The exponential x(t) = e^at for t ≥ 0 and zero otherwise is a very common continuous-time signal. Likewise, y(n) = αⁿ for integers n ≥ 0 and zero otherwise is a very common discrete-time signal. Let us see how they are related. Do the following using MATLAB:

(a) Let a = − 0.5, plot x(t)

(b) Let a = − 1, plot the corresponding signal x(t). Does this signal go to zero faster than the exponential for a = − 0.5?

(c) Suppose we sample the signal x(t) using T_s = 1 what would be x(nT_s) and how can it be related to y(n), i.e., what is the value of αthat would make the two equal?

(d) Suppose that a current x(t) = e^−0.5t for t ≥ 0 and zero otherwise is applied to a discharged capacitor of capacitance C = 1 F at t = 0. What would be the voltage in the capacitor at t = 1 sec?

(e) How would you obtain an approximate result to the above problem using a computer? Explain.