1. In an impulse invariance design, show that if Ha(s) is a transfer function of a stable system, the corresponding H[z] is also a transfer function of a stable system.
2. First-order backward differences provide the transformation rule s = (1- z-1)/T
a. Show that this transformation maps the ω axis in the s plane to a circle of radius 1/2 centered at (1/2, 0) in the z plane.
b. Show that this transformation maps the left-half s plane to the interior of the unit circle in the z plane, which ensures that stability is preserved.