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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.

Answer

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Solution

Assume, \(\omega_{c}\) be the carrier wave frequency
Signal wave frequency=\(\omega_{s}\)
Signal received, \(V = V _{1} \cos \left(\omega_{c}+\omega_{s}\right) t\)
Instantaneous voltage of the carrier wave, \(V_{in}=V_{c} \cos \omega_{c} t\)
\(V \cdot V _{ in }= V _{1} \cos \left(\omega_{c}+\omega_{s}\right) t \cdot\left( V _{ c } \cos \omega_{c} t \right)\)
\(= V _{1} V _{ c }\left[\cos \left(\omega_{c}+\omega_{s}\right) t \cdot \cos \omega_{c} t \right]\)
\(=\frac{V_{1} V_{c}}{2}\left[\cos \left(\omega_{c}+\omega_{s}\right) t+\omega_{c} t+\cos \left(\omega_{c}+\omega_{s}\right) t-\omega_{c} t\right]\)
Only the high frequency signals are allowed to pass through the low pass filter. The low frequency signal \(\omega_{s}\) is obstructed by it.
Therefore, at the receiving station, we can record the modulating signal,
\(\frac{V_{1} V_{c}}{2}\left[\cos \left(2 \omega_{c}+\omega_{s}\right) t+\cos \omega_{s} t\right]\) which is the signal frequency.
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