Question

Due to economic reasons, only the upper side band of an AM wave is transmitted, but at the receiving station, there is a facility for generating the carrier. Show that if a device is available which can multiply two signals, then it is possible to recover the modulating signal at the receiver station.

Answer 1

Upper side band:

$y_u\left(t\right)=\frac{AM}{2}sin\left(\right(w_c+w_m\left)t\right)$

$c\left(t\right)=Asin\left(w_{c}t\right)$

$r\left(t\right)=y_t{c}_t=\frac{A^{2}M}{2}sin\left(w_{c}t\right)sin\left(\right(w_c+w_m\left)\right)$

$=\frac{A^{2}M}{2}\left[cos\right(w_{m}t)-cos[(w_m+2w_c)t\left]\right]$

By passing $r\left(t\right)$ through a low-pass filter, the higher frequency is removed.

$⟹r^{\prime }\left(t\right)=\frac{A^{\prime }{A}^{2}M}{4}cos\left(w_{m}t\right)$

Hence, it is possible to recover the modulating signal at the receiver station from the upper side band of transmitted AM wave from the carrier signal.