Hybrid Euler-Bilinear Mapping

We can easily interpolate between Backward Euler and Bilinear Transform:

$$\zbox{s\to\frac{1+\alpha}{T}\frac{1-z^{-1}}{1+\alpha\,z^{-1}}}$$

• $\alpha=0$ gives Backward Euler (high-frequency modes artificially damped)
• $\alpha=1$ gives Bilinear Transform (high-frequency modes artificially squeezed in frequency)
• Intermediate $\alpha$ allows optimization of another consideration, such as decay time
• Low-frequency response approximately invariant,
  dc maps to dc in every case