Due to economic reasons, only the upper sideband of an AM wave is transmitted,
Question:

Due to economic reasons, only the upper sideband 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.

Solution:

Let ωc and ωs be the respective frequencies of the carrier and signal waves.

Signal received at the receiving station, V = V1 cos (ωc + ωs)t

Instantaneous voltage of the carrier wave, Vin = Vc cos ωct

$\therefore V V_{\mathrm{in}}=V_{1} \cos \left(\omega_{\mathrm{c}}+\omega_{\mathrm{s}}\right) t .\left(V_{\mathrm{c}} \cos \omega_{\mathrm{c}} t\right)$

$=V_{1} V_{\mathrm{c}}\left[\cos \left(\omega_{\mathrm{c}}+\omega_{\mathrm{s}}\right) t \cdot \cos \omega_{\mathrm{c}} t\right]$

$=\frac{V_{1} V_{\mathrm{c}}}{2}\left[2 \cos \left(\omega_{\mathrm{c}}+\omega_{\mathrm{s}}\right) t \cdot \cos \omega_{\mathrm{c}} t\right]$

$=\frac{V_{1} V_{\mathrm{c}}}{2}\left[\cos \left\{\left(\omega_{\mathrm{c}}+\omega_{\mathrm{s}}\right) t+\omega_{\mathrm{c}} t\right\}+\cos \left\{\left(\omega_{\mathrm{c}}+\omega_{\mathrm{s}}\right) t-\omega_{\mathrm{c}} t\right\}\right]$

$=\frac{V_{1} V_{\mathrm{c}}}{2}\left[\cos \left\{\left(2 \omega_{\mathrm{c}}+\omega_{\mathrm{s}}\right) t+\cos \omega_{\mathrm{s}} t\right\}\right]$

At the receiving station, the low-pass filter allows only high frequency signals to pass through it. It obstructs the low frequency signal $\omega_{\mathrm{s}}$. Thus, at the receiving station, one can record the modulating signal $\frac{V_{1} V_{\mathrm{c}}}{2} \cos \omega_{\mathrm{s}} t$, which is the signal frequency.