[SOLVED] ECE5610 Homework 2

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.

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EECE 5610 : Homework #2

Problem 2

Problem 2

A function e(t) is sampled, and the resultant sequence has the z-transform E(z) = z − b

z3 −cz2 +d

Find the z-transform of exp(akT)e(kT).
Hint: Solve this problem using E(z) and the properties of the z-transform.

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Problem 3

For the number sequence {e(k)},

EECE 5610 : Homework #2

Problem 3

E(z) =

z , (z − 1)2

(a) Apply the final-value theorem to E(z). (b) Find the z-transform of e(k) = k(−1)k. (c) Explain how parts(a) and (b) are related?

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