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 and 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 article, 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 input 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 an objective function based on the normalized magnitude or phase error. The effectiveness of the proposed method is validated through the design and testing of a low-pass filter and a quasi-resonant controller, demonstrating strong agreement between theoretical predictions and experimental results.