Question

A certain function time, f(t) has a Fourier transform,

$F\left(j\omega \right)=\frac{1}{{\omega }^2+1}{e}^{2{\omega }^2/({\omega }^2+1)}$

Then the Fourier transform of

$f(t-2).e^{f\left(t\right)}$ is

• A. $\frac{e^{j2(\omega +1)}}{{\omega }^2+2\omega +2}.e^{\frac{-2({\omega }^2+1+2\omega )}{{\omega }^2+2\omega +2}}$
• B. $\frac{e^{-j2(\omega -1)}}{{\omega }^2-2\omega +2}.e^{\frac{+2({\omega }^2+1-2\omega )}{{\omega }^2-2\omega +2}}$
• C. $\frac{e^{-j2(\omega +1)}}{{\omega }^2+2\omega +2}.e^{\frac{-2({\omega }^2+1+2\omega )}{{\omega }^2+2\omega +2}}$
• D. $\frac{e^{j2(\omega -1)}}{{\omega }^2-2\omega +2}.e^{\frac{+2({\omega }^2+1-2\omega )}{{\omega }^2-2\omega +2}}$

Answer 1

The correct option is B $\frac{e^{-j2(\omega -1)}}{{\omega }^2-2\omega +2}.e^{\frac{+2({\omega }^2+1-2\omega )}{{\omega }^2-2\omega +2}}$

$f\left(t\right)\stackrel{F.T}{⟷}F\left(j\omega \right)$

From time delay property

$f(t-t_0)\stackrel{F.T.}{⟷}{e}^{-j\omega t_0}F\left(j\omega \right)$

$\therefore f(t-2)\stackrel{F.T.}{⟷}\frac{e^{-j2\omega }}{{\omega }^2+1}{e}^{2{\omega }^2/({\omega }^2+1)}$

From frequency shifting property

$f\left(t\right)e^{j{\omega }_0}\stackrel{F.T}{⟷}F\left(j\right(\omega -{\omega }_0\left)\right)$

$\therefore f(t-2)e^{jt}\stackrel{F.T}{⟷}\frac{e^{-j2(\omega -1)}}{(\omega -1{)}^2+1}.e^{\frac{2(\omega -1{)}^2}{(\omega -1{)}^2+1}}$

$f(t-2)e^{jt}\stackrel{F.T.}{⟷}\frac{e^{-j2(\omega -1)}}{{\omega }^2-2\omega +2}.e^{\frac{2({\omega }^2+1-2\omega )}{{\omega }^2-2\omega +2}}$