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) |
![]() |
(3.34) |
In z transform notation, the
operator can be expressed as
[287]
![]() |
(3.35) |
![]() |
(3.36) |
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) |