An Alternative Derivation and Optimal Design Method of the Generalized Bilinear Transformation for Discretizing Analog Systems

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, $\alpha$, 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 $\alpha$ as a shape factor. We reveal, for the first time, the physical meaning of $\alpha$ 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 $\alpha$, we observe two distinct distortion modes, i.e., the magnitude and phase distortion. We further develop an optimal design method for $\alpha$ 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.