Optimized Design of the Generalized Bilinear Transformation for Discretizing Analog Systems
Shen Chen, Yanlong Li, Jiamin Cui, Wei Yao, Jisong Wang, Yixin Tian, Chaohou Liu, Yang Yang, Jiaxi Ying, Zeng Liu, Jinjun Liu
TL;DR
This work clarifies the physical meaning of the Generalized Bilinear Transformation (GBT) parameter $\alpha$ by introducing a hexagonal-area error approximation and proving a stable range of $[0.5,1]$. It presents a novel hexagonal derivation of the GBT, relates it to existing discretization methods, and defines $\alpha$ as the backward-rectangular area ratio, unifying Euler and Tustin within a single framework. An explicit five-step optimal design method for $\alpha$ is proposed and applied to discretize a low-pass filter (LPF), with experimental results validating reduced magnitude and phase distortion and illustrating the impact of sampling frequency on distortion. The findings offer a practical pathway to tailor discretization accuracy for digital control systems, particularly where phase fidelity or magnitude response is critical, and highlight the benefit of higher sampling rates to mitigate discretization-induced errors.
Abstract
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.