Question

The Fourier transform of the signal e^-|t| in [-2,2] is _____________

(a) \(\frac{2 + 2e^{-2} cos⁡2ω + 2ωe^{-2} \,sin⁡2ω}{1-ω^2}\)

(b) \(\frac{2 – 2e^{-2} cos⁡2ω – 2ωe^{-2} \,sin⁡2ω}{1+ω^2}\)

(c) \(\frac{2 – 2e^{-2} cos⁡2ω + 2ωe^{-2} \,sin⁡2ω}{1+ω^2}\)

(d) \(\frac{2 + 2e^{-2} cos⁡2ω + 2ωe^{-2} \,sin⁡2ω}{1-ω^2}\)

I have been asked this question in my homework.

This key question is from Common Fourier Transforms topic in section Fourier Transform of Signals and Systems

Answer 1

Correct option is (c) \(\frac{2 – 2e^{-2} cos⁡2ω + 2ωe^{-2} \,sin⁡2ω}{1+ω^2}\)

Easiest explanation: X (jω) = \(\int_{-2}^0 e^t e^{-jωt} \,dt + \int_0^2 e^{-t} e^{-jωt} \,dt\)

= \(\frac{1-e^{-(1-jω)^2}}{1-jω} + \frac{1-e^{-(1+jω)^2}}{1+jω}\)

= \(\frac{2 – 2e^{-2} cos⁡2ω + 2ωe^{-2} \,sin⁡2ω}{1+ω^2}\).