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प्रश्न
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.
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उत्तर
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
`:.VV_"in" = V_1cos(omega_c +omega_s)t.(V_c cos omega_ct)`
`= V_1V_c[cos(omega_c + omega_s)t.cos omega_ct]`
`= (V_1V_c)/2 [2cos(omega_c + omega_s)t.cos omega_ct]`
`= V_1V_c/2 [cos{(omega_c + omega_s)t + omega_ct} + cos{(omega_c + omega_s)t - omega_ct}]`
`= (V_1V_c)/2 [cos{(2omega_c + omega_s)t + cos omega_s t}]`
At the receiving station, the low-pass filter allows only high frequency signals to pass through it. It obstructs the low frequency signal ωs. Thus, at the receiving station, one can record the modulating signal `(V_1V_c)/2 cos omega_st` which is the signal frequency.
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