13. Convolution Theorem

Try these first

1. 1.

   Define convolution in your own words.

   Hint

   Think about how one function influences another.

2. 2.

   What does the Dirac delta function represent in convolution?

   Hint

   Remember what an identity element does in other mathematical operations.

3. 3.

   What is the result of convolving two functions in the time domain?

   • A new function
   • Their sum
   • Their product

   Hint

   Think beyond simple arithmetic; consider the integral's role.

4. 4.

   True or False: The convolution of two functions is commutative.

   • True
   • False

   Hint

   Look at how classical operations like addition behave.

5. 5.

   Given f(t) = e^(-2t) and g(t) = cos(t), compute the convolution (f ∗ g)(t).

   Hint

   Recall the relationship between the convolution and how it simplifies through Laplace.

6. 6.

   In a structural engineering context, explain how the convolution theorem can be utilized to analyze seismic responses of buildings.

   Hint

   Consider the effects of changing load inputs through time.