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.

Answer 1

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

Signal received at the receiving station, V = V₁ cos (ω_c + ω_s)t

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

`:.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 ω_s. Thus, at the receiving station, one can record the modulating signal `(V_1V_c)/2 cos omega_st` which is the signal frequency.