11. Fourier Transform and Properties

Try these first

1. 1.

   Define the Fourier Transform.

   Hint

   Think about the original on the time scale.

2. 2.

   What is the purpose of the Inverse Fourier Transform?

   Hint

   What do you need to return to the original function?

3. 3.

   What does the Fourier Transform do?

   • Transforms time-domain signals to frequency domain
   • Transforms spatial-domain signals to time domain
   • Both time and spatial transformations

   Hint

   Focus on what domain the signals originate from.

4. 4.

   True or False: The Inverse Fourier Transform allows us to obtain the original signal from its frequency representation.

   • True
   • False

   Hint

   Consider it a way to revert back to the original function.

5. 5.

   Given a continuous function f(t) = cos(t) defined for all real numbers, compute its Fourier Transform and explain the result.

   Hint

   Focus on how trigonometric functions are represented in frequency.

6. 6.

   Using Parseval’s Theorem, prove that the total energy of a signal in time equals its total energy in frequency.

   Hint

   Use the relationship defined in Parseval's Theorem explicitly.