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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2511.03403 |
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| _version_ | 1866916056817205248 |
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| author | Chen, Shen Li, Yanlong Cui, Jiamin Yao, Wei Wang, Jisong Tian, Yixin Liu, Chaohou Yang, Yang Ying, Jiaxi Liu, Zeng Liu, Jinjun |
| author_facet | Chen, Shen Li, Yanlong Cui, Jiamin Yao, Wei Wang, Jisong Tian, Yixin Liu, Chaohou Yang, Yang Ying, Jiaxi Liu, Zeng Liu, Jinjun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimized Design of the Generalized Bilinear Transformation for Discretizing Analog Systems Chen, Shen Li, Yanlong Cui, Jiamin Yao, Wei Wang, Jisong Tian, Yixin Liu, Chaohou Yang, Yang Ying, Jiaxi Liu, Zeng Liu, Jinjun Systems and Control 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. |
| title | Optimized Design of the Generalized Bilinear Transformation for Discretizing Analog Systems |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2511.03403 |