Comprehensive Analysis and Exclusion Hypothesis of $α$-Approximation Method for Discretizing Analog Systems

A popular method for designing digital models is transforming the transfer function of the corresponding analog models from continuous domain (s-domain) into discrete domain (z-domain) using the s-to-z transformation. The alpha-approximation is a generalized form of these transformations. When alpha is set to 0.5, the result is the well-known Tustin transformation or bi-linear transformation. In this paper, we provided a comprehensive analysis of the alpha-approximation method, including mathematical interpretation, stability analysis and distortion analysis. Through mathematical interpretation, we revealed that it can be derived by numerically integrating the error function We defined this as the hexagonal approximation. We demonstrated that the stable range of alpha was [0.5, 1] by doing stability analysis. Through distortion analysis, we found that minimizing amplitude and phase distortion simultaneously seemed impossible by regulating alpha alone. Finally, We proposed an exclusion hypothesis hypothesizing that there is no single parameter alpha to minimize the amplitude distortion and phase distortion simultaneously across all frequency points within the Nyquist frequency range. This paper demonstrates that designing parameter alpha involves balancing amplitude and phase distortion.