### Downsampling (Decimation) Operator

Figure 11.3 shows the symbol for downsampling by the factor . The downsampler selects every th sample and discards the rest:

In the frequency domain, we have

Thus, the frequency axis is expanded by the factor , wrapping times around the unit circle, adding to itself times. For , two partial spectra are summed, as indicated in Fig.11.4.

Using the common twiddle factor notation

 (12.1)

the aliasing expression can be written as

#### Example: Downsampling by 2

For , downsampling by 2 can be expressed as , so that (since )

#### Example: Upsampling by 2

For , upsampling (stretching) by 2 can be expressed as
, so that

 (12.2)

as discussed more fully in §2.3.11.

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Filtering and Downsampling
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Upsampling (Stretch) Operator