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

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

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 .

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Let `omega_(c)` be the angular frequency of carrier waves & `omega_(m)` be the angular frequency of signal waves .
Let the signal received at the receiving station be `e = E_(1)* cos(omega_(c) + omega_m)t`
Let the instantaneous voltage of carrier wave `e_(c) = E_(0) cos omega_(c) t` is available at receiving station .
Multiplying these two signals , we get
`e xx e_(c) = E_(1) E_(c) cos omega_(c) t . cos (omega_(c) + omega_(m))t`
`E = (E_(1) E_(c))/(2) * 2 . cos omega_(c) t . cos (omega_(c) + omega_(m)) t " " ("Let" e xx e_(c) = E)`
`=(E_(1)E_c)/(2) [ cos (omega_(c) + omega_(c) + omega_(m)) t + cos (omega_(c) + omega_(m)- omega_(c))t]`
`because 2[ cos A cos B = cos (A + B) + cos (A - B)]`
`(E_(1) E_(c))/(2) =[cos (2omega_(c) + omega_(m))t + cos omega_(m) t]`
Now , at the receiving end as the signal passes through filter , it will pass the high frequency `(2omega_(c) + omega_(m))` but obstract the frequency `omega_(m)` . so we can record the modulating signal `(E_(1) E_(c))/(2) ""cos omega_(m) t` which is a signal of angular frequency `omega_(m)`.