13.1.9. Summary

Try these first

1. 1.

   What is the definition of convolution?

   Hint

   Think about combining two functions in an integral.

2. 2.

   List one property of convolution.

   Hint

   Consider symmetry in functions.

3. 3.

   What is the equation for convolution of two functions?

   • \\(\\int_0^t f(\\tau) g(t - \\tau) d\\tau\\)
   • \\(f(t) + g(t)\\)
   • \\(f(t) * g(t)\\)

   Hint

   Focus on the integral form.

4. 4.

   The Convolution Theorem applies to which transforms?

   • True
   • False

   Hint

   Consider what the theorem states.

5. 5.

   Determine the inverse Laplace transform of $\frac{s^2}{s^3 + 2s^2 + s}$. Show the steps and utilize the Convolution Theorem.

   Hint

   Identify factors of the denominator to separate the functions.

6. 6.

   Analyze a system with two input signals defined by their Laplace transforms. Use convolution to find the system's overall response function.

   Hint

   Sketch the inputs and leverage properties of convolution.