Topic: Signals and Systems
Let $x(t)=rect(t-\frac{1}{2})$ for $-\frac{1}{2}\le t\le\frac{1}{2}$ and zero otherwise, then Fourier Transform of $x(t)+x(-t)$ will be given by
- A. $sinc(\frac{\omega}{2})$
- B. $2~sinc(\frac{\omega}{2})$
- C. $2sinc(\frac{\omega}{2})cos(\frac{\omega}{2})$
- D. $2~sinc(\frac{\omega}{2})sin(\frac{\omega}{2})$
Answer: C. $2sinc(\frac{\omega}{2})cos(\frac{\omega}{2})$
Explanation: Note: Based on the official key, the question is marked as having a discrepancy (*). The mathematical derivation yields $Y(\omega) = 2sinc(\frac{\omega}{2\pi})cos(\frac{\omega}{2})$. If the sinc function is defined as $sinc(x) = \frac{\sin(x)}{x}$, the transform involves $2sinc(\frac{\omega}{2})cos(\frac{\omega}{2})$. None of the options perfectly matched standard conventions in the official key, but derived logic points towards form (c).