Question

The Fourier transform of the signal e^-4|t| is ____________

(a) \(\frac{8}{16+ω^2}\)

(b) \(\frac{-8}{16+ω^2}\)

(c) \(\frac{4}{16+ω^2}\)

(d) \(\frac{-4}{16+ω^2}\)

I have been asked this question during an interview.

I'd like to ask this question from Exponential Fourier Series and Fourier Transforms in division Fourier Series of Signals and Systems

Answer 1

Correct choice is (a) \(\frac{8}{16+ω^2}\)

Explanation: X (jω) = \(\int_{-∞}^∞ e^{-4|t|} e^{-jωt} \,dt\)

Now, e^-4|t| = e^-4t, when t>0 and

 e^4t, when t<0

∴X (jω) = \(\int_{-∞}^0 e^{4t} e^{-jωt} \,dt + \int_0^∞ e^{-4t} e^{-jωt} \,dt\)

= \(\frac{8}{16+ω^2}\).