### 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 .

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 .

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Alias Operator

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Repeat Operator