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    $\begingroup$ The generalized Nyquist Theorem is "You need at least two samples per Hz of bandwidth". If your signal goes from 999000Hz to 1001000 Hz, the required sample rate is only 4kHz even if all frequencies involved are much higher. $\endgroup$ Commented Mar 22, 2012 at 13:12
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    $\begingroup$ I now see that "per Hz of bandwidth" is the key here that I was missing. $\endgroup$ Commented Mar 22, 2012 at 13:34