Topic: Signals and Systems
Let $x(t)=rect(t/2)$, where $rect(t/2)$ is the rectangular pulse of width 2 centered at $t=0$, i.e., $x(t)=1$ for $|t|\le1$ and 0 otherwise. What is the Fourier Transform $X(w)$ of $x(t)$?
- A. $X(w)=2~sinc(w)$
- B. $X(w)=sinc(w)$
- C. $X(w)=2~sinc(w/2)$
- D. $X(w)=sinc(w/2)$
Answer: A. $X(w)=2~sinc(w)$
Explanation: The Fourier transform of a rectangular pulse of width $T$ centered at the origin is $X(\omega) = \int_{-T/2}^{T/2} 1 \cdot e^{-j\omega t} dt = T \frac{\sin(\omega T / 2)}{\omega T / 2}$.
Here, the width $T = 2$. Substituting $T = 2$ gives $X(\omega) = 2 \frac{\sin(\omega)}{\omega}$.
Using the standard unnormalized sinc function definition $sinc(x) = \frac{\sin(x)}{x}$, this is exactly $2~sinc(\omega)$.