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
.
The following example of
is illustrated in Fig.7.10:
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 resampling
using bandlimited interpolation. 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
.