More than Meets the Eye

The questions below are due on Thursday March 13, 2025; 02:00:00 PM.

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Part 1

Consider the function x_1 described by the following expression and plot:

x_1(t) = 3te^{-2|t|}

Determine the Fourier transform X_1(\omega) of this signal. Express your answer in closed form.
X_1(\omega) =~

Part 2

Now consider a signal x_2 whose Fourier transform X_2 is given by the following expression:

X_2(\omega) = 3\omega e^{-2|\omega|}

Determine a closed-form expression for x_2(t).
x_2(t) =~

Part 3 (not graded)

Compare your answers to parts 1 and 2. How are these answers related?