### Upsampling (Stretch) Operator

Figure 11.1 shows the graphical symbol for a digital upsampler
by the factor
. To upsample by the integer factor
, we simply
insert
zeros between
and
for all
. In other
words, the upsampler implements the *stretch operator* defined
in §2.3.9:

In the frequency domain, we have, by the stretch (repeat) theorem for DTFTs:

Plugging in
, we see that the spectrum on
contracts by the factor
, and
*images* appear around the
unit circle. For
, this is depicted in Fig.11.2.

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Downsampling (Decimation) Operator

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Optimal Band Filters