Question

The inverse Fourier transform of $X\left(j\omega \right)=\frac{-2j\omega }{9+{\omega }^2}$ is

• A. $2e^{-3|t|}sgn(-t)$
• B. 2$e^{-3|t|}sgn\left(t\right)$
• C. $e^{-3|t|}sgn\left(t\right)$
• D. $e^{3|t|}sgn(-t)$

Answer 1

The correct option is C $e^{-3|t|}sgn\left(t\right)$

$X\left(j\omega \right)=\frac{-2j\omega }{9+j{\omega }^2}=\frac{-2j\omega }{(3+j\omega )(3-j\omega )}$

$=\frac{(3-j\omega )-(3+j\omega )}{(3+j\omega )(3-j\omega )}$

$X\left(j\omega \right)=\frac{1}{3+j\omega }-\frac{1}{3-j\omega }$

Applying inverse fourier transform, we get

$x\left(t\right)=e^{-3t}u\left(t\right)-e^{3t}u(-t)$

$x\left(t\right)=\{\begin{array}{ccc}{e}^{-3t}& for& t>0\\ -e^{3t}& for& t<0\end{array}$

$x\left(t\right)=e^{-3|t|}\{\begin{array}{ccc}1& for& t>0\\ -1& for& t<0\end{array}$

$x\left(t\right)=e^{-3|t|}sgn\left(t\right)$