Question

Which of the following signal is the inverse Fourier transform of $X\left(\omega \right)=\frac{1}{(a+j\omega {)}^4}$

• A. $x\left(t\right)=e^{-4at}u\left(t\right),a>0$
• B. $x\left(t\right)=e^{-at}u\left(t\right)\ast e^{-at}u\left(t\right),a>0$
• C. $x\left(t\right)=t\cdot e^{-at}u\left(t\right)\ast t{e}^{-at}u\left(t\right),a>0$
• D. $x\left(t\right)=t\cdot e^{-at}u\left(t\right)\ast e^{-at}u\left(t\right),a>0$

Answer 1

The correct option is C $x\left(t\right)=t\cdot e^{-at}u\left(t\right)\ast t{e}^{-at}u\left(t\right),a>0$

$t\cdot e^{-at}u\left(t\right)$ $\stackrel{F.T.}{\to }$ $\frac{1}{(a+j\omega {)}^2}Fora>0$

$t\cdot e^{-at}u\left(t\right)\ast t{e}^{-at}u\left(t\right)$ $\stackrel{F.T.}{\to }$ $\frac{1}{(a+j\omega {)}^4}Fora>0$