Nyquist rate relative to sampling

The Nyquist rate is the minimum sampling rate required to avoid aliasing, equal to twice the highest frequency contained within the signal.

where is the highest frequency at which the signal can have nonzero energy.
To avoid aliasing, the sampling rate must exceed the Nyquist rate:

A signal whose positive-frequency range of significant energy is (0, B), as depicted above, is called baseband or lowpass. But when the frequency range is (A, B), for some A > B − A, it is called bandpass. In that case, aliasing is not necessarily detrimental, and sampling below the Nyquist rate, called bandpass sampling, is sometimes done. With careful design, a rate as low as 2(B − A) may be achievable, and it is equivalent to mixing (heterodyne) the signal into the frequency range (0, B − A), whose Nyquist rate is 2(B − A). An even lower Nyquist rate can be achieved for a bandpass signal, such as amplitude modulation, whose energy distribution in (A, B) is symmetrical. In that case, the homodyne mixer translates the signal frequencies by (A + B)/2, with a synchronized phase, which moves the highest component to (B − A)/2 and the Nyquist rate to just (B − A).

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