Question

Determine the Laplace transform and the associated region of convergence and pole-zero plot for each of the following functions of time: x(t) = e⁻²ᵗ u(t) + e⁻³ᵗ u(t)

Answer 1

The Laplace transform of the given function x(t) = e⁻²ᵗ u(t) + e⁻³ᵗ u(t) is 1/(s + 2) + 1/(s + 3). The region of convergence is Re(s) > -2 and Re(s) > -3. The pole-zero plot has two poles at s = -2 and s = -3.

Step-by-step explanation:

To find the Laplace transform of the given function x(t) = e⁻²ᵗ u(t) + e⁻³ᵗ u(t), we can use the linearity property of the Laplace transform. The Laplace transform of e⁻²ᵗ u(t) is 1/(s + 2) and the Laplace transform of e⁻³ᵗ u(t) is 1/(s + 3). Therefore, the Laplace transform of x(t) is 1/(s + 2) + 1/(s + 3). The associated region of convergence is the set of values of s for which the Laplace transform converges. In this case, the region of convergence is Re(s) > -2 and Re(s) > -3, which means the Laplace transform converges for s > -2 and s > -3.

The pole-zero plot is a graphical representation of the poles and zeros of the Laplace transform. In this case, the Laplace transform has two simple poles at s = -2 and s = -3. Therefore, the pole-zero plot consists of two poles at these locations.