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.

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.

Philoid · Accepted Answer

Let ω_c and ω_s be the respective frequencies of the carrier and signal waves.

Signal received at the receiving station, V = V₁cos(ω_c + ω_s)t

Instantaneous Voltage of the carrier wave, V_in = V_c cos ω_ct

V.V_in = V_1.V_c{cos(ω_c + ω_s)t. cos ω_ct}

= ( V_1.V_c)/2 {2cos(ω_c + ω_s)t. cosω_ct}

= ( V_1.V_c)/2 [cos{(ω_c + ω_s)t + ω_ct} + cos {{(ω_c + ω_s)t - ω_ct }

= (V_1.V_c)/2 [cos{(2ω_c + ω_s)t} + cos ω_st ]

As the receiving centre allows only high frequency signals to pass through it. Obstructs the low frequency signal ω_s Thus, at the receiving station, one can record the modulating signal (V₁V_c/2).cosω_st , which is the signal frequency.