13.1.2. Definition of Convolution

Try these first

1. 1.

   Define convolution in your own words.

   Hint

   Think about how we can mix two different functions.

2. 2.

   What does the notation $(f \ast g)(t)$ represent?

   Hint

   Look at how we express the product of functions in convolution.

3. 3.

   What is the definition of convolution?

   Hint

   Think about integrating the product of two time-based functions.

4. 4.

   The Commutative property states that $(f \ast g)(t)$ equals which of the following?

   • (g \\ast f)(t)
   • (f \\ast g)(t)
   • (f + g)(t)

   Hint

   Consider if changing the order matters during addition.

5. 5.

   Using the functions $f(t) = sin(t)$ and $g(t) = e^{-t}$, compute the convolution $(f \ast g)(t)$. Show all steps.

   Hint

   Use the identity of sine and consider integration by parts.

6. 6.

   Illustrate the behavior of convolution through a graphical representation of $(f \ast g)(t)$ for $f(t) = t$ and $g(t) = u(t)$ over [0, 5].

   Hint

   Draw graphs and look at how the areas combine. Use numerical integration if needed.