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 –

Let the carrier wave frequency be represented by \[{{\omega }_{c}}\] and let \[{{\omega }_{s}}\]be the signal wave frequency.

Then the received signal will be given by

\[V={{V}_{1}}cos({{\omega }_{c}}+{{\omega }_{s}})t\]

Also, the instantaneous voltage of the carrier wave is given by –

\[{{V}_{in}}={{V}_{c}}cos{{\omega }_{c}}t\]

Now upon solving, we get –                                     

  $$V.{{V}_{in}}={{V}_{1}}cos({{\omega }_{c}}+{{\omega }_{s}})t.({{V}_{c}}cos{{\omega }_{c}}t)$$

 $$ ={{V}_{1}}{{V}_{c}}[cos({{\omega }_{c}}+{{\omega }_{s}})t.cos{{\omega }_{c}}t]$$

 $$ =\frac{{{V}_{1}}{{V}_{c}}}{2}\left[ cos({{\omega }_{c}}+{{\omega }_{s}})t+{{\omega }_{c}}t+cos({{\omega }_{c}}+{{\omega }_{s}})t-{{\omega }_{c}}t \right] $$

 $$ =\frac{{{V}_{1}}{{V}_{c}}}{2}\left[ cos(2{{\omega }_{c}}+{{\omega }_{s}})t+cos{{\omega }_{s}}t \right] $$

Only the high frequency signals are allowed  to pass through a low pass filter. The low frequency signal  ​ \[{{\omega }_{s}}\] is obstructed by it.

Thus, we record the modulating signal \[2{{V}_{1}}{{V}_{c}}cos{{\omega }_{s}}t\] at the receiving station which is the signal frequency.