# Nyquist Sampling Theory

In each graph, the solid thick blue line represents the original time series. The red squares represent the sampled data, starting with a random time. The thin red line connects with sampled points with a cubic spline curve.

 This sampling rate accurately captures the periodicity of the original time series. This sampling, while at a rate less than the period of the original time series, does not accurately capture the periodicity of the original time series. In fact the data appears to have an erroneous period longer than the true time series. This is known as aliasing. This sampling  is at a rate greater than the period of the original time series. Not only does it miss the periodicity of the original time series, but it would appear that the time series has a periodicity much longer than the true period.  This is another case of aliasing.

The Nyquist critical frequency is 1 / (2 * sampling interval). You need at least two sampled points in every period that you want to capture. If there is significant power that you do not capture, for instance in shorter periods, you will have problems with aliasing.

Any power outside the range of Nyquist critical frequency will be aliased (falsely transformed) into the range of frequencies correctly sampled.

Critical sampling of a sine wave requires at least two points per cycle.

Last revision 2/2/2009