11.3. Properties of the Fourier Transform

Try these first

1. 1.

   What does the linearity property of the Fourier Transform imply?

   Hint

   Consider if simpler functions can be combined.

2. 2.

   State the Time Shifting property of the Fourier Transform.

   Hint

   Think about what happens to a signal when it's delayed.

3. 3.

   What is the linearity property of the Fourier Transform?

   • Only works for sinusoidal functions.
   • Allows superposition of signals.
   • Dependent on time domain only.

   Hint

   Think about how different functions add together.

4. 4.

   True or False: Shifting a function in time results in a corresponding phase shift in frequency.

   • True
   • False

   Hint

   Visualize how a delayed signal impacts its frequency representation.

5. 5.

   Show how the convolution of two square pulses affects their Fourier Transforms by using the Convolution Theorem.

   Hint

   Integrate carefully and remember the properties of convolutions.

6. 6.

   Given the signal f(t) = e^{-at}u(t) (where u(t) is the unit step), compute its Fourier transform and interpret the result concerning energy.

   Hint

   Focus on rearranging into standard form for easier integration.