Question

Given a causal) signal (t) with Laplace transform X(s), find the Laplace trans- form of y(t) = r(t)ent using the definition of the Laplace transform. Determine the ROC of y() using the ROC of a(t).

Answer 1

To find the Laplace transform of y(t) = r(t)e^nt, we multiply the Laplace transforms of r(t) and e^nt together. The ROC of y(t) depends on the ROC of the individual Laplace transforms used.

Step-by-step explanation:

To find the Laplace transform of y(t)=r(t)e^nt, we need to first find the Laplace transform of r(t) and e^nt individually and then multiply them together.

The Laplace transform of r(t) is given by R(s) = 1/s where s is the complex variable of the Laplace transform. The Laplace transform of e^nt is given by E(s) = 1/(s-n).

So, the Laplace transform of y(t) = r(t)e^nt is Y(s) = R(s) * E(s) = 1/s * 1/(s-n).

The ROC (Region of Convergence) of y(t) will depend on the ROC of the individual Laplace transforms used. Since the ROC of r(t) is the entire complex plane except for the origin, and the ROC of e^nt is Re(s) > n, the ROC of y(t) will be Re(s) > n.