<s>
In	O
signal	O
processing	O
and	O
related	O
disciplines	O
,	O
aliasing	B-Error_Name
is	O
an	O
effect	O
that	O
causes	O
different	O
signals	O
to	O
become	O
indistinguishable	O
(	O
or	O
aliases	O
of	O
one	O
another	O
)	O
when	O
sampled	B-Algorithm
.	O
</s>
<s>
Aliasing	B-Error_Name
that	O
occurs	O
in	O
signals	O
sampled	B-Algorithm
in	O
time	O
,	O
for	O
instance	O
in	O
digital	O
audio	O
or	O
the	O
stroboscopic	O
effect	O
,	O
is	O
referred	O
to	O
as	O
temporal	B-Error_Name
aliasing	I-Error_Name
.	O
</s>
<s>
Aliasing	B-Error_Name
in	O
spatially	O
sampled	B-Algorithm
signals	O
(	O
e.g.	O
</s>
<s>
moiré	O
patterns	O
in	O
digital	B-Algorithm
images	I-Algorithm
)	O
is	O
referred	O
to	O
as	O
spatial	B-Error_Name
aliasing	I-Error_Name
.	O
</s>
<s>
Aliasing	B-Error_Name
is	O
generally	O
avoided	O
by	O
applying	O
low-pass	B-Algorithm
filters	I-Algorithm
or	O
anti-aliasing	O
filters	O
(	O
AAF	O
)	O
to	O
the	O
input	O
signal	O
before	O
sampling	O
and	O
when	O
converting	O
a	O
signal	O
from	O
a	O
higher	O
to	O
a	O
lower	O
sampling	O
rate	O
.	O
</s>
<s>
Suitable	O
reconstruction	B-Algorithm
filtering	O
should	O
then	O
be	O
used	O
when	O
restoring	O
the	O
sampled	B-Algorithm
signal	O
to	O
the	O
continuous	O
domain	O
or	O
converting	O
a	O
signal	O
from	O
a	O
lower	O
to	O
a	O
higher	O
sampling	O
rate	O
.	O
</s>
<s>
For	O
spatial	B-Algorithm
anti-aliasing	I-Algorithm
,	O
the	O
types	O
of	O
anti-aliasing	O
include	O
fast	B-Algorithm
approximate	I-Algorithm
anti-aliasing	I-Algorithm
(	O
FXAA	B-Algorithm
)	O
,	O
multisample	B-Algorithm
anti-aliasing	I-Algorithm
,	O
and	O
supersampling	B-Algorithm
.	O
</s>
<s>
When	O
a	O
digital	B-Algorithm
image	I-Algorithm
is	O
viewed	O
,	O
a	O
reconstruction	B-Algorithm
is	O
performed	O
by	O
a	O
display	O
or	O
printer	O
device	O
,	O
and	O
by	O
the	O
eyes	O
and	O
the	O
brain	O
.	O
</s>
<s>
If	O
the	O
image	O
data	O
is	O
processed	O
in	O
some	O
way	O
during	O
sampling	O
or	O
reconstruction	B-Algorithm
,	O
the	O
reconstructed	O
image	O
will	O
differ	O
from	O
the	O
original	O
image	O
,	O
and	O
an	O
alias	O
is	O
seen	O
.	O
</s>
<s>
An	O
example	O
of	O
spatial	B-Error_Name
aliasing	I-Error_Name
is	O
the	O
moiré	O
pattern	O
observed	O
in	O
a	O
poorly	O
pixelized	O
image	O
of	O
a	O
brick	O
wall	O
.	O
</s>
<s>
Spatial	B-Algorithm
anti-aliasing	I-Algorithm
techniques	O
avoid	O
such	O
poor	O
pixelizations	O
.	O
</s>
<s>
Aliasing	B-Error_Name
can	O
be	O
caused	O
either	O
by	O
the	O
sampling	O
stage	O
or	O
the	O
reconstruction	B-Algorithm
stage	O
;	O
these	O
may	O
be	O
distinguished	O
by	O
calling	O
sampling	O
aliasing	B-Error_Name
prealiasing	O
and	O
reconstruction	B-Algorithm
aliasing	B-Error_Name
postaliasing	O
.	O
</s>
<s>
Temporal	B-Error_Name
aliasing	I-Error_Name
is	O
a	O
major	O
concern	O
in	O
the	O
sampling	O
of	O
video	O
and	O
audio	O
signals	O
.	O
</s>
<s>
If	O
a	O
piece	O
of	O
music	O
is	O
sampled	B-Algorithm
at	O
32,000	O
samples	O
per	O
second	O
(	O
Hz	O
)	O
,	O
any	O
frequency	O
components	O
at	O
or	O
above	O
16,000	O
Hz	O
(	O
the	O
Nyquist	O
frequency	O
for	O
this	O
sampling	O
rate	O
)	O
will	O
cause	O
aliasing	B-Error_Name
when	O
the	O
music	O
is	O
reproduced	O
by	O
a	O
digital-to-analog	O
converter	O
(	O
DAC	O
)	O
.	O
</s>
<s>
The	O
high	O
frequencies	O
in	O
the	O
analog	O
signal	O
will	O
appear	O
as	O
lower	O
frequencies	O
(	O
wrong	O
alias	O
)	O
in	O
the	O
recorded	O
digital	B-Algorithm
sample	I-Algorithm
and	O
,	O
hence	O
,	O
cannot	O
be	O
reproduced	O
by	O
the	O
DAC	O
.	O
</s>
<s>
To	O
prevent	O
this	O
,	O
an	O
anti-aliasing	O
filter	O
is	O
used	O
to	O
remove	O
components	O
above	O
the	O
Nyquist	O
frequency	O
prior	O
to	O
sampling	O
.	O
</s>
<s>
In	O
video	O
or	O
cinematography	O
,	O
temporal	B-Error_Name
aliasing	I-Error_Name
results	O
from	O
the	O
limited	O
frame	O
rate	O
,	O
and	O
causes	O
the	O
wagon-wheel	O
effect	O
,	O
whereby	O
a	O
spoked	O
wheel	O
appears	O
to	O
rotate	O
too	O
slowly	O
or	O
even	O
backwards	O
.	O
</s>
<s>
Aliasing	B-Error_Name
has	O
changed	O
its	O
apparent	O
frequency	O
of	O
rotation	O
.	O
</s>
<s>
Temporal	B-Error_Name
aliasing	I-Error_Name
frequencies	O
in	O
video	O
and	O
cinematography	O
are	O
determined	O
by	O
the	O
frame	O
rate	O
of	O
the	O
camera	O
,	O
but	O
the	O
relative	O
intensity	O
of	O
the	O
aliased	O
frequencies	O
is	O
determined	O
by	O
the	O
shutter	O
timing	O
(	O
exposure	O
time	O
)	O
or	O
the	O
use	O
of	O
a	O
temporal	B-Error_Name
aliasing	I-Error_Name
reduction	O
filter	O
during	O
filming	O
.	O
</s>
<s>
Like	O
the	O
video	O
camera	O
,	O
most	O
sampling	O
schemes	O
are	O
periodic	O
;	O
that	O
is	O
,	O
they	O
have	O
a	O
characteristic	O
sampling	B-Algorithm
frequency	I-Algorithm
in	O
time	O
or	O
in	O
space	O
.	O
</s>
<s>
Digital	O
cameras	O
provide	O
a	O
certain	O
number	O
of	O
samples	O
(	O
pixels	B-Algorithm
)	O
per	O
degree	O
or	O
per	O
radian	O
,	O
or	O
samples	O
per	O
mm	O
in	O
the	O
focal	O
plane	O
of	O
the	O
camera	O
.	O
</s>
<s>
Audio	O
signals	O
are	O
sampled	B-Algorithm
(	O
digitized	O
)	O
with	O
an	O
analog-to-digital	O
converter	O
,	O
which	O
produces	O
a	O
constant	O
number	O
of	O
samples	O
per	O
second	O
.	O
</s>
<s>
Some	O
of	O
the	O
most	O
dramatic	O
and	O
subtle	O
examples	O
of	O
aliasing	B-Error_Name
occur	O
when	O
the	O
signal	O
being	O
sampled	B-Algorithm
also	O
has	O
periodic	O
content	O
.	O
</s>
<s>
Actual	O
signals	O
have	O
a	O
finite	O
duration	O
and	O
their	O
frequency	O
content	O
,	O
as	O
defined	O
by	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
has	O
no	O
upper	O
bound	O
.	O
</s>
<s>
Some	O
amount	O
of	O
aliasing	B-Error_Name
always	O
occurs	O
when	O
such	O
functions	O
are	O
sampled	B-Algorithm
.	O
</s>
<s>
If	O
sampled	B-Algorithm
at	O
a	O
high	O
enough	O
rate	O
,	O
determined	O
by	O
the	O
bandwidth	O
,	O
the	O
original	O
function	O
can	O
,	O
in	O
theory	O
,	O
be	O
perfectly	O
reconstructed	O
from	O
the	O
infinite	O
set	O
of	O
samples	O
.	O
</s>
<s>
Sometimes	O
aliasing	B-Error_Name
is	O
used	O
intentionally	O
on	O
signals	O
with	O
no	O
low-frequency	O
content	O
,	O
called	O
bandpass	O
signals	O
.	O
</s>
<s>
Undersampling	B-Algorithm
,	O
which	O
creates	O
low-frequency	O
aliases	O
,	O
can	O
produce	O
the	O
same	O
result	O
,	O
with	O
less	O
effort	O
,	O
as	O
frequency-shifting	O
the	O
signal	O
to	O
lower	O
frequencies	O
before	O
sampling	O
at	O
the	O
lower	O
rate	O
.	O
</s>
<s>
Some	O
digital	O
channelizers	O
exploit	O
aliasing	B-Error_Name
in	O
this	O
way	O
for	O
computational	O
efficiency	O
.	O
</s>
<s>
Sinusoids	O
are	O
an	O
important	O
type	O
of	O
periodic	O
function	O
,	O
because	O
realistic	O
signals	O
are	O
often	O
modeled	O
as	O
the	O
summation	O
of	O
many	O
sinusoids	O
of	O
different	O
frequencies	O
and	O
different	O
amplitudes	O
(	O
for	O
example	O
,	O
with	O
a	O
Fourier	O
series	O
or	O
transform	B-Algorithm
)	O
.	O
</s>
<s>
Understanding	O
what	O
aliasing	B-Error_Name
does	O
to	O
the	O
individual	O
sinusoids	O
is	O
useful	O
in	O
understanding	O
what	O
happens	O
to	O
their	O
sum	O
.	O
</s>
<s>
Aliasing	B-Error_Name
matters	O
when	O
one	O
attempts	O
to	O
reconstruct	O
the	O
original	O
waveform	O
from	O
its	O
samples	O
.	O
</s>
<s>
The	O
most	O
common	O
reconstruction	B-Algorithm
technique	O
produces	O
the	O
smallest	O
of	O
the	O
frequencies	O
.	O
</s>
<s>
The	O
lower	O
left	O
frame	O
of	O
Fig.2	O
depicts	O
the	O
typical	O
reconstruction	B-Algorithm
result	O
of	O
the	O
available	O
samples	O
.	O
</s>
<s>
Until	O
exceeds	O
the	O
Nyquist	O
frequency	O
,	O
the	O
reconstruction	B-Algorithm
matches	O
the	O
actual	O
waveform	O
(	O
upper	O
left	O
frame	O
)	O
.	O
</s>
<s>
The	O
figures	O
below	O
offer	O
additional	O
depictions	O
of	O
aliasing	B-Error_Name
,	O
due	O
to	O
sampling	O
.	O
</s>
<s>
The	O
dashed	O
red	O
lines	O
are	O
the	O
corresponding	O
paths	O
of	O
the	O
aliases.thumb	O
|x180px|	O
Fig.4	O
:	O
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
music	O
sampled	B-Algorithm
at	O
44,100	O
samples/sec	O
exhibits	O
symmetry	O
(	O
called	O
"	O
folding	O
"	O
)	O
around	O
the	O
Nyquist	O
frequency	O
(	O
22,050	O
Hz	O
)	O
.thumb	O
|x180px|	O
Fig.5	O
:	O
Graph	O
of	O
frequency	O
aliasing	B-Error_Name
,	O
showing	O
folding	O
frequency	O
and	O
periodicity	O
.	O
</s>
<s>
That	O
is	O
typically	O
approximated	O
by	O
filtering	O
the	O
original	O
signal	O
to	O
attenuate	O
high	O
frequency	O
components	O
before	O
it	O
is	O
sampled	B-Algorithm
.	O
</s>
<s>
A	O
filter	O
chosen	O
in	O
anticipation	O
of	O
a	O
certain	O
sample	B-Algorithm
frequency	I-Algorithm
is	O
called	O
an	O
anti-aliasing	O
filter	O
.	O
</s>
<s>
Most	O
sampled	B-Algorithm
signals	O
are	O
not	O
simply	O
stored	O
and	O
reconstructed	O
.	O
</s>
<s>
But	O
the	O
fidelity	O
of	O
a	O
theoretical	O
reconstruction	B-Algorithm
(	O
via	O
the	O
Whittaker	O
–	O
Shannon	O
interpolation	O
formula	O
)	O
is	O
a	O
customary	O
measure	O
of	O
the	O
effectiveness	O
of	O
sampling	O
.	O
</s>
<s>
Historically	O
the	O
term	O
aliasing	B-Error_Name
evolved	O
from	O
radio	O
engineering	O
because	O
of	O
the	O
action	O
of	O
superheterodyne	O
receivers	O
.	O
</s>
<s>
When	O
the	O
receiver	O
shifts	O
multiple	O
signals	O
down	O
to	O
lower	O
frequencies	O
,	O
from	O
RF	O
to	O
IF	O
by	O
heterodyning	B-Algorithm
,	O
an	O
unwanted	O
signal	O
,	O
from	O
an	O
RF	O
frequency	O
equally	O
far	O
from	O
the	O
local	O
oscillator	O
(	O
LO	O
)	O
frequency	O
as	O
the	O
desired	O
signal	O
,	O
but	O
on	O
the	O
wrong	O
side	O
of	O
the	O
LO	O
,	O
can	O
end	O
up	O
at	O
the	O
same	O
IF	O
frequency	O
as	O
the	O
wanted	O
one	O
.	O
</s>
<s>
That	O
paper	O
includes	O
an	O
example	O
of	O
frequency	O
aliasing	B-Error_Name
dating	O
back	O
to	O
1922	O
.	O
</s>
<s>
The	O
first	O
published	O
use	O
of	O
the	O
term	O
"	O
aliasing	B-Error_Name
"	O
in	O
this	O
context	O
is	O
due	O
to	O
Blackman	O
and	O
Tukey	O
in	O
1958	O
.	O
</s>
<s>
In	O
their	O
preface	O
to	O
the	O
Dover	O
reprint	O
of	O
this	O
paper	O
,	O
they	O
point	O
out	O
that	O
the	O
idea	O
of	O
aliasing	B-Error_Name
had	O
been	O
illustrated	O
graphically	O
by	O
Stumpf	O
ten	O
years	O
prior	O
.	O
</s>
<s>
The	O
1949	O
Bell	O
technical	O
report	O
refers	O
to	O
aliasing	B-Error_Name
as	O
though	O
it	O
is	O
a	O
well-known	O
concept	O
,	O
but	O
does	O
not	O
offer	O
a	O
source	O
for	O
the	O
term	O
.	O
</s>
<s>
Aliasing	B-Error_Name
occurs	O
whenever	O
the	O
use	O
of	O
discrete	O
elements	O
to	O
capture	O
or	O
produce	O
a	O
continuous	O
signal	O
causes	O
frequency	O
ambiguity	O
.	O
</s>
<s>
Spatial	B-Error_Name
aliasing	I-Error_Name
,	O
particular	O
of	O
angular	O
frequency	O
,	O
can	O
occur	O
when	O
reproducing	O
a	O
light	O
field	O
or	O
sound	O
field	O
with	O
discrete	O
elements	O
,	O
as	O
in	O
3D	B-Algorithm
displays	I-Algorithm
or	O
wave	O
field	O
synthesis	O
of	O
sound	O
.	O
</s>
<s>
This	O
aliasing	B-Error_Name
is	O
visible	O
in	O
images	O
such	O
as	O
posters	O
with	O
lenticular	O
printing	O
:	O
if	O
they	O
have	O
low	O
angular	O
resolution	B-General_Concept
,	O
then	O
as	O
one	O
moves	O
past	O
them	O
,	O
say	O
from	O
left-to-right	O
,	O
the	O
2D	O
image	O
does	O
not	O
initially	O
change	O
(	O
so	O
it	O
appears	O
to	O
move	O
left	O
)	O
,	O
then	O
as	O
one	O
moves	O
to	O
the	O
next	O
angular	O
image	O
,	O
the	O
image	O
suddenly	O
changes	O
(	O
so	O
it	O
jumps	O
right	O
)	O
–	O
and	O
the	O
frequency	O
and	O
amplitude	O
of	O
this	O
side-to-side	O
movement	O
corresponds	O
to	O
the	O
angular	O
resolution	B-General_Concept
of	O
the	O
image	O
(	O
and	O
,	O
for	O
frequency	O
,	O
the	O
speed	O
of	O
the	O
viewer	O
's	O
lateral	O
movement	O
)	O
,	O
which	O
is	O
the	O
angular	O
aliasing	B-Error_Name
of	O
the	O
4D	O
light	O
field	O
.	O
</s>
<s>
The	O
lack	O
of	O
parallax	B-Device
on	O
viewer	O
movement	O
in	O
2D	O
images	O
and	O
in	O
3-D	B-Algorithm
film	I-Algorithm
produced	O
by	O
stereoscopic	B-Algorithm
glasses	O
(	O
in	O
3D	B-Algorithm
films	I-Algorithm
the	O
effect	O
is	O
called	O
"	O
yawing	O
"	O
,	O
as	O
the	O
image	O
appears	O
to	O
rotate	O
on	O
its	O
axis	O
)	O
can	O
similarly	O
be	O
seen	O
as	O
loss	O
of	O
angular	O
resolution	B-General_Concept
,	O
all	O
angular	O
frequencies	O
being	O
aliased	O
to	O
0	O
(	O
constant	O
)	O
.	O
</s>
<s>
The	O
qualitative	O
effects	O
of	O
aliasing	B-Error_Name
can	O
be	O
heard	O
in	O
the	O
following	O
audio	O
demonstration	O
.	O
</s>
<s>
The	O
aliasing	B-Error_Name
distortion	O
in	O
the	O
lower	O
frequencies	O
is	O
increasingly	O
obvious	O
with	O
higher	O
fundamental	O
frequencies	O
,	O
and	O
while	O
the	O
bandlimited	O
sawtooth	O
is	O
still	O
clear	O
at	O
1760Hz	O
,	O
the	O
aliased	O
sawtooth	O
is	O
degraded	O
and	O
harsh	O
with	O
a	O
buzzing	O
audible	O
at	O
frequencies	O
lower	O
than	O
the	O
fundamental	O
.	O
</s>
<s>
A	O
form	O
of	O
spatial	B-Error_Name
aliasing	I-Error_Name
can	O
also	O
occur	O
in	O
antenna	O
arrays	O
or	O
microphone	O
arrays	O
used	O
to	O
estimate	O
the	O
direction	O
of	O
arrival	O
of	O
a	O
wave	O
signal	O
,	O
as	O
in	O
geophysical	O
exploration	O
by	O
seismic	O
waves	O
.	O
</s>
<s>
Waves	O
must	O
be	O
sampled	B-Algorithm
more	O
densely	O
than	O
two	O
points	O
per	O
wavelength	O
,	O
or	O
the	O
wave	O
arrival	O
direction	O
becomes	O
ambiguous	O
.	O
</s>
