Problem 3 [28 points, 4 points each] Consider the continuous-time linear time-invariant

system with transfer-function:

Answer the following questions:

a. What is the differential equation associated with the above transfer-function?

b. Calculate the poles and zeros of G(s).

c. Is G(s) asymptotically stable?

d. Use the bilinear transformation (aka Tustin transformation)

S =

2 z 1

Tsz + 1

to calculate the corresponding discretized transfer-function Ga(z).

e. Calculate the poles and zeros of Ga(z).

f. Is Ga(z) asymptotically stable? (A discrete-time system is asympotically stable if the poles

satisfy |pi|< 1.)

g. Use a computer program or calculator to sketch the magnitude and the phase of the frequency

response G(jw). Now sketch the magnitude and phase of Gd(es) as a function of w when

Ts = {0.01, 0.1, 1}s. Compare all the obtained responses. What is the role of Ts?

Fig: 1