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Home » 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.
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.
CBSE | Class 12 | Communication Systems | Guides | NCERT | Physics
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.

i

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