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In A3E transmission, I wonder how a single antenna is used to transmit a carrier wave, upper sideband and lower sideband altogether at the same time?

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The antenna does not do Fourier analysis and has no idea that there is a carrier signal and an upper sideband and lower sideband. Whatever the signal applied to the antenna via electrical wiring or coax cable or transmission line, the flow of current in the antenna causes an electromagnetic wave (coupled time-varying electrical and magnetic fields) to be created and propagated out into the medium. Antennas do have frequency responses and input signals at different frequencies are not radiated equally and so for double-sided AM the antenna is designed such that the frequency response is essentially constant over the bandwidth of the AM signal. In AM, the carrier frequency is usually very much larger than the bandwidth of the modulating signal and so the antenna needs to have essentially flat frequency response over only a relatively small band of frequencies, such as a $10$ kHz band centered at $25$ MHz.

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