
Downsampling Operator
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

.
Figure:
Illustration of
.
![\includegraphics[width=4in]{eps/downsamplex}](http://www.dsprelated.com/josimages_new/mdft/img1271.png) |
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

.
Next Section: Alias OperatorPrevious Section: Repeat Operator