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## QuestionPhysicsClass 12

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.

See analysis below
4.6     4.6     ## 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.          