Downsampling by (also called decimation by ) is defined for as taking every th sample, starting with sample zero:
The operator maps a length signal down to a length signal. It is the inverse of the operator (but not vice versa), i.e.,
The stretch and downsampling operations do not commute because they are linear time-varying operators. They can be modeled using time-varying switches controlled by the sample index .
An example of is shown in Fig.7.10. The example is
Note that the term ``downsampling'' may also refer to the more elaborate process of sampling-rate conversion to a lower sampling rate, in which a signal's sampling rate is lowered by means of bandlimited interpolation (discussed in Appendix D). To distinguish these cases, we can call this bandlimited downsampling, because a lowpass-filter is needed, in general, prior to downsampling so that aliasing is avoided. This topic is address in Appendix D. Early sampling-rate converters were in fact implemented using the operation, followed by an appropriate lowpass filter, followed by , in order to implement a sampling-rate conversion by the factor .