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I always wondered why transmission capacity depends on bandwidth. For example, let us say that there is an isolated island. In this island, people decide that all wireless networks use frequencies that are positive integer multiples of one base frequency. Then when we do Fourier transform, we can just use least common multiples of all used frequencies and data in each frequency will exactly be determined except background noises. Or am I wrong here? In this sense, bandwidth seems just to be useless....

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    $\begingroup$ You lost me at "least common multiples". What exactly do you mean? $\endgroup$ – jan Nov 19 '13 at 2:19
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    $\begingroup$ You'll still have finite capacity to this theoretical communication system because each information-carrying signal requires nonzero bandwidth. Specifically, the Shannon capacity theorem specifies the relationship between information rate, signal-to-noise ratio, and bandwidth that is required for reliable communication. You can coordinate frequency allocation all you want, but you'll still be eventually stymied by the Shannon limit. $\endgroup$ – Jason R Nov 19 '13 at 13:28
  • $\begingroup$ A perfect sinusoid at a pre-determined frequency has zero bits of entropy, and can therefore carry no information. I think that your question needs some more details to explain how you expect to convey any information using this system. $\endgroup$ – nispio Nov 20 '13 at 0:28
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It's not that easy to distinguish different frequencies when they are close to each other so you'll need to make sure that there is sufficient spacing between the frequencies. Also, you can't just keep transmitting higher and higher frequencies. At some point there will be a limit of the highest frequency that can go through a wire or whatever transmission mechanism you are using and also the fastest computer that you have to process it. So once you cap the highest allowed frequency your method comes down to dividing up the remaining space among a finite set of frequencies. In other words, you are proposing to allot a uniform bandwidth to each of your channels, which is exactly what you said you were trying to avoid.

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Each frequency carrier will still have to be modulated to carry any information (as in bits per second), and that modulation will take up some finite bandwidth in the spectrum. Imperfect clock synchronization and stability will also cause any signal to take up more bandwidth than zero.

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