# Nyquist frequency

The Nyquist frequency, named after the Nyquist-Shannon sampling theorem, is half the sampling frequency for a signal. It is sometimes called the critical frequency. The sampling theorem tells us that aliasing can be avoided iff the Nyquist frequency is at least as large as the bandwidth of the signal being sampled (or the maximum frequency if the signal is a baseband signal).

In principle, a Nyquist frequency equal to the signal bandwidth is sufficient to allow perfect reconstruction of the signal from the samples. However, this reconstruction requires an unrealizable filter that passes some frequencies unchanged while suppressing all others completely. When realizable filters are used, oversampling is necessary to accommodate the practical constraints on anti-aliasing filters. Even with oversampling, the Nyquist frequency is half the sampling frequency.

For example, audio CDs have a sampling frequency of 44,100 Hz. The Nyquist frequency is therefore 22,050 Hz, which represents the highest frequency the data can produce (again if the anti-aliasing filter is perfect). For example, if the chosen anti-aliasing filter (a low-pass filter in this case) has a transition band of 2,000 Hz then the cut-off frequency should be at 20,050 Hz to yield a signal with no power at frequencies of 22,050 Hz and greater. (Again, because realizable filters are not perfect, the frequencies greater than 22,050 will still have power except the aliasing they will produce is minimal.)

It should be noted that the Nyquist frequency itself should not be contained within the signal. If the signal contains a frequency at the Nyquist frequency then the phase between the signal and the sampler will determine the level of the frequency contained within the discrete signal.

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