9.12. Parseval’s Theorem for Fourier Integrals

Try these first

1. 1.

   Explain Parseval's theorem in your own words.

   Hint

   Think about the relationship between time and frequency.

2. 2.

   What is meant by an absolutely integrable function?

   Hint

   Focus on the meaning of 'integrable' and how it applies.

3. 3.

   What does Parseval's theorem relate?

   • Time domain and frequency domain
   • Physical mass and energy
   • Two different structures

   Hint

   Think of how signals are analyzed.

4. 4.

   True or False: Parseval's theorem can only be applied to periodic functions.

   • True
   • False

   Hint

   Reflect on the types of functions discussed.

5. 5.

   Calculate the Energy for the Function f(x) = cos(x) over the interval (-∞, ∞) and verify Parseval's theorem.

   Hint

   Think about how Parseval's formula applies and compute carefully.

6. 6.

   Explain the implications of Parseval's theorem in structural monitoring systems.

   Hint

   Consider ways in which engineers extract signal data for analysis.