Question

Find the Fourier transform of the given signal \(x(t) = e^{-t}u(t - 2)\)

• A. \(\dfrac{e^{-2j\omega}}{1 + j\omega}\)
• B. \(\dfrac{e^{2j\omega}}{1 + 2j\omega}\)
• C. \(\dfrac{\omega^2}{1 + j\omega}\)
• D. \(\dfrac{-\omega^2}{1 + 2j\omega}\)

Hint:

For \(x(t - a)\), multiply the Fourier transform by \(e^{-j\omega a}\). Shift in time → exponential factor in frequency.

Answer 1

The Correct Option is A

Step 1: Time-shifting property of Fourier Transform: 
Let \(f(t) = e^{-t}u(t)\), whose Fourier transform is: \[ F(\omega) = \dfrac{1}{1 + j\omega} \] Now for \(x(t) = f(t - 2) = e^{-(t - 2)}u(t - 2)\), the Fourier transform is: \[ X(\omega) = e^{-j\omega \cdot 2} \cdot \dfrac{1}{1 + j\omega} = \dfrac{e^{-2j\omega}}{1 + j\omega} \]