### 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.

**Next Section:**

Filtering and Downsampling

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Upsampling (Stretch) Operator