Conversion of the analog signal to a series of digital values by measurement at a set of particular times; this utilizes the analog to digital converter. If the rate of sampling is less than twice the highest frequency in the signal, aliasing will occur. The duration of sampling determines how small a difference of frequencies can be separated.

See also Aliasing.

Oversampling is the increase in data to avoid aliasing and wrap around artifacts. Aliasing is the incorrectly mapping of tissue signal from outside the FOV to a location inside the FOV. This is caused by the fact, that the acquired k-space frequency data is not sampled density enough.
Oversampling in frequency direction, done by increasing the sampling frequency, prevents this aliasing artifact. The proper frequency based on the sampling theorem (Shannon sampling theorem/Nyquist sampling theorem) must be at least twice the frequency of each frequency component in the incoming signal. All frequency components above this limit will be aliased to frequencies between zero and half of the sampling frequency and combined with the proper signal information, which creates the artifact. Oversampling creates a larger field of view, more data needs to be stored and processed, but this is for modern MRI systems not a real problem. Oversampling in phase direction (no phase wrap), to eliminate wrap around artifacts, by increasing the number of phase encoding steps, results in longer scan/processing times.

Undersampling is the decrease in data to increase image acquisition speed (shorter scan times without loss of quality - increased productivity - reduced cost of equipment). There are different strategies to decrease data without losing quality (e.g. reduction of the FOV in one or more spatial directions - RFOV).
Reduction in data normally is associated with an increase in aliasing (degradation of the SNR through backfolding of the entire noise spectrum), or with other artifact caused by missing data, which results in fine lines.

Cartesian sampling is used to refer to data collection with a fixed value of the phase encoding gradient. In 2D Fourier imaging with common Cartesian sampling of k-space sensitivity encoding by means of a receiver array enables to reduce the number of Fourier encoding steps. This is achieved by increasing the distance of sampling positions in k-space while maintaining the maximum k-values.
The Cartesian coordinates are obtained from the polar coordinates by the operations
x = r sin f
y = r cos f
using the trigonometric functions sine and cosine.

The data acquisition beyond the FOV in phase encoding direction, with doubling the number of acquisitions and the scan time.

See also Oversampling and Aliasing Artifact.