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I'm trying to wrap my head around radio carrier frequencies

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I'm trying to wrap my head around radio carrier frequencies and data transmission rates.

For analog signals, they can have a carrier frequency.

From what I've picked up, the carrier signal must be greater than the passband width of the analog signals being sent. And the data transmission rate is proportional to the frequency of the carrier signal. So higher frequency carrier signals = more data transmission = more bandwidth of original signals being transmitted.

Is this correct? Why is the carrier signal frequency proportional to the input bandwidth?

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