Downsampling and Aliasing

The *downsampling* operator
selects every
sample of a signal:

(3.32) |

The *aliasing theorem* states that downsampling in time
corresponds to *aliasing* in the frequency domain:

(3.33) |

where the operator is defined as

(3.34) |

for . The summation terms for are called

In *z* transform notation, the
operator can be expressed as
[287]

(3.35) |

where is a common notation for the primitive th root of unity. On the unit circle of the plane, this becomes

(3.36) |

The frequency scaling corresponds to having a sampling interval of after downsampling, which corresponds to the interval prior to downsampling.

The aliasing theorem makes it clear that, in order to downsample by factor without aliasing, we must first lowpass-filter the spectrum to . This filtering (when ideal) zeroes out the spectral regions which alias upon downsampling.

Note that any rational sampling-rate conversion factor
may be implemented as an upsampling by the factor
followed by
downsampling by the factor
[50,287].
Conceptually, a stretch-by-
is followed by a lowpass filter cutting
off at
, followed by
downsample-by-
, *i.e.*,

(3.37) |

In practice, there are more efficient algorithms for sampling-rate conversion [270,135,78] based on a more direct approach to

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University