Practice Exercises (9.15) - Fourier Integrals - Mathematics (Civil Engineering -1)

Practise this section

9 questions

Flashcard drill
3 cards from this section. Quick, good the night before a test.
Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
Sign up to start the exercise
Quiz
3 questions · about 2 min · pass mark 70%. Hints available.
Sign up to start the quiz

6 sample questions

Question 1.
What is the Fourier integral formula for f(x) = xe^{-x}, x > 0?
Show hintHide hint
Consider applying integration by parts.
Question 2.
What bounds do we use when evaluating the Fourier sine integral of a step function?
Show hintHide hint
Focus on where f(x) is defined.
3Question 3.
What is the main purpose of the Fourier cosine integral?
- To analyze periodic functions
- To represent non-periodic functions
- To calculate Fourier coefficients
Show hintHide hint
Recall the main objective of Fourier transforms.
4Question 4.
True or False: The Fourier sine integral is defined only for odd functions.
- True
- False
Show hintHide hint
Think about the properties of sine functions.
Question 5.
Using the Fourier integral representation, prove the Parseval's theorem for a function defined over an infinite domain.
Show hintHide hint
Start by expressing both sides using the integral definitions.
Question 6.
Consider the function f(x) = e^{-a|x|}. Derive its full Fourier transform and discuss the significance of the results obtained in signal processing.
Show hintHide hint
Use properties of even functions to aid in the derivation.

5 more questions in this section. Enroll free to practise them all.