Nyquist frequency
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The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system.[1][2] It is sometimes called the folding frequency, or the cut-off frequency of a sampling system.[3]
The sampling theorem shows that aliasing can be avoided if the Nyquist frequency is greater than the bandwidth, or maximum component frequency, of the signal being sampled.
The Nyquist frequency should not be confused with the Nyquist rate, which is the lower bound of the sampling frequency that satisfies the Nyquist sampling criterion for a given signal or family of signals. This lower bound is twice the bandwidth or maximum component frequency of the signal. Nyquist rate, as commonly used with respect to sampling, is a property of a continuous-time signal, not of a system, whereas Nyquist frequency is a property of a discrete-time system, not of a signal. The domain of the signals does not have to be time, though that is common, leading to Nyquist frequency in hertz; for example, an image sampling system has a Nyquist frequency expressed in units such as cycles per meter.
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[edit] The aliasing problem
In principle, a Nyquist frequency just larger than the signal bandwidth is sufficient to allow perfect reconstruction of the signal from the samples. However, this reconstruction requires an unattainable filter that passes some frequencies unchanged while suppressing all others completely (commonly called a brickwall filter). When attainable filters are used, some degree of oversampling is necessary to accommodate the practical constraints on anti-aliasing filters. That is, frequencies close to the Nyquist frequency may be distorted in the sampling and reconstruction process, so the bandwidth should be kept below the Nyquist frequency by some margin that depends on the actual filters used.
For example, audio CDs have a sampling frequency of 44100 Hz. The Nyquist frequency is therefore 22050 Hz, which is an upper bound on the highest frequency the data can unambiguously represent. If the chosen anti-aliasing filter (a low-pass filter in this case) has a transition band of 2000 Hz, then the cut-off frequency should be no higher than 20050 Hz to yield a signal with negligible power at frequencies of 22050 Hz and greater.
To avoid aliasing, the Nyquist frequency must be strictly greater than the maximum frequency component within the signal. If the signal contains a frequency component at precisely the Nyquist frequency then the corresponding component of the sample values cannot have sufficient information to reconstruct the Nyquist component in the continuous-time signal because of phase ambiguity. In such a case, there would be an infinite number of possible and different sinusoids (of varying amplitude and phase) of the Nyquist frequency component that are represented by the discrete samples.
[edit] Other meanings
Early uses of the term Nyquist Frequency, such as those cited above, are all consistent with the definition presented in this article. Some later publications, including some respectable textbooks, call the signal bandwidth[4] or twice the signal bandwidth[5] the Nyquist frequency; still others refer to the Nyquist rate (twice the signal bandwidth) as Nyquist frequency;[6][7] these are distinctly minority usages.
[edit] References
- ^ Ulf. Grenander (1959). Probability and Statistics: The Harald Cramér Volume. Wiley. “The Nyquist frequency is that frequency whose period is two sampling intervals.”
- ^ Harry L. Stiltz (1961). Aerospace Telemetry. Prentice-Hall. “the existence of power in the continuous signal spectrum at frequencies higher than the Nyquist frequency is the cause of aliasing error”
- ^ B. V. Korvin-Kroukovsky (1961). Theory of Seakeeping. Society of Naval Architects and Marine Engineers. “The Nyquist frequency is often called the folding frequency or cut-off frequency”
- ^ Michael J. Roberts (2004). Signals and Systems: Analysis Using Transform Methods and MATLAB. McGraw-Hill Professional. ISBN 0072499427.
- ^ Uwe Windhorst and Håkan Johansson (1999). Modern Techniques in Neuroscience Research. Springer. ISBN 3540644601.
- ^ Jonathan M. Blackledge (2003). Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications. Horwood Publishing. ISBN 1898563489.
- ^ Paulo Sergio Ramirez Diniz, Eduardo A. B. Da Silva, Sergio L. Netto (2002). Digital Signal Processing: System Analysis and Design. Cambridge University Press. ISBN 0521781752.
