13. Laplace Transforms & Applications

Try these first

1. 1.

   Define convolution in your own words.

   Hint

   Think about how functions can overlap in a time period.

2. 2.

   What does the convolution theorem state?

   Hint

   Recall the relationship between Laplace transforms and convolution.

3. 3.

   What is the convolution of f(t) = t and g(t) = exp(-t)?

   Hint

   Look up integral properties to help solve.

4. 4.

   True or False: The convolution operation is always associative.

   Hint

   Recall how often you can regroup terms in math.

5. 5.

   For functions f(t) = e^(-2t) and g(t) = t^2, find the convolution and describe its meaning in the context of signal processing.

   Hint

   Set up the integral and pay attention to limits of integration based on the definitions of f and g.

6. 6.

   Demonstrate how to utilize the theorem to analyze a system's response to input over time, using a case where f(t) is a unit step function and g(t) a decaying exponential.

   Hint

   Establish the integral and substitute appropriately for the functions defined.