular ratio of the proposed hexagonal shape. Through domain mapping, the stable range of is rigorously established to be [0.5, 1]. Depending on the operating frequency and the chosen $α$, we observe two distinct distortion modes, i.e., the magnitude and phase distortion. We further develop an optimal design method for $α$ by minimizing a normalized magnitude or phase error objective function. The effectiveness of the proposed method is validated through the design and testing of a low-pass filter (LPF), demonstrating strong agreement between theoretical predictions and experimental results.