11.8. Example Problems

Try these first

1. 1.

   What is the Fourier Transform of f(t) = e^{-3t}u(t)?

   Hint

   Use the similar approach we applied to e^{-2t}.

2. 2.

   Describe the function represented by u(t).

   Hint

   Think about its behavior for negative and positive values of t.

3. 3.

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

   Hint

   Think about the general form of the Fourier Transform for exponential functions.

4. 4.

   Using the time-scaling property, what is F{rect(3t)}?

   • (1/3)sinc(ω/3)
   • (1/3)sinc(ω/6)
   • (1/3)sinc(ω/9)

   Hint

   Recall the effect of a time compression or expansion on the sinc function.

5. 5.

   Explore the Fourier Transform of a combination function f(t) = e^{-t}u(t) + sin(t). How would you approach this?

   Hint

   Break down the function into simpler parts before combining.

6. 6.

   Using f(t) = rect(t-1), determine how the Fourier Transform shifts in frequency domain.

   Hint

   Consider the time-shifting property for insight into this shift.