The bilinear transform, given by Eq. (8.9), is actually a one-to-one mapping relationship between the s-plane and the z-plane. (a) Show that the bilinear transform maps the jω-axis in the s-plane to...

The bilinear transform, given by Eq. (8.9), is actually a one-to-one mapping relationship between the s-plane and the z-plane.

(a) Show that the bilinear transform maps the jω-axis in the s-plane to the unit circle in the z-plane.

(b) Show that the bilinear transform maps the LHP and the RHP of the s-plane map to the interior and the exterior, respectively, of the unit circle in the z-plane.

(c) Show that the bilinear transform maps a stable continuous-time system H_c(s) to a stable discrete-time system H(z).