Question

The Fourier transform of the following signal x(t) is

• A. $j4\frac{\sin \omega \cos \omega }{\omega }$
• B. $j2\left(\frac{1-cos\left(2\omega \right)}{\omega }\right)$
• C. $j4\left(\frac{1-cos\left(2\omega \right)}{\omega }\right)$
• D. $j2\frac{si{n}^2\omega }{\omega }$

Answer 1

The correct option is B $j2\left(\frac{1-cos\left(2\omega \right)}{\omega }\right)$

$x\left(t\right)=rect\left(\frac{t+1}{2}\right)-rect\left(\frac{t-1}{2}\right)$

$rect\left(t\right)⟷\frac{2sin\left(\omega /2\right)}{\omega }$

$rect\left(\frac{t}{2}\right)⟷\frac{4sin\left(2\omega /2\right)}{2\omega }=\frac{2sin\omega }{\omega }$

$rect\left(\frac{t+1}{2}\right)=\frac{2sin\omega }{\omega }{e}^{j\omega }$

$rect\left(\frac{t-1}{2}\right)=\frac{2sin\omega }{\omega }{e}^{-j\omega }$

$\Rightarrow X\left(\omega \right)=\frac{2sin\omega }{\omega }(e^{j\omega }-e^{-j\omega })=\frac{j4\sin \omega \sin \omega }{\omega }=\frac{j2(1-cos(2\omega \left)\right)}{\omega }$