11.7.2. Fourier Transform of Exponential Decay

Try these first

1. 1.

   What is the Fourier Transform of f(t) = e^{-2t}u(t)?

   Hint

   Use the transformation formula and evaluate the integral.

2. 2.

   Define the unit step function.

   Hint

   Think about the switching behavior of the function.

3. 3.

   What is the general form of the Fourier Transform?

   • ∫ f(t)e^{-iωt} dt
   • ∫ e^{at} dt
   • f(t) + g(t)

   Hint

   Recall the formula we discussed during the lecture.

4. 4.

   True or False: The Fourier Transform of e^{-at}u(t) results in a complex function.

   • True
   • False

   Hint

   Think about how imaginary units appear in the transformation.

5. 5.

   Given the function f(t) = e^{-2t}u(t), derive the Fourier Transform and explain how it can be used in practical applications such as signal processing.

   Hint

   Use the steps of integration and pay attention to how limits affect your result.

6. 6.

   Explore the implications of changing 'a' in the function f(t) = e^{-at}u(t) on the resultant Fourier Transform. Discuss at least two scenarios.

   Hint

   Consider values of 'a' such as 1, 2, and how it affects the transform's output.