How to find the phase spectrum of a rectangular pulse?
The Fourier transform of a rectangular pulse
$$ x(t) = \begin{cases} 1, & \text{for $|t| \le \tau /2$ } \\ 0, & \text{otherwise} \end{cases} $$
is given by:
$$F[x(t)]=\tau [\frac{\sin \omega(\tau /2)}{\omega (\tau /2)}]$$
In general, the Fourier Transform, $X(\omega)$ is id a complex valued function of $\omega$. Therefore, $X(\omega)$ can be written as:
$$X(\omega)=X_R(\omega)+jX_I(\omega)$$
The magnitude of $X(\omega)$ is given by
$$\vert X(\omega) \vert= \sqrt {(X_R(\omega))^2+(X_I(\omega))^2}$$
The phase of $X(\omega)$ is given by
$$\angle{X(\omega)}=\tan^{-1}\frac{X_I(\omega)}{X_R(\omega}$$
Question:
1)How can we find the phase spectrum of a rectangular pulse as there doesn't appear to be any imaginary part in $X(\omega)$?
2)Why is the phase spectrum changing the way it is when we are time shifting the rectangular pulse from the origin?
3) Any other insight regarding the magnitude and phase spectrum is welcome.