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
![$ \hbox{\sc Alias}$](http://www.dsprelated.com/josimages_new/sasp2/img214.png)
![]() |
(3.34) |
for
![$ \omega\in[-\pi,\pi)$](http://www.dsprelated.com/josimages_new/sasp2/img94.png)
![$ k\neq 0$](http://www.dsprelated.com/josimages_new/sasp2/img216.png)
In z transform notation, the
operator can be expressed as
[287]
![]() |
(3.35) |
where
![$ W_M\isdeftext e^{j2\pi/M}$](http://www.dsprelated.com/josimages_new/sasp2/img218.png)
![$ M$](http://www.dsprelated.com/josimages_new/sasp2/img26.png)
![$ z$](http://www.dsprelated.com/josimages_new/sasp2/img3.png)
![]() |
(3.36) |
The frequency scaling corresponds to having a sampling interval of
![$ T=1$](http://www.dsprelated.com/josimages_new/sasp2/img220.png)
![$ T=1/M$](http://www.dsprelated.com/josimages_new/sasp2/img221.png)
The aliasing theorem makes it clear that, in order to downsample by
factor
without aliasing, we must first lowpass-filter the spectrum
to
. This filtering (when ideal) zeroes out the
spectral regions which alias upon downsampling.
Note that any rational sampling-rate conversion factor
may be implemented as an upsampling by the factor
followed by
downsampling by the factor
[50,287].
Conceptually, a stretch-by-
is followed by a lowpass filter cutting
off at
, followed by
downsample-by-
, i.e.,
![]() |
(3.37) |
In practice, there are more efficient algorithms for sampling-rate conversion [270,135,78] based on a more direct approach to bandlimited interpolation.
Proof of Aliasing Theorem
To show:
![$\displaystyle \zbox {\hbox{\sc Downsample}_N(x) \;\longleftrightarrow\;\frac{1}{N} \hbox{\sc Alias}_N(X)}
$](http://www.dsprelated.com/josimages_new/sasp2/img226.png)
or
![\fbox{$x(nN) \;\longleftrightarrow\;\dfrac{1}{N} \displaystyle\sum_{m=0}^{N-1} X\left(e^{j2\pi m/N} z^{1/N}\right)$}](http://www.dsprelated.com/josimages_new/sasp2/img227.png)
From the DFT case [264], we know this is true when
and
are each complex sequences of length
, in which case
and
are length
. Thus,
![]() |
(3.38) |
where we have chosen to keep frequency samples
![$ \omega_k$](http://www.dsprelated.com/josimages_new/sasp2/img100.png)
![$ \omega_k =
2\pi k/ N_s$](http://www.dsprelated.com/josimages_new/sasp2/img232.png)
![$ X$](http://www.dsprelated.com/josimages_new/sasp2/img119.png)
![$ Y$](http://www.dsprelated.com/josimages_new/sasp2/img229.png)
![$ N_s\to\infty$](http://www.dsprelated.com/josimages_new/sasp2/img233.png)
![$ \omega_k$](http://www.dsprelated.com/josimages_new/sasp2/img100.png)
![$ \omega$](http://www.dsprelated.com/josimages_new/sasp2/img89.png)
![]() |
(3.39) |
Replacing
![$ \omega$](http://www.dsprelated.com/josimages_new/sasp2/img89.png)
![$ \omega^\prime =\omega N$](http://www.dsprelated.com/josimages_new/sasp2/img235.png)
![$ z$](http://www.dsprelated.com/josimages_new/sasp2/img3.png)
![$ X(z)$](http://www.dsprelated.com/josimages_new/sasp2/img236.png)
![$ X(\omega)$](http://www.dsprelated.com/josimages_new/sasp2/img162.png)
![$ z=e^{j\omega^\prime }$](http://www.dsprelated.com/josimages_new/sasp2/img237.png)
Next Section:
Differentiation Theorem Dual
Previous Section:
Stretch/Repeat (Scaling) Theorem