<s>
However	O
,	O
filters	O
do	O
not	O
exclusively	O
act	O
in	O
the	O
frequency	O
domain	O
;	O
especially	O
in	O
the	O
field	O
of	O
image	B-Algorithm
processing	I-Algorithm
many	O
other	O
targets	O
for	O
filtering	O
exist	O
.	O
</s>
<s>
Filters	O
are	O
widely	O
used	O
in	O
electronics	O
and	O
telecommunication	O
,	O
in	O
radio	O
,	O
television	O
,	O
audio	O
recording	O
,	O
radar	B-Application
,	O
control	O
systems	O
,	O
music	O
synthesis	O
,	O
image	B-Algorithm
processing	I-Algorithm
,	O
computer	O
graphics	O
,	O
and	O
structural	B-Algorithm
dynamics	I-Algorithm
.	O
</s>
<s>
time-variant	O
or	O
time-invariant	B-Algorithm
,	O
also	O
known	O
as	O
shift	O
invariance	O
.	O
</s>
<s>
Filters	O
processing	O
time-domain	O
signals	O
in	O
real	B-General_Concept
time	I-General_Concept
must	O
be	O
causal	O
,	O
but	O
not	O
filters	O
acting	O
on	O
spatial	O
domain	O
signals	O
or	O
deferred-time	O
processing	O
of	O
time-domain	O
signals	O
.	O
</s>
<s>
The	O
difference	O
between	O
these	O
filter	O
families	O
is	O
that	O
they	O
all	O
use	O
a	O
different	O
polynomial	O
function	O
to	O
approximate	O
to	O
the	O
ideal	B-Algorithm
filter	I-Algorithm
response	O
.	O
</s>
<s>
The	O
frequency	O
response	O
can	O
be	O
classified	O
into	O
a	O
number	O
of	O
different	O
bandforms	O
describing	O
which	O
frequency	O
bands	O
the	O
filter	O
passes	O
(	O
the	O
passband	B-Algorithm
)	O
and	O
which	O
it	O
rejects	O
(	O
the	O
stopband	O
)	O
:	O
</s>
<s>
Low-pass	B-Algorithm
filter	I-Algorithm
low	O
frequencies	O
are	O
passed	O
,	O
high	O
frequencies	O
are	O
attenuated	O
.	O
</s>
<s>
Comb	B-Algorithm
filter	I-Algorithm
has	O
multiple	O
regularly	O
spaced	O
narrow	O
passbands	B-Algorithm
giving	O
the	O
bandform	O
the	O
appearance	O
of	O
a	O
comb	O
.	O
</s>
<s>
Roll-off	B-Algorithm
is	O
the	O
rate	O
at	O
which	O
attenuation	O
increases	O
beyond	O
the	O
cut-off	O
frequency	O
.	O
</s>
<s>
Transition	B-Algorithm
band	I-Algorithm
,	O
the	O
(	O
usually	O
narrow	O
)	O
band	O
of	O
frequencies	O
between	O
a	O
passband	B-Algorithm
and	O
stopband	O
.	O
</s>
<s>
Ripple	O
is	O
the	O
variation	O
of	O
the	O
filter	O
's	O
insertion	O
loss	O
in	O
the	O
passband	B-Algorithm
.	O
</s>
<s>
Increasing	O
order	O
increases	O
roll-off	B-Algorithm
and	O
brings	O
the	O
filter	O
closer	O
to	O
the	O
ideal	O
response	O
.	O
</s>
<s>
Many	O
telecommunication	O
systems	O
use	O
frequency-division	B-Protocol
multiplexing	I-Protocol
,	O
where	O
the	O
system	O
designers	O
divide	O
a	O
wide	O
frequency	O
band	O
into	O
many	O
narrower	O
frequency	O
bands	O
called	O
"	O
slots	O
"	O
or	O
"	O
channels	O
"	O
,	O
and	O
each	O
stream	O
of	O
information	O
is	O
allocated	O
one	O
of	O
those	O
channels	O
.	O
</s>
<s>
Multilevel	O
and	O
multiphase	B-General_Concept
digital	O
modulation	O
systems	O
require	O
filters	O
that	O
have	O
flat	O
phase	O
delay	O
—	O
are	O
linear	O
phase	O
in	O
the	O
passband	B-Algorithm
—	O
to	O
preserve	O
pulse	O
integrity	O
in	O
the	O
time	O
domain	O
,	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
analog	O
audio	O
systems	O
using	O
analog	B-Protocol
transmission	I-Protocol
can	O
tolerate	O
much	O
larger	O
ripples	O
in	O
phase	O
delay	O
,	O
and	O
so	O
designers	O
of	O
such	O
systems	O
often	O
deliberately	O
sacrifice	O
linear	O
phase	O
to	O
get	O
filters	O
that	O
are	O
better	O
in	O
other	O
ways	O
—	O
better	O
stop-band	O
rejection	O
,	O
lower	O
passband	B-Algorithm
amplitude	O
ripple	O
,	O
lower	O
cost	O
,	O
etc	O
.	O
</s>
<s>
Likewise	O
,	O
there	O
are	O
corresponding	O
components	O
in	O
distributed-element	B-Algorithm
filters	I-Algorithm
.	O
</s>
<s>
Electronic	B-Architecture
filters	I-Architecture
were	O
originally	O
entirely	O
passive	O
consisting	O
of	O
resistance	O
,	O
inductance	O
and	O
capacitance	O
.	O
</s>
<s>
In	O
the	O
vast	O
majority	O
of	O
cases	O
they	O
are	O
used	O
to	O
process	O
an	O
electronic	O
signal	O
and	O
transducers	B-Algorithm
are	O
provided	O
to	O
convert	O
this	O
to	O
and	O
from	O
a	O
mechanical	O
vibration	O
.	O
</s>
<s>
Distributed-element	B-Algorithm
filters	I-Algorithm
are	O
constructed	O
out	O
of	O
components	O
made	O
from	O
small	O
pieces	O
of	O
transmission	O
line	O
or	O
other	O
distributed	O
elements	O
.	O
</s>
<s>
There	O
are	O
structures	O
in	O
distributed-element	B-Algorithm
filters	I-Algorithm
that	O
directly	O
correspond	O
to	O
the	O
lumped	O
elements	O
of	O
electronic	B-Architecture
filters	I-Architecture
,	O
and	O
others	O
that	O
are	O
unique	O
to	O
this	O
class	O
of	O
technology	O
.	O
</s>
<s>
Waveguide	B-Algorithm
filters	I-Algorithm
consist	O
of	O
waveguide	O
components	O
or	O
components	O
inserted	O
in	O
the	O
waveguide	O
.	O
</s>
<s>
Waveguides	O
are	O
a	O
class	O
of	O
transmission	O
line	O
and	O
many	O
structures	O
of	O
distributed-element	B-Algorithm
filters	I-Algorithm
,	O
for	O
instance	O
the	O
stub	O
,	O
can	O
also	O
be	O
implemented	O
in	O
waveguides	O
.	O
</s>
<s>
With	O
the	O
rise	O
of	O
optical	B-Architecture
fiber	I-Architecture
technology	O
,	O
however	O
,	O
optical	O
filters	O
increasingly	O
find	O
signal	O
processing	O
applications	O
and	O
signal	O
processing	O
filter	O
terminology	O
,	O
such	O
as	O
longpass	O
and	O
shortpass	O
,	O
are	O
entering	O
the	O
field	O
.	O
</s>
<s>
Transversal	O
filter	O
,	O
or	O
delay	O
line	B-Algorithm
filter	I-Algorithm
,	O
works	O
by	O
summing	O
copies	O
of	O
the	O
input	O
after	O
various	O
time	O
delays	O
.	O
</s>
<s>
This	O
can	O
be	O
implemented	O
with	O
various	O
technologies	O
including	O
analog	O
delay	O
lines	O
,	O
active	O
circuitry	O
,	O
CCD	B-Algorithm
delay	O
lines	O
,	O
or	O
entirely	O
in	O
the	O
digital	O
domain	O
.	O
</s>
<s>
Digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
allows	O
the	O
inexpensive	O
construction	O
of	O
a	O
wide	O
variety	O
of	O
filters	O
.	O
</s>
<s>
A	O
computer	O
program	O
running	O
on	O
a	O
CPU	B-General_Concept
or	O
a	O
specialized	O
DSP	B-Architecture
(	O
or	O
less	O
often	O
running	O
on	O
a	O
hardware	O
implementation	O
of	O
the	O
algorithm	O
)	O
calculates	O
an	O
output	O
number	O
stream	O
.	O
</s>
<s>
Due	O
to	O
the	O
sampling	O
involved	O
,	O
the	O
input	O
signal	O
must	O
be	O
of	O
limited	O
frequency	O
content	O
or	O
aliasing	B-Error_Name
will	O
occur	O
.	O
</s>
<s>
In	O
the	O
late	O
1930s	O
,	O
engineers	O
realized	O
that	O
small	O
mechanical	O
systems	O
made	O
of	O
rigid	O
materials	O
such	O
as	O
quartz	B-General_Concept
would	O
acoustically	O
resonate	O
at	O
radio	O
frequencies	O
,	O
i.e.	O
</s>
<s>
Some	O
early	O
resonators	O
were	O
made	O
of	O
steel	O
,	O
but	O
quartz	B-General_Concept
quickly	O
became	O
favored	O
.	O
</s>
<s>
The	O
biggest	O
advantage	O
of	O
quartz	B-General_Concept
is	O
that	O
it	O
is	O
piezoelectric	O
.	O
</s>
<s>
This	O
means	O
that	O
quartz	B-General_Concept
resonators	O
can	O
directly	O
convert	O
their	O
own	O
mechanical	O
motion	O
into	O
electrical	O
signals	O
.	O
</s>
<s>
Quartz	B-General_Concept
also	O
has	O
a	O
very	O
low	O
coefficient	O
of	O
thermal	O
expansion	O
which	O
means	O
that	O
quartz	B-General_Concept
resonators	O
can	O
produce	O
stable	O
frequencies	O
over	O
a	O
wide	O
temperature	O
range	O
.	O
</s>
<s>
Quartz	B-General_Concept
crystal	O
filters	O
have	O
much	O
higher	O
quality	O
factors	O
than	O
LCR	O
filters	O
.	O
</s>
<s>
A	O
large	O
number	O
of	O
crystals	O
can	O
be	O
collapsed	O
into	O
a	O
single	O
component	O
,	O
by	O
mounting	O
comb-shaped	O
evaporations	O
of	O
metal	O
on	O
a	O
quartz	B-General_Concept
crystal	O
.	O
</s>
<s>
In	O
this	O
scheme	O
,	O
a	O
"	O
tapped	O
delay	O
line	O
"	O
reinforces	O
the	O
desired	O
frequencies	O
as	O
the	O
sound	O
waves	O
flow	O
across	O
the	O
surface	O
of	O
the	O
quartz	B-General_Concept
crystal	O
.	O
</s>
<s>
SAW	B-Algorithm
filters	I-Algorithm
are	O
limited	O
to	O
frequencies	O
up	O
to	O
3GHz	O
.	O
</s>
<s>
BAW	O
filters	O
typically	O
operate	O
at	O
frequencies	O
from	O
around	O
2	O
to	O
around	O
16GHz	O
,	O
and	O
may	O
be	O
smaller	O
or	O
thinner	O
than	O
equivalent	O
SAW	B-Algorithm
filters	I-Algorithm
.	O
</s>
<s>
The	O
garnet	O
sits	O
on	O
a	O
strip	O
of	O
metal	O
driven	O
by	O
a	O
transistor	B-Application
,	O
and	O
a	O
small	O
loop	O
antenna	O
touches	O
the	O
top	O
of	O
the	O
sphere	O
.	O
</s>
<s>
Another	O
method	O
,	O
used	O
at	O
high	O
,	O
fixed	O
frequencies	O
with	O
very	O
weak	O
radio	O
signals	O
,	O
is	O
to	O
use	O
a	O
ruby	B-Language
maser	O
tapped	O
delay	O
line	O
.	O
</s>
<s>
Distributed-element	B-Algorithm
filters	I-Algorithm
do	O
not	O
,	O
in	O
general	O
,	O
have	O
rational-function	O
transfer	O
functions	O
,	O
but	O
can	O
approximate	O
them	O
.	O
</s>
<s>
An	O
alternative	O
to	O
transfer	O
functions	O
is	O
to	O
give	O
the	O
behavior	O
of	O
the	O
filter	O
as	O
a	O
convolution	B-Language
of	O
the	O
time-domain	O
input	O
with	O
the	O
filter	O
's	O
impulse	O
response	O
.	O
</s>
<s>
The	O
convolution	B-Language
theorem	O
,	O
which	O
holds	O
for	O
Laplace	O
transforms	O
,	O
guarantees	O
equivalence	O
with	O
transfer	O
functions	O
.	O
</s>
<s>
An	O
ideal	B-Algorithm
filter	I-Algorithm
has	O
full	O
transmission	O
in	O
the	O
pass	B-Algorithm
band	I-Algorithm
,	O
complete	O
attenuation	O
in	O
the	O
stop	O
band	O
,	O
and	O
an	O
abrupt	O
transition	O
between	O
the	O
two	O
bands	O
,	O
but	O
this	O
filter	O
has	O
infinite	O
order	O
(	O
i.e.	O
,	O
the	O
response	O
cannot	O
be	O
expressed	O
as	O
a	O
linear	O
differential	O
equation	O
with	O
a	O
finite	O
sum	O
)	O
and	O
infinite	O
latency	O
(	O
i.e.	O
,	O
its	O
compact	O
support	O
in	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
forces	O
its	O
time	O
response	O
to	O
be	O
ever	O
lasting	O
)	O
.	O
</s>
<s>
The	O
filters	O
in	O
this	O
illustration	O
are	O
all	O
fifth-order	O
low-pass	B-Algorithm
filters	I-Algorithm
.	O
</s>
<s>
Any	O
family	O
can	O
be	O
used	O
to	O
implement	O
a	O
particular	O
bandform	O
of	O
which	O
frequencies	O
are	O
transmitted	O
,	O
and	O
which	O
,	O
outside	O
the	O
passband	B-Algorithm
,	O
are	O
more	O
or	O
less	O
attenuated	O
.	O
</s>
<s>
A	O
particular	O
bandform	O
of	O
filter	O
can	O
be	O
obtained	O
by	O
transformation	B-Algorithm
of	O
a	O
prototype	O
filter	O
of	O
that	O
family	O
.	O
</s>
<s>
When	O
implemented	O
in	O
a	O
distributed-element	O
format	O
,	O
these	O
devices	O
can	O
take	O
the	O
form	O
of	O
a	O
distributed-element	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
The	O
inverse	O
is	O
also	O
true	O
:	O
distributed-element	B-Algorithm
filters	I-Algorithm
can	O
take	O
the	O
form	O
of	O
coupled	O
lines	O
.	O
</s>
