<s>
In	O
mathematics	O
,	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	O
)	O
converts	O
a	O
finite	O
sequence	O
of	O
equally-spaced	O
samples	B-Algorithm
of	O
a	O
function	O
into	O
a	O
same-length	O
sequence	O
of	O
equally-spaced	O
samples	B-Algorithm
of	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
,	O
which	O
is	O
a	O
complex-valued	O
function	O
of	O
frequency	O
.	O
</s>
<s>
An	O
inverse	O
DFT	O
(	O
IDFT	O
)	O
is	O
a	O
Fourier	O
series	O
,	O
using	O
the	O
DTFT	O
samples	B-Algorithm
as	O
coefficients	O
of	O
complex	O
sinusoids	O
at	O
the	O
corresponding	O
DTFT	O
frequencies	O
.	O
</s>
<s>
If	O
the	O
original	O
sequence	O
spans	O
all	O
the	O
non-zero	O
values	O
of	O
a	O
function	O
,	O
its	O
DTFT	O
is	O
continuous	O
(	O
and	O
periodic	O
)	O
,	O
and	O
the	O
DFT	O
provides	O
discrete	O
samples	B-Algorithm
of	O
one	O
cycle	O
.	O
</s>
<s>
In	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
,	O
the	O
function	O
is	O
any	O
quantity	O
or	O
signal	O
that	O
varies	O
over	O
time	O
,	O
such	O
as	O
the	O
pressure	O
of	O
a	O
sound	O
wave	O
,	O
a	O
radio	O
signal	O
,	O
or	O
daily	O
temperature	O
readings	O
,	O
sampled	O
over	O
a	O
finite	O
time	O
interval	O
(	O
often	O
defined	O
by	O
a	O
window	B-Language
function	I-Language
)	O
.	O
</s>
<s>
In	O
image	B-Algorithm
processing	I-Algorithm
,	O
the	O
samples	B-Algorithm
can	O
be	O
the	O
values	O
of	O
pixels	B-Algorithm
along	O
a	O
row	O
or	O
column	O
of	O
a	O
raster	B-Algorithm
image	I-Algorithm
.	O
</s>
<s>
The	O
DFT	O
is	O
also	O
used	O
to	O
efficiently	O
solve	O
partial	O
differential	O
equations	O
,	O
and	O
to	O
perform	O
other	O
operations	O
such	O
as	O
convolutions	B-Language
or	O
multiplying	O
large	O
integers	O
.	O
</s>
<s>
Since	O
it	O
deals	O
with	O
a	O
finite	O
amount	O
of	O
data	O
,	O
it	O
can	O
be	O
implemented	O
in	O
computers	O
by	O
numerical	B-General_Concept
algorithms	I-General_Concept
or	O
even	O
dedicated	O
hardware	O
.	O
</s>
<s>
These	O
implementations	O
usually	O
employ	O
efficient	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
algorithms	B-Algorithm
;	O
so	O
much	O
so	O
that	O
the	O
terms	O
"	O
FFT	O
"	O
and	O
"	O
DFT	O
"	O
are	O
often	O
used	O
interchangeably	O
.	O
</s>
<s>
Prior	O
to	O
its	O
current	O
usage	O
,	O
the	O
"	O
FFT	O
"	O
initialism	O
may	O
have	O
also	O
been	O
used	O
for	O
the	O
ambiguous	O
term	O
"	O
finite	O
Fourier	B-Algorithm
transform	I-Algorithm
"	O
.	O
</s>
<s>
But	O
physically	O
it	O
can	O
be	O
related	O
to	O
signal	B-Algorithm
processing	I-Algorithm
as	O
a	O
discrete	O
version	O
(	O
i.e.	O
</s>
<s>
samples	B-Algorithm
)	O
of	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
,	O
which	O
is	O
a	O
continuous	O
and	O
periodic	O
function	O
.	O
</s>
<s>
The	O
DFT	O
computes	O
N	O
equally-spaced	O
samples	B-Algorithm
of	O
one	O
cycle	O
of	O
the	O
DTFT	O
.	O
</s>
<s>
A	O
normalization	O
of	O
for	O
both	O
the	O
DFT	O
and	O
IDFT	O
,	O
for	O
instance	O
,	O
makes	O
the	O
transforms	O
unitary	B-Algorithm
.	O
</s>
<s>
Its	O
inverse	O
is	O
known	O
as	O
Inverse	B-Algorithm
Discrete	I-Algorithm
Fourier	I-Algorithm
Transform	I-Algorithm
(	O
IDFT	O
)	O
.	O
</s>
<s>
The	O
DFT	O
is	O
a	O
linear	B-Architecture
transform	I-Architecture
,	O
i.e.	O
</s>
<s>
The	O
vectors	O
form	O
an	O
orthogonal	B-Algorithm
basis	I-Algorithm
over	O
the	O
set	O
of	O
N-dimensional	O
complex	O
vectors	O
:	O
</s>
<s>
This	O
orthogonality	B-Application
condition	O
can	O
be	O
used	O
to	O
derive	O
the	O
formula	O
for	O
the	O
IDFT	O
from	O
the	O
definition	O
of	O
the	O
DFT	O
,	O
and	O
is	O
equivalent	O
to	O
the	O
unitarity	O
property	O
below	O
.	O
</s>
<s>
These	O
theorems	O
are	O
also	O
equivalent	O
to	O
the	O
unitary	B-Algorithm
condition	O
below	O
.	O
</s>
<s>
The	O
convolution	B-Language
theorem	O
for	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
indicates	O
that	O
a	O
convolution	B-Language
of	O
two	O
sequences	O
can	O
be	O
obtained	O
as	O
the	O
inverse	O
transform	O
of	O
the	O
product	O
of	O
the	O
individual	O
transforms	O
.	O
</s>
<s>
where	O
is	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
the	O
sequence	O
:	O
</s>
<s>
Then	O
the	O
convolution	B-Language
can	O
be	O
written	O
as	O
:	O
</s>
<s>
which	O
gives	O
rise	O
to	O
the	O
interpretation	O
as	O
a	O
circular	O
convolution	B-Language
of	O
and	O
It	O
is	O
often	O
used	O
to	O
efficiently	O
compute	O
their	O
linear	O
convolution	B-Language
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
any	O
linear	B-Architecture
transform	I-Architecture
that	O
turns	O
convolution	B-Language
into	O
pointwise	O
product	O
is	O
the	O
DFT	O
(	O
up	O
to	O
a	O
permutation	O
of	O
coefficients	O
)	O
.	O
</s>
<s>
which	O
is	O
the	O
circular	O
convolution	B-Language
of	O
and	O
.	O
</s>
<s>
This	O
interpolation	O
is	O
not	O
unique	O
:	O
aliasing	B-Error_Name
implies	O
that	O
one	O
could	O
add	O
N	O
to	O
any	O
of	O
the	O
complex-sinusoid	O
frequencies	O
(	O
e.g.	O
</s>
<s>
With	O
unitary	B-Algorithm
normalization	O
constants	O
,	O
the	O
DFT	O
becomes	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
,	O
defined	O
by	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
In	O
a	O
real	O
vector	O
space	O
,	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
can	O
be	O
thought	O
of	O
as	O
simply	O
a	O
rigid	O
rotation	O
of	O
the	O
coordinate	O
system	O
,	O
and	O
all	O
of	O
the	O
properties	O
of	O
a	O
rigid	O
rotation	O
can	O
be	O
found	O
in	O
the	O
unitary	B-Algorithm
DFT	O
.	O
</s>
<s>
The	O
orthogonality	B-Application
of	O
the	O
DFT	O
is	O
now	O
expressed	O
as	O
an	O
orthonormality	B-Algorithm
condition	O
(	O
which	O
arises	O
in	O
many	O
areas	O
of	O
mathematics	O
as	O
described	O
in	O
root	O
of	O
unity	O
)	O
:	O
</s>
<s>
If	O
we	O
view	O
the	O
DFT	O
as	O
just	O
a	O
coordinate	O
transformation	O
which	O
simply	O
specifies	O
the	O
components	O
of	O
a	O
vector	O
in	O
a	O
new	O
coordinate	O
system	O
,	O
then	O
the	O
above	O
is	O
just	O
the	O
statement	O
that	O
the	O
dot	O
product	O
of	O
two	O
vectors	O
is	O
preserved	O
under	O
a	O
unitary	B-Algorithm
DFT	O
transformation	O
.	O
</s>
<s>
A	O
consequence	O
of	O
the	O
circular	O
convolution	B-Language
theorem	O
is	O
that	O
the	O
DFT	O
matrix	O
diagonalizes	O
any	O
circulant	O
matrix	O
.	O
</s>
<s>
(	O
For	O
example	O
,	O
in	O
computations	O
,	O
it	O
is	O
often	O
convenient	O
to	O
only	O
implement	O
a	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
corresponding	O
to	O
one	O
transform	O
direction	O
and	O
then	O
to	O
get	O
the	O
other	O
transform	O
direction	O
from	O
the	O
first	O
.	O
)	O
</s>
<s>
The	O
conjugation	O
trick	O
can	O
also	O
be	O
used	O
to	O
define	O
a	O
new	O
transform	O
,	O
closely	O
related	O
to	O
the	O
DFT	O
,	O
that	O
is	O
involutory	B-Algorithm
—	O
that	O
is	O
,	O
which	O
is	O
its	O
own	O
inverse	O
.	O
</s>
<s>
A	O
closely	O
related	O
involutory	B-Algorithm
transformation	O
(	O
by	O
a	O
factor	O
of	O
)	O
is	O
,	O
since	O
the	O
factors	O
in	O
cancel	O
the	O
2	O
.	O
</s>
<s>
For	O
real	O
inputs	O
,	O
the	O
real	O
part	O
of	O
is	O
none	O
other	O
than	O
the	O
discrete	O
Hartley	O
transform	O
,	O
which	O
is	O
also	O
involutory	B-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
seen	O
from	O
the	O
inverse	O
properties	O
above	O
:	O
operating	O
twice	O
gives	O
the	O
original	O
data	O
in	O
reverse	O
order	O
,	O
so	O
operating	O
four	O
times	O
gives	O
back	O
the	O
original	O
data	O
and	O
is	O
thus	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
(	O
There	O
are	O
N	O
independent	O
eigenvectors	O
;	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
is	O
never	O
defective	O
.	O
)	O
</s>
<s>
Multiplicities	O
of	O
the	O
eigenvalues	O
λ	O
of	O
the	O
unitary	B-Algorithm
DFT	O
matrix	O
U	O
as	O
a	O
function	O
of	O
the	O
transform	O
size	O
N	O
(	O
in	O
terms	O
of	O
an	O
integer	O
m	O
)	O
.	O
</s>
<s>
Various	O
researchers	O
have	O
proposed	O
different	O
choices	O
of	O
eigenvectors	O
,	O
selected	O
to	O
satisfy	O
useful	O
properties	O
like	O
orthogonality	B-Application
and	O
to	O
have	O
"	O
simple	O
"	O
forms	O
(	O
e.g.	O
,	O
McClellan	O
and	O
Parks	O
,	O
1972	O
;	O
Dickinson	O
and	O
Steiglitz	O
,	O
1982	O
;	O
Grünbaum	O
,	O
1982	O
;	O
Atakishiyev	O
and	O
Wolf	O
,	O
1997	O
;	O
Candan	O
et	O
al.	O
,	O
2000	O
;	O
Hanna	O
et	O
al.	O
,	O
2004	O
;	O
Gurevich	O
and	O
Hadani	O
,	O
2008	O
)	O
.	O
</s>
<s>
A	O
straightforward	O
approach	O
is	O
to	O
discretize	B-Algorithm
an	O
eigenfunction	O
of	O
the	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
</s>
<s>
and	O
discretization	B-Algorithm
means	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
the	O
spectrum	O
,	O
</s>
<s>
the	O
discretized	B-Algorithm
and	O
periodically	O
summed	O
Gaussian	O
function	O
yields	O
an	O
eigenvector	O
of	O
the	O
discrete	O
transform	O
:	O
</s>
<s>
The	O
choice	O
of	O
eigenvectors	O
of	O
the	O
DFT	O
matrix	O
has	O
become	O
important	O
in	O
recent	O
years	O
in	O
order	O
to	O
define	O
a	O
discrete	O
analogue	O
of	O
the	O
fractional	O
Fourier	B-Algorithm
transform	I-Algorithm
—	O
the	O
DFT	O
matrix	O
can	O
be	O
taken	O
to	O
fractional	O
powers	O
by	O
exponentiating	O
the	O
eigenvalues	O
(	O
e.g.	O
,	O
Rubio	O
and	O
Santhanam	O
,	O
2005	O
)	O
.	O
</s>
<s>
For	O
the	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
the	O
natural	O
orthogonal	O
eigenfunctions	O
are	O
the	O
Hermite	O
functions	O
,	O
so	O
various	O
discrete	O
analogues	O
of	O
these	O
have	O
been	O
employed	O
as	O
the	O
eigenvectors	O
of	O
the	O
DFT	O
,	O
such	O
as	O
the	O
Kravchuk	O
polynomials	O
(	O
Atakishiyev	O
and	O
Wolf	O
,	O
1997	O
)	O
.	O
</s>
<s>
The	O
"	O
best	O
"	O
choice	O
of	O
eigenvectors	O
to	O
define	O
a	O
fractional	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
remains	O
an	O
open	O
question	O
,	O
however	O
.	O
</s>
<s>
This	O
is	O
sometimes	O
known	O
as	O
a	O
generalized	O
DFT	O
(	O
or	O
GDFT	O
)	O
,	O
also	O
called	O
the	O
shifted	B-Algorithm
DFT	I-Algorithm
or	O
offset	B-Algorithm
DFT	I-Algorithm
,	O
and	O
has	O
analogous	O
properties	O
to	O
the	O
ordinary	O
DFT	O
:	O
</s>
<s>
Thus	O
,	O
the	O
specific	O
case	O
of	O
is	O
known	O
as	O
an	O
odd-time	O
odd-frequency	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
or	O
O2	O
DFT	O
)	O
.	O
</s>
<s>
Such	O
shifted	O
transforms	O
are	O
most	O
often	O
used	O
for	O
symmetric	O
data	O
,	O
to	O
represent	O
different	O
boundary	O
symmetries	O
,	O
and	O
for	O
real-symmetric	O
data	O
they	O
correspond	O
to	O
different	O
forms	O
of	O
the	O
discrete	O
cosine	B-General_Concept
and	O
sine	O
transforms	O
.	O
</s>
<s>
Another	O
interesting	O
choice	O
is	O
,	O
which	O
is	O
called	O
the	O
centered	B-Algorithm
DFT	I-Algorithm
(	O
or	O
CDFT	O
)	O
.	O
</s>
<s>
The	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
be	O
viewed	O
as	O
a	O
special	O
case	O
of	O
the	O
z-transform	B-Algorithm
,	O
evaluated	O
on	O
the	O
unit	O
circle	O
in	O
the	O
complex	O
plane	O
;	O
more	O
general	O
z-transforms	B-Algorithm
correspond	O
to	O
complex	O
shifts	O
a	O
and	O
b	O
above	O
.	O
</s>
<s>
The	O
ordinary	O
DFT	O
transforms	O
a	O
one-dimensional	O
sequence	O
or	O
array	B-Architecture
that	O
is	O
a	O
function	O
of	O
exactly	O
one	O
discrete	O
variable	O
n	O
.	O
The	O
multidimensional	O
DFT	O
of	O
a	O
multidimensional	O
array	B-Architecture
that	O
is	O
a	O
function	O
of	O
d	O
discrete	O
variables	O
for	O
in	O
is	O
defined	O
by	O
:	O
</s>
<s>
This	O
decomposition	O
is	O
of	O
great	O
importance	O
for	O
everything	O
from	O
digital	B-Algorithm
image	I-Algorithm
processing	I-Algorithm
(	O
two-dimensional	O
)	O
to	O
solving	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
The	O
multidimensional	O
DFT	O
can	O
be	O
computed	O
by	O
the	O
composition	B-Application
of	O
a	O
sequence	O
of	O
one-dimensional	O
DFTs	O
along	O
each	O
dimension	O
.	O
</s>
<s>
In	O
the	O
two-dimensional	O
case	O
the	O
independent	O
DFTs	O
of	O
the	O
rows	O
(	O
i.e.	O
,	O
along	O
)	O
are	O
computed	O
first	O
to	O
form	O
a	O
new	O
array	B-Architecture
.	O
</s>
<s>
There	O
are	O
also	O
intrinsically	O
multidimensional	O
FFT	O
algorithms	B-Algorithm
.	O
</s>
<s>
All	O
applications	O
of	O
the	O
DFT	O
depend	O
crucially	O
on	O
the	O
availability	O
of	O
a	O
fast	O
algorithm	O
to	O
compute	O
discrete	B-Algorithm
Fourier	I-Algorithm
transforms	I-Algorithm
and	O
their	O
inverses	O
,	O
a	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
When	O
the	O
DFT	O
is	O
used	O
for	O
signal	O
spectral	O
analysis	O
,	O
the	O
sequence	O
usually	O
represents	O
a	O
finite	O
set	O
of	O
uniformly	O
spaced	O
time-samples	O
of	O
some	O
signal	O
,	O
where	O
represents	O
time	O
.	O
</s>
<s>
The	O
conversion	O
from	O
continuous	O
time	O
to	O
samples	B-Algorithm
(	O
discrete-time	O
)	O
changes	O
the	O
underlying	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
into	O
a	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
,	O
which	O
generally	O
entails	O
a	O
type	O
of	O
distortion	O
called	O
aliasing	B-Error_Name
.	O
</s>
<s>
The	O
procedure	O
is	O
sometimes	O
referred	O
to	O
as	O
zero-padding	O
,	O
which	O
is	O
a	O
particular	O
implementation	O
used	O
in	O
conjunction	O
with	O
the	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
algorithm	O
.	O
</s>
<s>
The	O
inefficiency	O
of	O
performing	O
multiplications	O
and	O
additions	O
with	O
zero-valued	O
"	O
samples	B-Algorithm
"	O
is	O
more	O
than	O
offset	O
by	O
the	O
inherent	O
efficiency	O
of	O
the	O
FFT	O
.	O
</s>
<s>
The	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
widely	O
used	O
with	O
spatial	O
frequencies	O
in	O
modeling	O
the	O
way	O
that	O
light	O
,	O
electrons	O
,	O
and	O
other	O
probes	O
travel	O
through	O
optical	O
systems	O
and	O
scatter	O
from	O
objects	O
in	O
two	O
and	O
three	O
dimensions	O
.	O
</s>
<s>
The	O
dual	O
(	O
direct/reciprocal	O
)	O
vector	O
space	O
of	O
three	O
dimensional	O
objects	O
further	O
makes	O
available	O
a	O
three	O
dimensional	O
reciprocal	O
lattice	O
,	O
whose	O
construction	O
from	O
translucent	O
object	O
shadows	O
(	O
via	O
the	O
Fourier	B-Algorithm
slice	I-Algorithm
theorem	I-Algorithm
)	O
allows	O
tomographic	O
reconstruction	O
of	O
three	O
dimensional	O
objects	O
with	O
a	O
wide	O
range	O
of	O
applications	O
e.g.	O
</s>
<s>
The	O
field	O
of	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
relies	O
heavily	O
on	O
operations	O
in	O
the	O
frequency	O
domain	O
(	O
i.e.	O
</s>
<s>
on	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
)	O
.	O
</s>
<s>
For	O
example	O
,	O
several	O
lossy	B-Algorithm
image	O
and	O
sound	O
compression	O
methods	O
employ	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
:	O
the	O
signal	O
is	O
cut	O
into	O
short	O
segments	O
,	O
each	O
is	O
transformed	O
,	O
and	O
then	O
the	O
Fourier	O
coefficients	O
of	O
high	O
frequencies	O
,	O
which	O
are	O
assumed	O
to	O
be	O
unnoticeable	O
,	O
are	O
discarded	O
.	O
</s>
<s>
(	O
Compression	O
applications	O
often	O
use	O
a	O
specialized	O
form	O
of	O
the	O
DFT	O
,	O
the	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
or	O
sometimes	O
the	O
modified	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
.	O
)	O
</s>
<s>
Some	O
relatively	O
recent	O
compression	O
algorithms	B-Algorithm
,	O
however	O
,	O
use	O
wavelet	B-Algorithm
transforms	I-Algorithm
,	O
which	O
give	O
a	O
more	O
uniform	O
compromise	O
between	O
time	O
and	O
frequency	O
domain	O
than	O
obtained	O
by	O
chopping	O
data	O
into	O
segments	O
and	O
transforming	O
each	O
segment	O
.	O
</s>
<s>
Discrete	B-Algorithm
Fourier	I-Algorithm
transforms	I-Algorithm
are	O
often	O
used	O
to	O
solve	O
partial	O
differential	O
equations	O
,	O
where	O
again	O
the	O
DFT	O
is	O
used	O
as	O
an	O
approximation	O
for	O
the	O
Fourier	O
series	O
(	O
which	O
is	O
recovered	O
in	O
the	O
limit	O
of	O
infinite	O
N	O
)	O
.	O
</s>
<s>
(	O
However	O
,	O
the	O
choice	O
of	O
is	O
not	O
unique	O
due	O
to	O
aliasing	B-Error_Name
;	O
for	O
the	O
method	O
to	O
be	O
convergent	O
,	O
a	O
choice	O
similar	O
to	O
that	O
in	O
the	O
trigonometric	O
interpolation	O
section	O
above	O
should	O
be	O
used	O
.	O
)	O
</s>
<s>
Such	O
an	O
approach	O
is	O
called	O
a	O
spectral	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
ordinary	O
product	O
expression	O
for	O
the	O
coefficients	O
of	O
c	O
involves	O
a	O
linear	O
(	O
acyclic	O
)	O
convolution	B-Language
,	O
where	O
indices	O
do	O
not	O
"	O
wrap	O
around.	O
"	O
</s>
<s>
This	O
can	O
be	O
rewritten	O
as	O
a	O
cyclic	O
convolution	B-Language
by	O
taking	O
the	O
coefficient	O
vectors	O
for	O
a(x )	O
and	O
b(x )	O
with	O
constant	O
term	O
first	O
,	O
then	O
appending	O
zeros	O
so	O
that	O
the	O
resultant	O
coefficient	O
vectors	O
a	O
and	O
b	O
have	O
dimension	O
.	O
</s>
<s>
But	O
convolution	B-Language
becomes	O
multiplication	O
under	O
the	O
DFT	O
:	O
</s>
<s>
With	O
a	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
the	O
resulting	O
algorithm	O
takes	O
O(NlogN )	O
arithmetic	O
operations	O
.	O
</s>
<s>
Due	O
to	O
its	O
simplicity	O
and	O
speed	O
,	O
the	O
Cooley	B-Algorithm
–	I-Algorithm
Tukey	I-Algorithm
FFT	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
is	O
limited	O
to	O
composite	O
sizes	O
,	O
is	O
often	O
chosen	O
for	O
the	O
transform	O
operation	O
.	O
</s>
<s>
The	O
fastest	O
known	O
algorithms	B-Algorithm
for	O
the	O
multiplication	O
of	O
very	O
large	O
integers	O
use	O
the	O
polynomial	O
multiplication	O
method	O
outlined	O
above	O
.	O
</s>
<s>
When	O
data	O
is	O
convolved	B-Language
with	O
a	O
function	O
with	O
wide	O
support	O
,	O
such	O
as	O
for	O
downsampling	O
by	O
a	O
large	O
sampling	O
ratio	O
,	O
because	O
of	O
the	O
Convolution	B-Language
theorem	O
and	O
the	O
FFT	O
algorithm	O
,	O
it	O
may	O
be	O
faster	O
to	O
transform	O
it	O
,	O
multiply	O
pointwise	O
by	O
the	O
transform	O
of	O
the	O
filter	O
and	O
then	O
reverse	O
transform	O
it	O
.	O
</s>
<s>
+	O
Some	O
DFT	O
pairs	O
Note	O
Frequency	O
shift	O
theorem	O
Time	O
shift	O
theorem	O
Real	O
DFT	O
from	O
the	O
geometric	O
progression	O
formula	O
from	O
the	O
binomial	O
theorem	O
is	O
a	O
rectangular	O
window	B-Language
function	I-Language
of	O
W	O
points	O
centered	O
on	O
n	O
=	O
0	O
,	O
where	O
W	O
is	O
an	O
odd	O
integer	O
,	O
and	O
is	O
a	O
sinc-like	O
function	O
(	O
specifically	O
,	O
is	O
a	O
Dirichlet	O
kernel	O
)	O
Discretization	B-Algorithm
and	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
the	O
scaled	O
Gaussian	O
functions	O
for	O
.	O
</s>
<s>
Since	O
either	O
or	O
is	O
larger	O
than	O
one	O
and	O
thus	O
warrants	O
fast	O
convergence	O
of	O
one	O
of	O
the	O
two	O
series	O
,	O
for	O
large	O
you	O
may	O
choose	O
to	O
compute	O
the	O
frequency	O
spectrum	O
and	O
convert	O
to	O
the	O
time	O
domain	O
using	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Such	O
properties	O
include	O
the	O
completeness	O
,	O
orthogonality	B-Application
,	O
Plancherel/Parseval	O
,	O
periodicity	O
,	O
shift	O
,	O
convolution	B-Language
,	O
and	O
unitarity	O
properties	O
above	O
,	O
as	O
well	O
as	O
many	O
FFT	O
algorithms	B-Algorithm
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
be	O
defined	O
by	O
using	O
roots	O
of	O
unity	O
in	O
fields	O
other	O
than	O
the	O
complex	O
numbers	O
,	O
and	O
such	O
generalizations	O
are	O
commonly	O
called	O
number-theoretic	O
transforms	O
(	O
NTTs	O
)	O
in	O
the	O
case	O
of	O
finite	O
fields	O
.	O
</s>
<s>
For	O
more	O
information	O
,	O
see	O
number-theoretic	O
transform	O
and	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
general	O
)	O
.	O
</s>
<s>
This	O
suggests	O
the	O
generalization	O
to	O
Fourier	B-Algorithm
transforms	I-Algorithm
on	O
arbitrary	O
finite	O
groups	O
,	O
which	O
act	O
on	O
functions	O
G	O
→	O
C	O
where	O
G	O
is	O
a	O
finite	O
group	O
.	O
</s>
<s>
In	O
this	O
framework	O
,	O
the	O
standard	O
DFT	O
is	O
seen	O
as	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
a	O
cyclic	O
group	O
,	O
while	O
the	O
multidimensional	O
DFT	O
is	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
a	O
direct	O
sum	O
of	O
cyclic	O
groups	O
.	O
</s>
<s>
Further	O
,	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
on	O
cosets	O
of	O
a	O
group	O
.	O
</s>
<s>
The	O
analog	O
of	O
the	O
DFT	O
is	O
the	O
discrete	O
wavelet	B-Algorithm
transform	I-Algorithm
(	O
DWT	O
)	O
.	O
</s>
<s>
From	O
the	O
point	O
of	O
view	O
of	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
,	O
a	O
key	O
limitation	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
that	O
it	O
does	O
not	O
include	O
location	O
information	O
,	O
only	O
frequency	O
information	O
,	O
and	O
thus	O
has	O
difficulty	O
in	O
representing	O
transients	O
.	O
</s>
<s>
For	O
details	O
,	O
see	O
comparison	O
of	O
the	O
discrete	O
wavelet	B-Algorithm
transform	I-Algorithm
with	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
