Backward Difference Conformal Map

We saw that the backwards difference substitution can be seen as a conformal map taking the $s$ plane to the $z$ plane:

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

Look at the image of the $j\omega$ axis under this mapping:

The continuous-time frequency axis, $s=j\omega$ , is not mapped to the discrete-time frequency axis (unit circle):

• dc ($s=0$ ) mapped to dc ($z=1$ )
• infinite frequency mapped to ($z=0$ )

This means artificial damping will be introduced for high-frequency system resonances