Due to economic reasons, only the upper sideband of an AM wave istransmitted, but at the receiving station, there is a facility forgenerating the carrier. Show that if a device is available which canmultiply two signals, then it is possible to recover the modulatingsignal at the receiver station.
Let ω c be the frequency of the carrier wave and ω s be the frequency of signal wave.
Let V be the signal received at the receiving station
Thus,
V = V 1 cos ( ω c + ω s ) t
Where, V 1 is the amplitude and t is the time period of the system.
Let V C be the voltage of the carrier system.
Then, instantaneous voltage of the carrier wave is,
V in = V c cos ω c t
Hence,
VV in = V 1 cos ( ω c + ω s ) t ( V c cos ω c t ) = V 1 V c 2 [ cos ( ω c + ω s + ω c ) t + cos ( ω c + ω s − ω c ) t ] = V 1 V c 2 [ cos ( 2 ω c + ω s ) t+cos ( ω s ) t ]
The low-pass filter allows only high frequency signal to pass through it. It obstructs the low frequency signal ω s .
Hence, at receiving station, signal frequency will be,
f r = V 1 V c 2 cos ω s t
Let
Let
Thus,
Where,
Let
Then, instantaneous voltage of the carrier wave is,
Hence,
The low-pass filter allows only high frequency signal to pass through it. It obstructs the low frequency signal
Hence, at receiving station, signal frequency will be,
