13.1.5.2. Associative

Try these first

1. 1.

   What is the definition of convolution?

   Hint

   Think about how two functions are combined in the integral.

2. 2.

   State one property of convolution.

   Hint

   Recall the properties that allow rearranging or regrouping functions.

3. 3.

   What does the Convolution Theorem state?

   • The product of two functions is always zero.
   • The inverse Laplace transform of the product of two functions is their convolution.
   • Convolution operates on discrete functions only.

   Hint

   Think about how transformations relate in the Laplace domain.

4. 4.

   True or False: Convolution is always commutative.

   • True
   • False

   Hint

   Recall properties discussed earlier in class.

5. 5.

   Given $f(t) = e^{-t}$ and $g(t) = u(t)$, derive $(f * g)(t)$ using the definition of convolution.

   Hint

   Substituting u(t-\\tau) might simplify your calculation.

6. 6.

   Prove that convolution is commutative, meaning show $(f * g)(t) = (g * f)(t)$.

   Hint

   Start from the definitions and apply the properties of the integral.