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Why do we need carrier waves of very high frequency in the modulation of signals ?

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Why do we need carrier waves of very high frequency in the modulation of signals ? A carrier wave of peak voltage 20V is used to transmit a message signal. What should be the peak voltage of the modulating signal, in order to have a modulation index of 80%?

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High frequency waves require antenna of reasonable length and can travel long distances without any appreciable power loss; so we need high frequency carrier waves. 

If Em is the peak value of modulating signal and EC that of carrier wave.

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