11.7. Derivation of Fourier Transform of Common Functions

Try these first

1. 1.

   Define the Fourier Transform in your own words.

   Hint

   Think about what transformation helps analyze signals better.

2. 2.

   What does a rectangular pulse look like?

   Hint

   Visualize a wave that only exists for a limited time.

3. 3.

   What is the Fourier Transform of a rectangular pulse?

   • e^{-at}
   • T·sinc(ωT/2π)
   • 1/(a + iω)

   Hint

   Recall the formula we derived in class.

4. 4.

   True or False: The Fourier Transform can help analyze time-dependent signals.

   • True
   • False

   Hint

   Think about its applications in engineering.

5. 5.

   Analyze the impact of changing the width of a rectangular pulse on its Fourier Transform. Discuss what happens to the frequency components.

   Hint

   Consider how time duration relates to frequency content.

6. 6.

   Derive the inverse transform of the exponential decay function and discuss its significance.

   Hint

   Revisit the integration techniques employed earlier.