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


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.


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.

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