The s- to z-plane mapping for both the finite difference and bilinear transform is shown in Fig. 2.4. Note that the bilinear transform maps the axis exactly onto the unit circle.
Figure 12:
Finite Difference and Bilinear Transform to z-plane frequency mapping.
Both methods preserve order, stability, and are free from aliasing. Both methods provide an ideal frequency mapping at zero frequency but compressively warp higher frequencies.
Only the finite difference method introduces artificial damping at higher frequencies. Because of this, it is even possible that unstable s-plane poles could be mapped to stable z-plane poles.