Note on the Bilinear Transform

Recall the mapping $s\to\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$, which maps real frequencies to real frequencies. In fact, the mapping takes the RHP to the outside of the unit circle and the LHP to the inside. Thus:

• Under this particular bilinear transform, a stable minimum phase continuous time system is mapped to a stable minimum phase discrete time system.

• A function PR in the RHP is mapped to a function PR outside the unit circle.

• Thus, in some sense this mapping preserves passivity or energetic properties of the original system.

• This is the fundamental reason why it is used by Fettweis et al. as the basis for filter design and numerical integration.