Question

Which of the following is the inverse Fourier transform of $X\left(\omega \right)=\frac{sin\left(\frac{\omega -\pi }{2}\right)}{\omega -\pi }$

• A. $\frac{1}{2}{e}^{j\pi t}rect\left(t\right)$
• B. $2rect\left(\frac{t-\pi }{4}\right)$
• C. $\frac{1}{2}{e}^{j\pi t}rect\left(\frac{t}{\pi }\right)$
• D. $2rect\left(\frac{t-4}{\pi }\right)$

Answer 1

The correct option is A $\frac{1}{2}{e}^{j\pi t}rect\left(t\right)$

$\frac{sin\frac{\omega }{2}}{\omega /2}\stackrel{I.F.T}{⟷}rect\left(t\right)$

$2\frac{sin\frac{\omega }{2}}{\omega }\stackrel{I.F.T}{⟷}rect\left(t\right)$

$\frac{sin\frac{\omega }{2}}{\omega }\stackrel{I.F.T}{⟷}\frac{1}{2}rect\left(t\right)$

$\frac{sin\left(\frac{\omega -\pi }{2}\right)}{\omega -\pi }\stackrel{I.F.T}{⟷}\frac{1}{2}{e}^{j\pi t}rect\left(t\right)$