Problem 1
Find the z-transforms of the number sequences generated by sampling the following time functions every T seconds, beginning at t = 0. Express these transforms in closed form.
(a) e(t) = exp(−at)
(b) e(t) = exp(−t + T)u(t − T)
(c) e(t) = exp(−t + 5T)u(t − 5T)
Hint: Note u(t) is the unit step function and exp(x) = ex is the exponential function. First, you need to obtain associated discrete functions (e[k] = e(Tk)), and then you need to use the properties of the z-transform that we discussed in the class.
Problem 2
Problem 2
A function e(t) is sampled, and the resultant sequence has the z-transform
Find the z-transform of exp(akT)e(kT).
Hint: Solve this problem using E(z) and the properties of the z-transform.
Problem 3
Problem 3
For the number sequence {e(k)},
,
(a) Apply the final-value theorem to (b) Find the z-transform of e(k) = k(−1)k.
(c) Explain how parts(a) and (b) are related?
