Alias Operator
Aliasing occurs when a
signal is
undersampled. If the signal
sampling rate 
is too low, we get
frequency-domain
aliasing.

The topic of aliasing normally arises in the context of
sampling a continuous-time signal. The
sampling theorem
(Appendix
D) says that we will have no aliasing due to sampling
as long as the sampling rate is higher than twice the highest
frequency present in the signal being sampled.
In this chapter, we are considering only discrete-time signals, in
order to keep the math as simple as possible. Aliasing in this
context occurs when a discrete-time signal is
downsampled to reduce its
sampling rate. You can think of continuous-time sampling as the
limiting case for which the starting sampling rate is infinity.
An example of aliasing is shown in Fig.
7.11. In the figure,
the high-frequency
sinusoid is indistinguishable from the
lower-frequency
sinusoid due to aliasing. We say the higher frequency
aliases to the lower frequency.
Figure 7.11:
Example of
frequency-domain aliasing due to undersampling in the time domain.
![\includegraphics[scale=0.5]{eps/aliasing}](http://www.dsprelated.com/josimages_new/mdft/img1275.png) |
Undersampling in the frequency domain gives rise to
time-domain
aliasing. If time or frequency is not specified, the term
``aliasing'' normally means frequency-domain aliasing (due to
undersampling in the time domain).
The
aliasing operator for

-sample signals

is defined by
Like the

operator, the

operator maps a
length

signal down to a length

signal. A way to think of
it is to partition the original

samples into

blocks of length

, with the first block extending from sample 0 to sample

,
the second block from

to

, etc. Then just add up the blocks.
This process is called
aliasing. If the original signal

is
a time signal, it is called
time-domain aliasing; if it is a
spectrum, we call it
frequency-domain aliasing, or just
aliasing. Note that aliasing is
not invertible in general.
Once the blocks are added together, it is usually not possible to
recover the original blocks.
Example:
The alias operator is used to state the
Fourier theorem (§
7.4.11)
That is, when you downsample a signal by the factor

, its
spectrum is
aliased by the factor

.
Figure:
Illustration of aliasing in
the frequency domain.
a)
from Fig.7.9c.
b) First half of the original unit circle (0 to
) wrapped around the
new, smaller unit circle (which is magnified to the original size).
c) Second half (
to
), also wrapped around the new unit circle.
d) Overlay of components to be summed.
e) Sum of components (the aliased spectrum).
f) Both sum and overlay.
![\includegraphics[width=4.5in]{eps/aliasingfd}](http://www.dsprelated.com/josimages_new/mdft/img1282.png) |
Figure
7.12 shows the result of

applied to

from Figure
7.9c. Imagine the spectrum of
Fig.
7.12a as being plotted on a piece of paper rolled
to form a cylinder, with the edges of the paper meeting at

(upper
right corner of Fig.
7.12a). Then the

operation can be
simulated by rerolling the cylinder of paper to cut its circumference in
half. That is, reroll it so that at every point,
two sheets of paper
are in contact at all points on the new, narrower cylinder. Now, simply
add the values on the two overlapping sheets together, and you have the

of the original spectrum on the unit circle. To alias by

,
we would shrink the cylinder further until the paper edges again line up,
giving three layers of paper in the cylinder, and so on.
Figure
7.12b shows what is plotted on the first circular wrap of the
cylinder of paper, and Fig.
7.12c shows what is on the second wrap.
These are overlaid in Fig.
7.12d and added together in
Fig.
7.12e. Finally, Figure
7.12f shows both the addition
and the overlay of the two components. We say that the second component
(Fig.
7.12c) ``aliases'' to new frequency components, while the
first component (Fig.
7.12b) is considered to be at its original
frequencies. If the unit circle of Fig.
7.12a covers frequencies
0 to

, all other unit circles (Fig.
7.12b-c) cover
frequencies 0 to

.
In general, aliasing by the factor

corresponds to a
sampling-rate reduction by the factor

. To prevent aliasing
when reducing the sampling rate, an
anti-aliasing lowpass
filter is generally used. The
lowpass filter attenuates all signal
components at frequencies outside the interval
![$ [-f_s/(2K),f_s/(2K)]$](http://www.dsprelated.com/josimages_new/mdft/img1285.png)
so that all frequency components which would alias are first removed.
Conceptually, in the frequency domain, the unit circle is reduced by

to a unit circle
half the original size, where the two halves are summed. The inverse
of aliasing is then ``repeating'' which should be understood as
increasing the unit circle circumference using ``
periodic
extension'' to generate ``more spectrum'' for the larger unit circle.
In the time domain, on the other hand,
downsampling is the inverse of
the
stretch operator. We may interchange ``time'' and ``frequency''
and repeat these remarks. All of these relationships are precise only
for
integer stretch/downsampling/aliasing/repeat factors; in
continuous time and frequency, the restriction to integer factors is
removed, and we obtain the (simpler)
scaling theorem (proved
in §
C.2).
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