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

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.