13.1. Convolution Theorem

Try these first

1. 1.

   What is the definition of convolution?

   Hint

   Look for the integral definition.

2. 2.

   State one property of convolution.

   Hint

   Think about the order of functions.

3. 3.

   What is the formula for convolution of two functions?

   Hint

   Review the definition provided in this section.

4. 4.

   Is the Convolution theorem commutative?

   • True
   • False

   Hint

   Think about as addition works.

5. 5.

   Given $f(t) = e^{-3t}$ and $g(t) = t^2$, find $(f * g)(t)$.

   Hint

   Consider using substitution for \\( t - \\tau \\).

6. 6.

   Calculate the Laplace Transform of the convolution of two functions, where drivers are $f(t) = u(t)$ and $g(t) = u(t - 3)$ and verify the Convolution Theorem.

   Hint

   Remember the properties of the Heaviside unit step function.