### Downsampling (Decimation) Operator

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

*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