13.11. Example 3: Piecewise Convolution

Try these first

1. 1.

   Define a piecewise function using f(t) = {1, 0 ≤ t ≤ 2}, g(t) = {t, t ≤ 2}. What is the convolution for t=1?

   Hint

   Check the function definitions for limits.

2. 2.

   What does g(t) represent in our example?

   Hint

   It increases linearly.

3. 3.

   What is the convolution of two functions used for?

   • To modify input
   • To produce an output combining two functions
   • To find echoes

   Hint

   Remember the principle of convolution.

4. 4.

   For t ≤ 1, what is (f ∗ g)(t)?

   Hint

   Evaluate the integral properly!

5. 5.

   Given f(t) = {2, 0 ≤ t ≤ 3} and g(t) = {t, t ≤ 3}, compute (f ∗ g)(t) for t = 2.

   Hint

   Make sure to correctly align the limits according to the piecewise definitions.

6. 6.

   Design a piecewise function where f(t) = sin(t) for 0 ≤ t ≤ π; how does this affect convolution with g(t)?

   Hint

   Visualize the changes in the sine function.