<s>
In	O
signal	O
processing	O
,	O
a	O
sinc	B-Algorithm
filter	I-Algorithm
is	O
an	O
idealized	O
filter	O
that	O
removes	O
all	O
frequency	O
components	O
above	O
a	O
given	O
cutoff	O
frequency	O
,	O
without	O
affecting	O
lower	O
frequencies	O
,	O
and	O
has	O
linear	O
phase	O
response	O
.	O
</s>
<s>
The	O
filter	O
's	O
impulse	O
response	O
is	O
a	O
sinc	B-Algorithm
function	I-Algorithm
in	O
the	O
time	O
domain	O
and	O
its	O
frequency	O
response	O
is	O
a	O
rectangular	O
function	O
.	O
</s>
<s>
It	O
is	O
an	O
"	O
ideal	O
"	O
low-pass	B-Algorithm
filter	I-Algorithm
in	O
the	O
frequency	O
sense	O
,	O
perfectly	O
passing	O
low	O
frequencies	O
,	O
perfectly	O
cutting	O
high	O
frequencies	O
;	O
and	O
thus	O
may	O
be	O
considered	O
to	O
be	O
a	O
brick-wall	O
filter	O
.	O
</s>
<s>
Real-time	O
filters	O
can	O
only	O
approximate	O
this	O
ideal	O
,	O
since	O
an	O
ideal	O
sinc	B-Algorithm
filter	I-Algorithm
(	O
a.k.a.	O
</s>
<s>
rectangular	O
filter	O
)	O
is	O
non-causal	B-Algorithm
and	O
has	O
an	O
infinite	O
delay	O
,	O
but	O
it	O
is	O
commonly	O
found	O
in	O
conceptual	O
demonstrations	O
or	O
proofs	O
,	O
such	O
as	O
the	O
sampling	O
theorem	O
and	O
the	O
Whittaker	O
–	O
Shannon	O
interpolation	O
formula	O
.	O
</s>
<s>
where	O
sinc	B-Algorithm
is	O
the	O
normalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
As	O
the	O
sinc	B-Algorithm
filter	I-Algorithm
has	O
infinite	O
impulse	O
response	O
in	O
both	O
positive	O
and	O
negative	O
time	O
directions	O
,	O
it	O
must	O
be	O
approximated	O
for	O
real-world	O
(	O
non-abstract	O
)	O
applications	O
;	O
a	O
windowed	B-Language
sinc	B-Algorithm
filter	I-Algorithm
is	O
often	O
used	O
instead	O
.	O
</s>
<s>
Windowing	O
and	O
truncating	O
a	O
sinc	B-Algorithm
filter	I-Algorithm
kernel	O
in	O
order	O
to	O
use	O
it	O
on	O
any	O
practical	O
real	O
world	O
data	O
set	O
reduces	O
its	O
ideal	O
properties	O
.	O
</s>
<s>
An	O
idealized	O
electronic	B-Architecture
filter	I-Architecture
with	O
full	O
transmission	O
in	O
the	O
pass	O
band	O
,	O
complete	O
attenuation	O
in	O
the	O
stop	O
band	O
,	O
and	O
abrupt	O
transitions	O
is	O
known	O
colloquially	O
as	O
a	O
"	O
brick-wall	O
filter	O
"	O
(	O
in	O
reference	O
to	O
the	O
shape	O
of	O
the	O
transfer	O
function	O
)	O
.	O
</s>
<s>
The	O
sinc	B-Algorithm
filter	I-Algorithm
is	O
a	O
brick-wall	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
from	O
which	O
brick-wall	O
band-pass	O
filters	O
and	O
high-pass	O
filters	O
are	O
easily	O
constructed	O
.	O
</s>
<s>
The	O
lowpass	B-Algorithm
filter	I-Algorithm
with	O
brick-wall	O
cutoff	O
at	O
frequency	O
BL	O
has	O
impulse	O
response	O
and	O
transfer	O
function	O
given	O
by	O
:	O
</s>
<s>
The	O
band-pass	O
filter	O
with	O
lower	O
band	O
edge	O
BL	O
and	O
upper	O
band	O
edge	O
BH	O
is	O
just	O
the	O
difference	O
of	O
two	O
such	O
sinc	B-Algorithm
filters	I-Algorithm
(	O
since	O
the	O
filters	O
are	O
zero	O
phase	O
,	O
their	O
magnitude	O
responses	O
subtract	O
directly	O
)	O
:	O
</s>
<s>
The	O
high-pass	O
filter	O
with	O
lower	O
band	O
edge	O
BH	O
is	O
just	O
a	O
transparent	O
filter	O
minus	O
a	O
sinc	B-Algorithm
filter	I-Algorithm
,	O
which	O
makes	O
it	O
clear	O
that	O
the	O
Dirac	O
delta	O
function	O
is	O
the	O
limit	O
of	O
a	O
narrow-in-time	O
sinc	B-Algorithm
filter	I-Algorithm
:	O
</s>
<s>
The	O
name	O
"	O
sinc	B-Algorithm
filter	I-Algorithm
"	O
is	O
applied	O
also	O
to	O
the	O
filter	O
shape	O
that	O
is	O
rectangular	O
in	O
time	O
and	O
a	O
sinc	B-Algorithm
function	I-Algorithm
in	O
frequency	O
,	O
as	O
opposed	O
to	O
the	O
ideal	O
low-pass	B-Algorithm
sinc	B-Algorithm
filter	I-Algorithm
,	O
which	O
is	O
sinc	B-Algorithm
in	O
time	O
and	O
rectangular	O
in	O
frequency	O
.	O
</s>
<s>
In	O
case	O
of	O
confusion	O
,	O
one	O
may	O
refer	O
to	O
these	O
as	O
sinc-in-frequency	O
and	O
sinc-in-time	O
,	O
according	O
to	O
which	O
domain	O
the	O
filter	O
is	O
sinc	B-Algorithm
in	O
.	O
</s>
<s>
Sinc-in-frequency	O
Cascaded	O
integrator	O
–	O
comb	O
(	O
CIC	O
)	O
filters	O
,	O
among	O
many	O
other	O
applications	O
,	O
are	O
almost	O
universally	O
used	O
for	O
decimating	B-Algorithm
delta-sigma	O
ADCs	O
,	O
as	O
they	O
are	O
easy	O
to	O
implement	O
and	O
nearly	O
optimal	O
for	O
this	O
use	O
.	O
</s>
<s>
The	O
simplest	O
implementation	O
of	O
a	O
Sinc-in-frequency	O
filter	O
is	O
a	O
group-averaging	O
filter	O
,	O
also	O
known	O
as	O
accumulate-and-dump	O
filter	O
.	O
</s>
<s>
Thus	O
,	O
the	O
decimation	B-Algorithm
factor	O
of	O
this	O
filter	O
is	O
N	O
.	O
</s>
<s>
It	O
can	O
be	O
modeled	O
as	O
a	O
FIR	O
filter	O
with	O
all	O
N	O
coefficients	O
equal	O
,	O
followed	O
by	O
a	O
N-time	O
downsampling	B-Algorithm
block	O
.	O
</s>
<s>
The	O
sinc	B-Algorithm
filter	I-Algorithm
is	O
not	O
bounded-input	O
–	O
bounded-output	O
(	O
BIBO	O
)	O
stable	O
.	O
</s>
<s>
That	O
is	O
,	O
a	O
bounded	O
input	O
can	O
produce	O
an	O
unbounded	O
output	O
,	O
because	O
the	O
integral	O
of	O
the	O
absolute	O
value	O
of	O
the	O
sinc	B-Algorithm
function	I-Algorithm
is	O
infinite	O
.	O
</s>
<s>
A	O
bounded	O
input	O
that	O
produces	O
an	O
unbounded	O
output	O
is	O
sgn(sinc(t )	O
)	O
.	O
</s>
