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
aliasing components.
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
![$\displaystyle \hbox{\sc Alias}_{M,\omega}(X) \eqsp \sum_{k=0}^{M-1} X\left[e^{j\left(\frac{\omega}{M} + k\frac{2\pi}{M}\right)}\right], \quad -\pi\leq \omega < \pi.$](http://www.dsprelated.com/josimages_new/sasp2/img219.png) |
(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.,
![$\displaystyle x^\prime \eqsp \hbox{\sc Downsample}_M\{\hbox{\sc Lowpass}_{\omega_c}[\hbox{\sc Stretch}_L(x)]\}$](http://www.dsprelated.com/josimages_new/sasp2/img225.png) |
(3.37) |
In practice, there are more efficient algorithms for
sampling-rate
conversion [
270,
135,
78] based on a more
direct approach to
bandlimited
interpolation.
To show:
or
From the
DFT case [
264], we know this is true when

and

are each complex sequences of length

, in which case

and

are length

. Thus,
 |
(3.38) |
where we have chosen to keep frequency samples

in terms of
the original frequency axis prior to
downsampling,
i.e.,

for both

and

. This choice allows us to easily take
the limit as

by simply replacing

by

:
 |
(3.39) |
Replacing

by

and converting to

-transform
notation

instead of
Fourier transform notation

,
with

, yields the final result.
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