<s>
Circular	B-Algorithm
convolution	I-Algorithm
,	O
also	O
known	O
as	O
cyclic	B-Algorithm
convolution	I-Algorithm
,	O
is	O
a	O
special	O
case	O
of	O
periodic	B-Algorithm
convolution	I-Algorithm
,	O
which	O
is	O
the	O
convolution	B-Language
of	O
two	O
periodic	O
functions	O
that	O
have	O
the	O
same	O
period	O
.	O
</s>
<s>
Periodic	B-Algorithm
convolution	I-Algorithm
arises	O
,	O
for	O
example	O
,	O
in	O
the	O
context	O
of	O
the	O
discrete-time	O
Fourier	O
transform	O
(	O
DTFT	O
)	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
DTFT	O
of	O
the	O
product	O
of	O
two	O
discrete	O
sequences	O
is	O
the	O
periodic	B-Algorithm
convolution	I-Algorithm
of	O
the	O
DTFTs	O
of	O
the	O
individual	O
sequences	O
.	O
</s>
<s>
And	O
each	O
DTFT	O
is	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
a	O
continuous	O
Fourier	O
transform	O
function	O
(	O
see	O
)	O
.	O
</s>
<s>
Although	O
DTFTs	O
are	O
usually	O
continuous	O
functions	O
of	O
frequency	O
,	O
the	O
concepts	O
of	O
periodic	O
and	O
circular	B-Algorithm
convolution	I-Algorithm
are	O
also	O
directly	O
applicable	O
to	O
discrete	O
sequences	O
of	O
data	O
.	O
</s>
<s>
In	O
that	O
context	O
,	O
circular	B-Algorithm
convolution	I-Algorithm
plays	O
an	O
important	O
role	O
in	O
maximizing	O
the	O
efficiency	O
of	O
a	O
certain	O
kind	O
of	O
common	O
filtering	O
operation	O
.	O
</s>
<s>
The	O
periodic	B-Algorithm
convolution	I-Algorithm
of	O
two	O
T-periodic	O
functions	O
,	O
and	O
can	O
be	O
defined	O
as	O
:	O
</s>
<s>
An	O
alternative	O
definition	O
,	O
in	O
terms	O
of	O
the	O
notation	O
of	O
normal	O
linear	O
or	O
aperiodic	O
convolution	B-Language
,	O
follows	O
from	O
expressing	O
and	O
as	O
periodic	B-Algorithm
summations	I-Algorithm
of	O
aperiodic	O
components	O
and	O
,	O
i.e.	O
</s>
<s>
Both	O
forms	O
can	O
be	O
called	O
periodic	B-Algorithm
convolution	I-Algorithm
.	O
</s>
<s>
The	O
term	O
circular	B-Algorithm
convolution	I-Algorithm
arises	O
from	O
the	O
important	O
special	O
case	O
of	O
constraining	O
the	O
non-zero	O
portions	O
of	O
both	O
and	O
to	O
the	O
interval	O
Then	O
the	O
periodic	B-Algorithm
summation	I-Algorithm
becomes	O
a	O
periodic	O
extension	O
,	O
which	O
can	O
also	O
be	O
expressed	O
as	O
a	O
circular	O
function	O
:	O
</s>
<s>
Similarly	O
,	O
for	O
discrete	O
sequences	O
,	O
and	O
a	O
parameter	O
N	O
,	O
we	O
can	O
write	O
a	O
circular	B-Algorithm
convolution	I-Algorithm
of	O
aperiodic	O
functions	O
and	O
as	O
:	O
</s>
<s>
Then	O
many	O
of	O
the	O
values	O
of	O
the	O
circular	B-Algorithm
convolution	I-Algorithm
are	O
identical	O
to	O
values	O
of	O
x∗h	O
,	O
which	O
is	O
actually	O
the	O
desired	O
result	O
when	O
the	O
h	O
sequence	O
is	O
a	O
finite	O
impulse	O
response	O
(	O
FIR	O
)	O
filter	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
circular	B-Algorithm
convolution	I-Algorithm
is	O
very	O
efficient	O
to	O
compute	O
,	O
using	O
a	O
fast	O
Fourier	O
transform	O
(	O
FFT	O
)	O
algorithm	O
and	O
the	O
circular	B-Algorithm
convolution	I-Algorithm
theorem	O
.	O
</s>
<s>
We	O
describe	O
it	O
first	O
in	O
terms	O
of	O
normal	O
or	O
linear	O
convolution	B-Language
.	O
</s>
<s>
When	O
a	O
normal	O
convolution	B-Language
is	O
performed	O
on	O
each	O
block	O
,	O
there	O
are	O
start-up	O
and	O
decay	O
transients	O
at	O
the	O
block	O
edges	O
,	O
due	O
to	O
the	O
filter	O
latency	O
(	O
200-samples	O
)	O
.	O
</s>
<s>
Only	O
824	O
of	O
the	O
convolution	B-Language
outputs	O
are	O
unaffected	O
by	O
edge	O
effects	O
.	O
</s>
<s>
This	O
method	O
is	O
referred	O
to	O
as	O
overlap-save	B-Algorithm
,	O
although	O
the	O
method	O
we	O
describe	O
next	O
requires	O
a	O
similar	O
"	O
save	O
"	O
with	O
the	O
output	O
samples	O
.	O
</s>
<s>
When	O
an	O
FFT	O
is	O
used	O
to	O
compute	O
the	O
824	O
unaffected	O
DFT	O
samples	O
,	O
we	O
do	O
n't	O
have	O
the	O
option	O
of	O
not	O
computing	O
the	O
affected	O
samples	O
,	O
but	O
the	O
leading	O
and	O
trailing	O
edge-effects	O
are	O
overlapped	O
and	O
added	O
because	O
of	O
circular	B-Algorithm
convolution	I-Algorithm
.	O
</s>
<s>
To	O
illustrate	O
this	O
,	O
the	O
fourth	O
frame	O
of	O
the	O
figure	O
at	O
right	O
depicts	O
a	O
block	O
that	O
has	O
been	O
periodically	O
(	O
or	O
"	O
circularly	O
"	O
)	O
extended	O
,	O
and	O
the	O
fifth	O
frame	O
depicts	O
the	O
individual	O
components	O
of	O
a	O
linear	O
convolution	B-Language
performed	O
on	O
the	O
entire	O
sequence	O
.	O
</s>
<s>
This	O
method	O
is	O
known	O
as	O
overlap-add	B-Algorithm
.	O
</s>
<s>
Both	O
methods	O
advance	O
only	O
824	O
samples	O
per	O
1024-point	O
IFFT	O
,	O
but	O
overlap-save	B-Algorithm
avoids	O
the	O
initial	O
zero-padding	O
and	O
final	O
addition	O
.	O
</s>
