<s>
In	O
signal	O
processing	O
,	O
particularly	O
digital	B-Algorithm
image	I-Algorithm
processing	I-Algorithm
,	O
ringing	B-Algorithm
artifacts	I-Algorithm
are	O
artifacts	B-Algorithm
that	O
appear	O
as	O
spurious	O
signals	O
near	O
sharp	O
transitions	O
in	O
a	O
signal	O
.	O
</s>
<s>
The	O
term	O
"	O
ringing	B-Algorithm
"	O
is	O
because	O
the	O
output	O
signal	O
oscillates	O
at	O
a	O
fading	O
rate	O
around	O
a	O
sharp	O
transition	O
in	O
the	O
input	O
,	O
similar	O
to	O
a	O
bell	O
after	O
being	O
struck	O
.	O
</s>
<s>
As	O
with	O
other	O
artifacts	B-Algorithm
,	O
their	O
minimization	O
is	O
a	O
criterion	O
in	O
filter	O
design	O
.	O
</s>
<s>
The	O
main	O
cause	O
of	O
ringing	B-Algorithm
artifacts	I-Algorithm
is	O
due	O
to	O
a	O
signal	O
being	O
bandlimited	O
(	O
specifically	O
,	O
not	O
having	O
high	O
frequencies	O
)	O
or	O
passed	O
through	O
a	O
low-pass	B-Algorithm
filter	I-Algorithm
;	O
this	O
is	O
the	O
frequency	O
domain	O
description	O
.	O
</s>
<s>
In	O
terms	O
of	O
the	O
time	O
domain	O
,	O
the	O
cause	O
of	O
this	O
type	O
of	O
ringing	B-Algorithm
is	O
the	O
ripples	O
in	O
the	O
sinc	B-Algorithm
function	I-Algorithm
,	O
which	O
is	O
the	O
impulse	O
response	O
(	O
time	O
domain	O
representation	O
)	O
of	O
a	O
perfect	O
low-pass	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
One	O
may	O
distinguish	O
overshoot	B-Algorithm
(	O
and	O
undershoot	B-Algorithm
)	O
,	O
which	O
occurs	O
when	O
transitions	O
are	O
accentuated	O
–	O
the	O
output	O
is	O
higher	O
than	O
the	O
input	O
–	O
from	O
ringing	B-Algorithm
,	O
where	O
after	O
an	O
overshoot	B-Algorithm
,	O
the	O
signal	O
overcorrects	O
and	O
is	O
now	O
below	O
the	O
target	O
value	O
;	O
these	O
phenomena	O
often	O
occur	O
together	O
,	O
and	O
are	O
thus	O
often	O
conflated	O
and	O
jointly	O
referred	O
to	O
as	O
"	O
ringing	B-Algorithm
"	O
.	O
</s>
<s>
The	O
term	O
"	O
ringing	B-Algorithm
"	O
is	O
most	O
often	O
used	O
for	O
ripples	O
in	O
the	O
time	O
domain	O
,	O
though	O
it	O
is	O
also	O
sometimes	O
used	O
for	O
frequency	O
domain	O
effects	O
:	O
</s>
<s>
windowing	O
a	O
filter	O
in	O
the	O
time	O
domain	O
by	O
a	O
rectangular	O
function	O
causes	O
ripples	O
in	O
the	O
frequency	O
domain	O
for	O
the	O
same	O
reason	O
as	O
a	O
brick-wall	O
low	B-Algorithm
pass	I-Algorithm
filter	I-Algorithm
(	O
rectangular	O
function	O
in	O
the	O
frequency	O
domain	O
)	O
causes	O
ripples	O
in	O
the	O
time	O
domain	O
,	O
in	O
each	O
case	O
the	O
Fourier	O
transform	O
of	O
the	O
rectangular	O
function	O
being	O
the	O
sinc	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
There	O
are	O
related	O
artifacts	B-Algorithm
caused	O
by	O
other	O
frequency	O
domain	O
effects	O
,	O
</s>
<s>
and	O
similar	O
artifacts	B-Algorithm
due	O
to	O
unrelated	O
causes	O
.	O
</s>
<s>
By	O
definition	O
,	O
ringing	B-Algorithm
occurs	O
when	O
a	O
non-oscillating	O
input	O
yields	O
an	O
oscillating	O
output	O
:	O
formally	O
,	O
when	O
an	O
input	O
signal	O
which	O
is	O
monotonic	O
on	O
an	O
interval	O
has	O
output	O
response	O
which	O
is	O
not	O
monotonic	O
.	O
</s>
<s>
This	O
occurs	O
most	O
severely	O
when	O
the	O
impulse	O
response	O
or	O
step	B-Algorithm
response	I-Algorithm
of	O
a	O
filter	O
has	O
oscillations	O
–	O
less	O
formally	O
,	O
if	O
for	O
a	O
spike	O
input	O
,	O
respectively	O
a	O
step	O
input	O
(	O
a	O
sharp	O
transition	O
)	O
,	O
the	O
output	O
has	O
bumps	O
.	O
</s>
<s>
Ringing	B-Algorithm
most	O
commonly	O
refers	O
to	O
step	O
ringing	B-Algorithm
,	O
and	O
that	O
will	O
be	O
the	O
focus	O
.	O
</s>
<s>
Ringing	B-Algorithm
is	O
closely	O
related	O
to	O
overshoot	B-Algorithm
and	O
undershoot	B-Algorithm
,	O
which	O
is	O
when	O
the	O
output	O
takes	O
on	O
values	O
higher	O
than	O
the	O
maximum	O
(	O
respectively	O
,	O
lower	O
than	O
the	O
minimum	O
)	O
input	O
value	O
:	O
one	O
can	O
have	O
one	O
without	O
the	O
other	O
,	O
but	O
in	O
important	O
cases	O
,	O
such	O
as	O
a	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
one	O
first	O
has	O
overshoot	B-Algorithm
,	O
then	O
the	O
response	O
bounces	O
back	O
below	O
the	O
steady-state	O
level	O
,	O
causing	O
the	O
first	O
ring	O
,	O
and	O
then	O
oscillates	O
back	O
and	O
forth	O
above	O
and	O
below	O
the	O
steady-state	O
level	O
.	O
</s>
<s>
Thus	O
overshoot	B-Algorithm
is	O
the	O
first	O
step	O
of	O
the	O
phenomenon	O
,	O
while	O
ringing	B-Algorithm
is	O
the	O
second	O
and	O
subsequent	O
steps	O
.	O
</s>
<s>
Due	O
to	O
this	O
close	O
connection	O
,	O
the	O
terms	O
are	O
often	O
conflated	O
,	O
with	O
"	O
ringing	B-Algorithm
"	O
referring	O
to	O
both	O
the	O
initial	O
overshoot	B-Algorithm
and	O
the	O
subsequent	O
rings	O
.	O
</s>
<s>
If	O
one	O
has	O
a	O
linear	O
time	O
invariant	O
(	O
LTI	O
)	O
filter	O
,	O
then	O
one	O
can	O
understand	O
the	O
filter	O
and	O
ringing	B-Algorithm
in	O
terms	O
of	O
the	O
impulse	O
response	O
(	O
the	O
time	O
domain	O
view	O
)	O
,	O
or	O
in	O
terms	O
of	O
its	O
Fourier	O
transform	O
,	O
the	O
frequency	O
response	O
(	O
the	O
frequency	O
domain	O
view	O
)	O
.	O
</s>
<s>
Ringing	B-Algorithm
is	O
a	O
time	O
domain	O
artifact	B-Algorithm
,	O
and	O
in	O
filter	O
design	O
is	O
traded	O
off	O
with	O
desired	O
frequency	O
domain	O
characteristics	O
:	O
the	O
desired	O
frequency	O
response	O
may	O
cause	O
ringing	B-Algorithm
,	O
while	O
reducing	O
or	O
eliminating	O
ringing	B-Algorithm
may	O
worsen	O
the	O
frequency	O
response	O
.	O
</s>
<s>
The	O
central	O
example	O
,	O
and	O
often	O
what	O
is	O
meant	O
by	O
"	O
ringing	B-Algorithm
artifacts	I-Algorithm
"	O
,	O
is	O
the	O
ideal	O
(	O
brick-wall	O
)	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
the	O
sinc	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
This	O
has	O
an	O
oscillatory	O
impulse	O
response	O
function	O
,	O
as	O
illustrated	O
above	O
,	O
and	O
the	O
step	B-Algorithm
response	I-Algorithm
–	O
its	O
integral	O
,	O
the	O
sine	O
integral	O
–	O
thus	O
also	O
features	O
oscillations	O
,	O
as	O
illustrated	O
at	O
right	O
.	O
</s>
<s>
These	O
ringing	B-Algorithm
artifacts	I-Algorithm
are	O
not	O
results	O
of	O
imperfect	O
implementation	O
or	O
windowing	O
:	O
the	O
ideal	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
while	O
possessing	O
the	O
desired	O
frequency	O
response	O
,	O
necessarily	O
causes	O
ringing	B-Algorithm
artifacts	I-Algorithm
in	O
the	O
time	O
domain	O
.	O
</s>
<s>
In	O
terms	O
of	O
impulse	O
response	O
,	O
the	O
correspondence	O
between	O
these	O
artifacts	B-Algorithm
and	O
the	O
behavior	O
of	O
the	O
function	O
is	O
as	O
follows	O
:	O
</s>
<s>
impulse	O
undershoot	B-Algorithm
is	O
equivalent	O
to	O
the	O
impulse	O
response	O
having	O
negative	O
values	O
,	O
</s>
<s>
impulse	O
ringing	B-Algorithm
(	O
ringing	B-Algorithm
near	O
a	O
point	O
)	O
is	O
precisely	O
equivalent	O
to	O
the	O
impulse	O
response	O
having	O
oscillations	O
,	O
which	O
is	O
equivalent	O
to	O
the	O
derivative	O
of	O
the	O
impulse	O
response	O
alternating	O
between	O
negative	O
and	O
positive	O
values	O
,	O
</s>
<s>
and	O
there	O
is	O
no	O
notion	O
of	O
impulse	O
overshoot	B-Algorithm
,	O
as	O
the	O
unit	O
impulse	O
is	O
assumed	O
to	O
have	O
infinite	O
height	O
(	O
and	O
integral	O
1	O
–	O
a	O
Dirac	O
delta	O
function	O
)	O
,	O
and	O
thus	O
cannot	O
be	O
overshot	O
.	O
</s>
<s>
Turning	O
to	O
step	B-Algorithm
response	I-Algorithm
,	O
</s>
<s>
the	O
step	B-Algorithm
response	I-Algorithm
is	O
the	O
integral	O
of	O
the	O
impulse	O
response	O
;	O
formally	O
,	O
the	O
value	O
of	O
the	O
step	B-Algorithm
response	I-Algorithm
at	O
time	O
a	O
is	O
the	O
integral	O
of	O
the	O
impulse	O
response	O
.	O
</s>
<s>
Thus	O
values	O
of	O
the	O
step	B-Algorithm
response	I-Algorithm
can	O
be	O
understood	O
in	O
terms	O
of	O
tail	O
integrals	O
of	O
the	O
impulse	O
response	O
.	O
</s>
<s>
Step	O
undershoot	B-Algorithm
is	O
equivalent	O
to	O
a	O
tail	O
integral	O
being	O
negative	O
,	O
in	O
which	O
case	O
the	O
magnitude	O
of	O
the	O
undershoot	B-Algorithm
is	O
the	O
value	O
of	O
the	O
tail	O
integral	O
.	O
</s>
<s>
Step	O
overshoot	B-Algorithm
is	O
equivalent	O
to	O
a	O
tail	O
integral	O
being	O
greater	O
than	O
1	O
,	O
in	O
which	O
case	O
the	O
magnitude	O
of	O
the	O
overshoot	B-Algorithm
is	O
the	O
amount	O
by	O
which	O
the	O
tail	O
integral	O
exceeds	O
1	O
–	O
or	O
equivalently	O
the	O
value	O
of	O
the	O
tail	O
in	O
the	O
other	O
direction	O
,	O
since	O
these	O
add	O
up	O
to	O
1	O
.	O
</s>
<s>
Step	O
ringing	B-Algorithm
is	O
equivalent	O
to	O
tail	O
integrals	O
alternating	O
between	O
increasing	O
and	O
decreasing	O
–	O
taking	O
derivatives	O
,	O
this	O
is	O
equivalent	O
to	O
the	O
impulse	O
response	O
alternating	O
between	O
positive	O
and	O
negative	O
values	O
.	O
</s>
<s>
Note	O
that	O
if	O
the	O
impulse	O
response	O
has	O
small	O
negative	O
lobes	O
and	O
larger	O
positive	O
lobes	O
,	O
then	O
it	O
will	O
exhibit	O
ringing	B-Algorithm
but	O
not	O
undershoot	B-Algorithm
or	O
overshoot	B-Algorithm
:	O
the	O
tail	O
integral	O
will	O
always	O
be	O
between	O
0	O
and	O
1	O
,	O
but	O
will	O
oscillate	O
down	O
at	O
each	O
negative	O
lobe	O
.	O
</s>
<s>
However	O
,	O
in	O
the	O
sinc	B-Algorithm
filter	I-Algorithm
,	O
the	O
lobes	O
monotonically	O
decrease	O
in	O
magnitude	O
and	O
alternate	O
in	O
sign	O
,	O
as	O
in	O
the	O
alternating	O
harmonic	O
series	O
,	O
and	O
thus	O
tail	O
integrals	O
alternate	O
in	O
sign	O
as	O
well	O
,	O
so	O
it	O
exhibits	O
overshoot	B-Algorithm
as	O
well	O
as	O
ringing	B-Algorithm
.	O
</s>
<s>
Conversely	O
,	O
if	O
the	O
impulse	O
response	O
is	O
always	O
nonnegative	O
,	O
so	O
it	O
has	O
no	O
negative	O
lobes	O
–	O
the	O
function	O
is	O
a	O
probability	O
distribution	O
–	O
then	O
the	O
step	B-Algorithm
response	I-Algorithm
will	O
exhibit	O
neither	O
ringing	B-Algorithm
nor	O
overshoot	B-Algorithm
or	O
undershoot	B-Algorithm
–	O
it	O
will	O
be	O
a	O
monotonic	O
function	O
growing	O
from	O
0	O
to	O
1	O
,	O
like	O
a	O
cumulative	O
distribution	O
function	O
.	O
</s>
<s>
The	O
frequency	O
domain	O
perspective	O
is	O
that	O
ringing	B-Algorithm
is	O
caused	O
by	O
the	O
sharp	O
cut-off	O
in	O
the	O
rectangular	O
passband	B-Algorithm
in	O
the	O
frequency	O
domain	O
,	O
and	O
thus	O
is	O
reduced	O
by	O
smoother	O
roll-off	B-Algorithm
,	O
as	O
discussed	O
below	O
.	O
</s>
<s>
Solutions	O
depend	O
on	O
the	O
parameters	O
of	O
the	O
problem	O
:	O
if	O
the	O
cause	O
is	O
a	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
one	O
may	O
choose	O
a	O
different	O
filter	O
design	O
,	O
which	O
reduces	O
artifacts	B-Algorithm
at	O
the	O
expense	O
of	O
worse	O
frequency	O
domain	O
performance	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
the	O
cause	O
is	O
a	O
band-limited	O
signal	O
,	O
as	O
in	O
JPEG	O
,	O
one	O
cannot	O
simply	O
replace	O
a	O
filter	O
,	O
and	O
ringing	B-Algorithm
artifacts	I-Algorithm
may	O
prove	O
hard	O
to	O
fix	O
–	O
they	O
are	O
present	O
in	O
JPEG	O
2000	O
and	O
many	O
audio	O
compression	O
codecs	O
(	O
in	O
the	O
form	O
of	O
pre-echo	O
)	O
,	O
as	O
discussed	O
in	O
the	O
examples	O
.	O
</s>
<s>
If	O
the	O
cause	O
is	O
the	O
use	O
of	O
a	O
brick-wall	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
one	O
may	O
replace	O
the	O
filter	O
with	O
one	O
that	O
reduces	O
the	O
time	O
domain	O
artifacts	B-Algorithm
,	O
at	O
the	O
cost	O
of	O
frequency	O
domain	O
performance	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
a	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
the	O
Gaussian	O
filter	O
is	O
non-negative	O
and	O
non-oscillatory	O
,	O
hence	O
causes	O
no	O
ringing	B-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
not	O
as	O
good	O
as	O
a	O
low-pass	B-Algorithm
filter	I-Algorithm
:	O
it	O
rolls	O
off	O
in	O
the	O
passband	B-Algorithm
,	O
and	O
leaks	O
in	O
the	O
stopband	O
:	O
in	O
image	O
terms	O
,	O
a	O
Gaussian	O
filter	O
"	O
blurs	O
"	O
the	O
signal	O
,	O
which	O
reflects	O
the	O
attenuation	O
of	O
desired	O
higher	O
frequency	O
signals	O
in	O
the	O
passband	B-Algorithm
.	O
</s>
<s>
A	O
general	O
solution	O
is	O
to	O
use	O
a	O
window	B-Language
function	I-Language
on	O
the	O
sinc	B-Algorithm
filter	I-Algorithm
,	O
which	O
cuts	O
off	O
or	O
reduces	O
the	O
negative	O
lobes	O
:	O
these	O
respectively	O
eliminate	O
and	O
reduce	O
overshoot	B-Algorithm
and	O
ringing	B-Algorithm
.	O
</s>
<s>
Note	O
that	O
truncating	O
some	O
but	O
not	O
all	O
of	O
the	O
lobes	O
eliminates	O
the	O
ringing	B-Algorithm
beyond	O
that	O
point	O
,	O
but	O
does	O
not	O
reduce	O
the	O
amplitude	O
of	O
the	O
ringing	B-Algorithm
that	O
is	O
not	O
truncated	O
(	O
because	O
this	O
is	O
determined	O
by	O
the	O
size	O
of	O
the	O
lobe	O
)	O
,	O
and	O
increases	O
the	O
magnitude	O
of	O
the	O
overshoot	B-Algorithm
if	O
the	O
last	O
non-cut	O
lobe	O
is	O
negative	O
,	O
since	O
the	O
magnitude	O
of	O
the	O
overshoot	B-Algorithm
is	O
the	O
integral	O
of	O
the	O
tail	O
,	O
which	O
is	O
no	O
longer	O
canceled	O
by	O
positive	O
lobes	O
.	O
</s>
<s>
Further	O
,	O
in	O
practical	O
implementations	O
one	O
at	O
least	O
truncates	O
sinc	B-Algorithm
,	O
otherwise	O
one	O
must	O
use	O
infinitely	O
many	O
data	O
points	O
(	O
or	O
rather	O
,	O
all	O
points	O
of	O
the	O
signal	O
)	O
to	O
compute	O
every	O
point	O
of	O
the	O
output	O
–	O
truncation	O
corresponds	O
to	O
a	O
rectangular	O
window	O
,	O
and	O
makes	O
the	O
filter	O
practically	O
implementable	O
,	O
but	O
the	O
frequency	O
response	O
is	O
no	O
longer	O
perfect	O
.	O
</s>
<s>
In	O
fact	O
,	O
if	O
one	O
takes	O
a	O
brick	O
wall	O
low-pass	B-Algorithm
filter	I-Algorithm
(	O
sinc	B-Algorithm
in	O
time	O
domain	O
,	O
rectangular	O
in	O
frequency	O
domain	O
)	O
and	O
truncates	O
it	O
(	O
multiplies	O
with	O
a	O
rectangular	O
function	O
in	O
the	O
time	O
domain	O
)	O
,	O
this	O
convolves	O
the	O
frequency	O
domain	O
with	O
sinc	B-Algorithm
(	O
Fourier	O
transform	O
of	O
the	O
rectangular	O
function	O
)	O
and	O
causes	O
ringing	B-Algorithm
in	O
the	O
frequency	O
domain	O
,	O
which	O
is	O
referred	O
to	O
as	O
ripple	O
.	O
</s>
<s>
In	O
symbols	O
,	O
The	O
frequency	O
ringing	B-Algorithm
in	O
the	O
stopband	O
is	O
also	O
referred	O
to	O
as	O
side	O
lobes	O
.	O
</s>
<s>
Flat	O
response	O
in	O
the	O
passband	B-Algorithm
is	O
desirable	O
,	O
so	O
one	O
windows	O
with	O
functions	O
whose	O
Fourier	O
transform	O
has	O
fewer	O
oscillations	O
,	O
so	O
the	O
frequency	O
domain	O
behavior	O
is	O
better	O
.	O
</s>
<s>
Multiplication	O
in	O
the	O
time	O
domain	O
corresponds	O
to	O
convolution	O
in	O
the	O
frequency	O
domain	O
,	O
so	O
multiplying	O
a	O
filter	O
by	O
a	O
window	B-Language
function	I-Language
corresponds	O
to	O
convolving	O
the	O
Fourier	O
transform	O
of	O
the	O
original	O
filter	O
by	O
the	O
Fourier	O
transform	O
of	O
the	O
window	O
,	O
which	O
has	O
a	O
smoothing	O
effect	O
–	O
thus	O
windowing	O
in	O
the	O
time	O
domain	O
corresponds	O
to	O
smoothing	O
in	O
the	O
frequency	O
domain	O
,	O
and	O
reduces	O
or	O
eliminates	O
overshoot	B-Algorithm
and	O
ringing	B-Algorithm
.	O
</s>
<s>
In	O
the	O
frequency	O
domain	O
,	O
the	O
cause	O
can	O
be	O
interpreted	O
as	O
due	O
to	O
the	O
sharp	O
(	O
brick-wall	O
)	O
cut-off	O
,	O
and	O
ringing	B-Algorithm
reduced	O
by	O
using	O
a	O
filter	O
with	O
smoother	O
roll-off	B-Algorithm
.	O
</s>
<s>
In	O
electronic	B-Architecture
filters	I-Architecture
,	O
the	O
trade-off	O
between	O
frequency	O
domain	O
response	O
and	O
time	O
domain	O
ringing	B-Algorithm
artifacts	I-Algorithm
is	O
well-illustrated	O
by	O
the	O
Butterworth	O
filter	O
:	O
the	O
frequency	O
response	O
of	O
a	O
Butterworth	O
filter	O
slopes	O
down	O
linearly	O
on	O
the	O
log	O
scale	O
,	O
with	O
a	O
first-order	O
filter	O
having	O
slope	O
of	O
−6	O
dB	O
per	O
octave	B-Language
,	O
a	O
second-order	O
filter	O
–	O
12dB	O
per	O
octave	B-Language
,	O
and	O
an	O
nth	O
order	O
filter	O
having	O
slope	O
of	O
dB	O
per	O
octave	B-Language
–	O
in	O
the	O
limit	O
,	O
this	O
approaches	O
a	O
brick-wall	O
filter	O
.	O
</s>
<s>
Thus	O
,	O
among	O
these	O
the	O
,	O
first-order	O
filter	O
rolls	O
off	O
slowest	O
,	O
and	O
hence	O
exhibits	O
the	O
fewest	O
time	O
domain	O
artifacts	B-Algorithm
,	O
but	O
leaks	O
the	O
most	O
in	O
the	O
stopband	O
,	O
while	O
as	O
order	O
increases	O
,	O
the	O
leakage	O
decreases	O
,	O
but	O
artifacts	B-Algorithm
increase	O
.	O
</s>
<s>
While	O
ringing	B-Algorithm
artifacts	I-Algorithm
are	O
generally	O
considered	O
undesirable	O
,	O
the	O
initial	O
overshoot	B-Algorithm
(	O
haloing	B-Algorithm
)	O
at	O
transitions	O
increases	O
acutance	B-Algorithm
(	O
apparent	O
sharpness	O
)	O
by	O
increasing	O
the	O
derivative	O
across	O
the	O
transition	O
,	O
and	O
thus	O
can	O
be	O
considered	O
as	O
an	O
enhancement	O
.	O
</s>
<s>
Another	O
artifact	B-Algorithm
is	O
overshoot	B-Algorithm
(	O
and	O
undershoot	B-Algorithm
)	O
,	O
which	O
manifests	O
itself	O
not	O
as	O
rings	O
,	O
but	O
as	O
an	O
increased	O
jump	O
at	O
the	O
transition	O
.	O
</s>
<s>
It	O
is	O
related	O
to	O
ringing	B-Algorithm
,	O
and	O
often	O
occurs	O
in	O
combination	O
with	O
it	O
.	O
</s>
<s>
Overshoot	B-Algorithm
and	O
undershoot	B-Algorithm
are	O
caused	O
by	O
a	O
negative	O
tail	O
–	O
in	O
the	O
sinc	B-Algorithm
,	O
the	O
integral	O
from	O
the	O
first	O
zero	O
to	O
infinity	O
,	O
including	O
the	O
first	O
negative	O
lobe	O
.	O
</s>
<s>
While	O
ringing	B-Algorithm
is	O
caused	O
by	O
a	O
following	O
positive	O
tail	O
–	O
in	O
sinc	B-Algorithm
,	O
the	O
integral	O
from	O
the	O
second	O
zero	O
to	O
infinity	O
,	O
including	O
the	O
first	O
non-central	O
positive	O
lobe	O
.	O
</s>
<s>
Thus	O
overshoot	B-Algorithm
is	O
necessary	O
for	O
ringing	B-Algorithm
,	O
but	O
can	O
occur	O
separately	O
:	O
for	O
example	O
,	O
the	O
2-lobed	O
Lanczos	B-Algorithm
filter	I-Algorithm
has	O
only	O
a	O
single	O
negative	O
lobe	O
on	O
each	O
side	O
,	O
with	O
no	O
following	O
positive	O
lobe	O
,	O
and	O
thus	O
exhibits	O
overshoot	B-Algorithm
but	O
no	O
ringing	B-Algorithm
,	O
while	O
the	O
3-lobed	O
Lanczos	B-Algorithm
filter	I-Algorithm
exhibits	O
both	O
overshoot	B-Algorithm
and	O
ringing	B-Algorithm
,	O
though	O
the	O
windowing	O
reduces	O
this	O
compared	O
to	O
the	O
sinc	B-Algorithm
filter	I-Algorithm
or	O
the	O
truncated	O
sinc	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
Similarly	O
,	O
the	O
convolution	O
kernel	O
used	O
in	O
bicubic	B-Algorithm
interpolation	I-Algorithm
is	O
similar	O
to	O
a	O
2-lobe	O
windowed	O
sinc	B-Algorithm
,	O
taking	O
on	O
negative	O
values	O
,	O
and	O
thus	O
produces	O
overshoot	B-Algorithm
artifacts	B-Algorithm
,	O
which	O
appear	O
as	O
halos	O
at	O
transitions	O
.	O
</s>
<s>
Following	O
from	O
overshoot	B-Algorithm
and	O
undershoot	B-Algorithm
is	O
clipping	O
.	O
</s>
<s>
If	O
the	O
signal	O
is	O
bounded	O
,	O
for	O
instance	O
an	O
8-bit	O
or	O
16-bit	O
integer	O
,	O
this	O
overshoot	B-Algorithm
and	O
undershoot	B-Algorithm
can	O
exceed	O
the	O
range	O
of	O
permissible	O
values	O
,	O
thus	O
causing	O
clipping	O
.	O
</s>
<s>
Strictly	O
speaking	O
,	O
the	O
clipping	O
is	O
caused	O
by	O
the	O
combination	O
of	O
overshoot	B-Algorithm
and	O
limited	O
numerical	O
accuracy	O
,	O
but	O
it	O
is	O
closely	O
associated	O
with	O
ringing	B-Algorithm
,	O
and	O
often	O
occurs	O
in	O
combination	O
with	O
it	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
clipping	O
can	O
be	O
exploited	O
to	O
conceal	O
ringing	B-Algorithm
in	O
images	O
.	O
</s>
<s>
Some	O
modern	O
JPEG	O
codecs	O
,	O
such	O
as	O
mozjpeg	O
and	O
ISO	O
libjpeg	O
,	O
use	O
such	O
a	O
trick	O
to	O
reduce	O
ringing	B-Algorithm
by	O
deliberately	O
causing	O
overshoots	B-Algorithm
in	O
the	O
IDCT	B-General_Concept
results	O
.	O
</s>
<s>
In	O
signal	O
processing	O
and	O
related	O
fields	O
,	O
the	O
general	O
phenomenon	O
of	O
time	O
domain	O
oscillation	O
is	O
called	O
ringing	B-Algorithm
,	O
while	O
frequency	O
domain	O
oscillations	O
are	O
generally	O
called	O
ripple	O
,	O
though	O
generally	O
not	O
"	O
rippling	O
"	O
.	O
</s>
<s>
A	O
key	O
source	O
of	O
ripple	O
in	O
digital	O
signal	O
processing	O
is	O
the	O
use	O
of	O
window	B-Language
functions	I-Language
:	O
if	O
one	O
takes	O
an	O
infinite	O
impulse	O
response	O
(	O
IIR	O
)	O
filter	O
,	O
such	O
as	O
the	O
sinc	B-Algorithm
filter	I-Algorithm
,	O
and	O
windows	O
it	O
to	O
make	O
it	O
have	O
finite	O
impulse	O
response	O
,	O
as	O
in	O
the	O
window	O
design	O
method	O
,	O
then	O
the	O
frequency	O
response	O
of	O
the	O
resulting	O
filter	O
is	O
the	O
convolution	O
of	O
the	O
frequency	O
response	O
of	O
the	O
IIR	O
filter	O
with	O
the	O
frequency	O
response	O
of	O
the	O
window	B-Language
function	I-Language
.	O
</s>
<s>
Notably	O
,	O
the	O
frequency	O
response	O
of	O
the	O
rectangular	O
filter	O
is	O
the	O
sinc	B-Algorithm
function	I-Algorithm
(	O
the	O
rectangular	O
function	O
and	O
the	O
sinc	B-Algorithm
function	I-Algorithm
are	O
Fourier	O
dual	O
to	O
each	O
other	O
)	O
,	O
and	O
thus	O
truncation	O
of	O
a	O
filter	O
in	O
the	O
time	O
domain	O
corresponds	O
to	O
multiplication	O
by	O
the	O
rectangular	O
filter	O
,	O
thus	O
convolution	O
by	O
the	O
sinc	B-Algorithm
filter	I-Algorithm
in	O
the	O
frequency	O
domain	O
,	O
causing	O
ripple	O
.	O
</s>
<s>
In	O
symbols	O
,	O
the	O
frequency	O
response	O
of	O
is	O
In	O
particular	O
,	O
truncating	O
the	O
sinc	B-Algorithm
function	I-Algorithm
itself	O
yields	O
in	O
the	O
time	O
domain	O
,	O
and	O
in	O
the	O
frequency	O
domain	O
,	O
so	O
just	O
as	O
low-pass	B-Algorithm
filtering	I-Algorithm
(	O
truncating	O
in	O
the	O
frequency	O
domain	O
)	O
causes	O
ringing	B-Algorithm
in	O
the	O
time	O
domain	O
,	O
truncating	O
in	O
the	O
time	O
domain	O
(	O
windowing	O
by	O
a	O
rectangular	O
filter	O
)	O
causes	O
ripple	O
in	O
the	O
frequency	O
domain	O
.	O
</s>
<s>
JPEG	O
compression	O
can	O
introduce	O
ringing	B-Algorithm
artifacts	I-Algorithm
at	O
sharp	O
transitions	O
,	O
which	O
are	O
particularly	O
visible	O
in	O
text	O
.	O
</s>
<s>
This	O
is	O
a	O
due	O
to	O
loss	O
of	O
high	O
frequency	O
components	O
,	O
as	O
in	O
step	B-Algorithm
response	I-Algorithm
ringing	B-Algorithm
.	O
</s>
<s>
JPEG	O
uses	O
8×8	O
blocks	O
,	O
on	O
which	O
the	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
DCT	B-General_Concept
)	O
is	O
performed	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
is	O
a	O
Fourier-related	B-Algorithm
transform	I-Algorithm
,	O
and	O
ringing	B-Algorithm
occurs	O
because	O
of	O
loss	O
of	O
high	O
frequency	O
components	O
or	O
loss	O
of	O
precision	O
in	O
high	O
frequency	O
components	O
.	O
</s>
<s>
They	O
can	O
also	O
occur	O
at	O
the	O
edge	O
of	O
an	O
image	O
:	O
since	O
JPEG	O
splits	O
images	O
into	O
8×8	O
blocks	O
,	O
if	O
an	O
image	O
is	O
not	O
an	O
integer	O
number	O
of	O
blocks	O
,	O
the	O
edge	O
cannot	O
easily	O
be	O
encoded	O
,	O
and	O
solutions	O
such	O
as	O
filling	O
with	O
a	O
black	O
border	O
create	O
a	O
sharp	O
transition	O
in	O
the	O
source	O
,	O
hence	O
ringing	B-Algorithm
artifacts	I-Algorithm
in	O
the	O
encoded	O
image	O
.	O
</s>
<s>
Ringing	B-Algorithm
also	O
occurs	O
in	O
the	O
wavelet-based	O
JPEG	O
2000	O
.	O
</s>
<s>
JPEG	O
and	O
JPEG	O
2000	O
have	O
other	O
artifacts	B-Algorithm
,	O
as	O
illustrated	O
above	O
,	O
such	O
as	O
blocking	O
(	O
"	O
jaggies	B-Algorithm
"	O
)	O
and	O
edge	O
busyness	O
(	O
"	O
mosquito	O
noise	O
"	O
)	O
,	O
though	O
these	O
are	O
due	O
to	O
specifics	O
of	O
the	O
formats	O
,	O
and	O
are	O
not	O
ringing	B-Algorithm
as	O
discussed	O
here	O
.	O
</s>
<s>
Image	O
Lossless	O
compression	O
Lossy	O
compression	O
Original	O
100px	O
100px	O
Processed	O
byCanny	O
edge	O
detector	O
,	O
highlighting	O
artifacts	B-Algorithm
.	O
</s>
<s>
In	O
audio	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
,	O
ringing	B-Algorithm
can	O
cause	O
echoes	O
to	O
occur	O
before	O
and	O
after	O
transients	O
,	O
such	O
as	O
the	O
impulsive	O
sound	O
from	O
percussion	O
instruments	O
,	O
such	O
as	O
cymbals	O
(	O
this	O
is	O
impulse	O
ringing	B-Algorithm
)	O
.	O
</s>
<s>
Thus	O
only	O
the	O
(	O
anti-causal	B-Algorithm
)	O
echo	O
before	O
the	O
transient	O
is	O
heard	O
,	O
and	O
the	O
phenomenon	O
is	O
called	O
pre-echo	O
.	O
</s>
<s>
This	O
phenomenon	O
occurs	O
as	O
a	O
compression	B-Algorithm
artifact	I-Algorithm
in	O
audio	O
compression	O
algorithms	O
that	O
use	O
Fourier-related	B-Algorithm
transforms	I-Algorithm
,	O
such	O
as	O
MP3	B-Application
,	O
AAC	B-Application
,	O
and	O
Vorbis	B-Algorithm
.	O
</s>
<s>
Other	O
phenomena	O
have	O
similar	O
symptoms	O
to	O
ringing	B-Algorithm
,	O
but	O
are	O
otherwise	O
distinct	O
in	O
their	O
causes	O
.	O
</s>
<s>
In	O
cases	O
where	O
these	O
cause	O
circular	O
artifacts	B-Algorithm
around	O
point	O
sources	O
,	O
these	O
may	O
be	O
referred	O
to	O
as	O
"	O
rings	O
"	O
due	O
to	O
the	O
round	O
shape	O
(	O
formally	O
,	O
an	O
annulus	O
)	O
,	O
which	O
is	O
unrelated	O
to	O
the	O
"	O
ringing	B-Algorithm
"	O
(	O
oscillatory	O
decay	O
)	O
frequency	O
phenomenon	O
discussed	O
on	O
this	O
page	O
.	O
</s>
<s>
Edge	B-Algorithm
enhancement	I-Algorithm
,	O
which	O
aims	O
to	O
increase	O
edges	O
,	O
may	O
cause	O
ringing	B-Algorithm
phenomena	O
,	O
particularly	O
under	O
repeated	O
application	O
,	O
such	O
as	O
by	O
a	O
DVD	O
player	O
followed	O
by	O
a	O
television	O
.	O
</s>
<s>
This	O
may	O
be	O
done	O
by	O
high-pass	O
filtering	O
,	O
rather	O
than	O
low-pass	B-Algorithm
filtering	I-Algorithm
.	O
</s>
<s>
Many	O
special	O
functions	O
exhibit	O
oscillatory	O
decay	O
,	O
and	O
thus	O
convolving	O
with	O
such	O
a	O
function	O
yields	O
ringing	B-Algorithm
in	O
the	O
output	O
;	O
one	O
may	O
consider	O
these	O
ringing	B-Algorithm
,	O
or	O
restrict	O
the	O
term	O
to	O
unintended	O
artifacts	B-Algorithm
in	O
frequency	O
domain	O
signal	O
processing	O
.	O
</s>
<s>
Fraunhofer	O
diffraction	O
yields	O
the	O
Airy	O
disk	O
as	O
point	O
spread	O
function	O
,	O
which	O
has	O
a	O
ringing	B-Algorithm
pattern	O
.	O
</s>
<s>
In	O
cameras	O
,	O
a	O
combination	O
of	O
defocus	O
and	O
spherical	O
aberration	O
can	O
yield	O
circular	O
artifacts	B-Algorithm
(	O
"	O
ring	O
"	O
patterns	O
)	O
.	O
</s>
<s>
However	O
,	O
the	O
pattern	O
of	O
these	O
artifacts	B-Algorithm
need	O
not	O
be	O
similar	O
to	O
ringing	B-Algorithm
(	O
as	O
discussed	O
on	O
this	O
page	O
)	O
–	O
they	O
may	O
exhibit	O
oscillatory	O
decay	O
(	O
circles	O
of	O
decreasing	O
intensity	O
)	O
,	O
or	O
other	O
intensity	O
patterns	O
,	O
such	O
as	O
a	O
single	O
bright	O
band	O
.	O
</s>
<s>
Though	O
this	O
is	O
not	O
ringing	B-Algorithm
,	O
it	O
can	O
be	O
interpreted	O
as	O
convolution	O
with	O
a	O
function	O
,	O
which	O
is	O
1	O
at	O
the	O
origin	O
and	O
ε	O
(	O
the	O
intensity	O
of	O
the	O
ghost	O
)	O
at	O
some	O
distance	O
,	O
which	O
is	O
formally	O
similar	O
to	O
the	O
above	O
functions	O
(	O
a	O
single	O
discrete	O
peak	O
,	O
rather	O
than	O
continuous	O
oscillation	O
)	O
.	O
</s>
<s>
Visual	O
illusions	O
can	O
occur	O
at	O
transitions	O
,	O
as	O
in	O
Mach	O
bands	O
,	O
which	O
perceptually	O
exhibit	O
a	O
similar	O
undershoot/overshoot	O
to	O
the	O
Gibbs	O
phenomenon	O
.	O
</s>
