# Subsampling

Under the sub-scan ( English undersampling is) in the signal processing , the sampling of a signal waveform with less than twice the bandwidth understood. Under certain conditions, the conditions of the Nyquist-Shannon sampling theorem are not violated. In this case, the undersampling can serve to shift a high-frequency signal into an intermediate frequency range with a lower frequency, as in the function of a mixer . If the requirements of the sampling theorem are not met, then aliasing occurs as a result of undersampling and the associated loss of information occurs. The sub-sampling is the counterpart to the oversampling ( oversampling is).

## Procedure

A signal in the bandpass position generally has a bandwidth of signal components which are arranged symmetrically around the center frequency . In order not to violate the conditions of the Nyquist-Shannon sampling theorem, the signal must not have any frequency components outside the bandwidth. This can be ensured, among other things, by bandpass filters prior to undersampling. ${\ displaystyle B}$${\ displaystyle f_ {0}}$

All the sampling frequencies shift with the sampling frequency ${\ displaystyle f_ {A}}$

${\ displaystyle f_ {A} = {\ frac {f_ {0} -f '_ {0}} {r}}, \ qquad r \ in \ mathbb {Z}}$

the center frequency of the bandpass signal to the selectable frame rate in the baseband. The value represents the factor of the undersampling, with increasing sampling frequencies and thus usable baseband widths become smaller and smaller. ${\ displaystyle f_ {0}}$${\ displaystyle f '_ {0}}$${\ displaystyle r}$${\ displaystyle r}$

The frame rate in the baseband is usually set to the value for a symmetrical band spectrum . A choice is made for asymmetrical band spectra . ${\ displaystyle f '_ {0}}$${\ displaystyle f '_ {0} = 0}$${\ displaystyle f '_ {0} = {\ frac {f_ {A}} {4}}}$

### Undersampling with symmetrical band spectrum

With a symmetrical band spectrum , such as amplitude modulation , the information in the signal is available twice and symmetrically . Typically in this case it is chosen what the frequencies in the sampled signal through ${\ displaystyle f_ {0}}$${\ displaystyle f '_ {0} = 0}$

${\ displaystyle f_ {A} = {\ frac {f_ {0}} {r}}, \ qquad r \ in \ mathbb {Z} \ setminus \ {0 \}}$

given are. The redundant half of the band is mapped to negative frequencies, which makes demodulation particularly easy. The minimum sampling frequency must be greater than the bandwidth , which means that the factor can then be determined with this secondary condition : ${\ displaystyle B}$${\ displaystyle r}$

${\ displaystyle r = {\ frac {f_ {0}} {B}}}$

The undersampling in the symmetrical band spectrum of the demodulation thus corresponds to an amplitude modulation.

### Undersampling with asymmetrical band spectrum

In general, however, the signal is only shifted to a lower intermediate frequency position for further processing (function of a mixer ). To fulfill the conditions of the Nyquist-Shannon sampling theorem, it is chosen that the frequencies in the sampled signal are then: ${\ displaystyle f '_ {0} = f_ {A} / 4}$

${\ displaystyle f_ {A} = {\ frac {4f_ {0}} {4r + 1}}, \ qquad r \ in \ mathbb {Z}}$

The minimum sampling frequency must be greater than twice the bandwidth , which means that the factor can then be determined with this secondary condition : ${\ displaystyle B}$${\ displaystyle r}$

${\ displaystyle r = {\ frac {f_ {0}} {2B}} - {\ frac {1} {4}}}$

## Non-band-limited signals

Effect of the sampling frequency in relation to the signal frequency

In the case of undersampling of signals that are not appropriately band-limited, the prerequisites for loss-free information acquisition mentioned in the Nyquist-Shannon sampling theorem are not met. Aliasing leads to the occurrence of image frequency components which superimpose parts of the useful signal.

The gray oscillation is the analog signal that is to be discretized (e.g. digitized). The blue numbers on the right indicate the value range of. A sample that falls within this range is assigned this digital number (quantization). The vertical lines (S 1 to S 25 ) indicate the points in time at which scanning takes place. The red × indicate in which range of values ​​the respective sample falls. The rectangular blue waveform represents the signal obtained from the digital data. (Before it is fed to a reconstruction filter.)

In the figure it can be seen that from sample 20 (S 20 ) the digitized values ​​no longer represent the sampled frequency. The signal is therefore reconstructed with a significantly lower frequency and thus incorrectly.

## literature

• Fernando Puente León, Uwe Kiencke, Holger Jäkel: Signals and Systems . 5th edition. Oldenbourg, 2011, ISBN 978-3-486-59748-6 .