<s>
This	O
is	O
a	O
list	O
of	O
linear	B-Architecture
transformations	I-Architecture
of	O
functions	O
related	O
to	O
Fourier	O
analysis	O
.	O
</s>
<s>
Such	O
transformations	O
map	B-Algorithm
a	O
function	O
to	O
a	O
set	O
of	O
coefficients	O
of	O
basis	O
functions	O
,	O
where	O
the	O
basis	O
functions	O
are	O
sinusoidal	O
and	O
are	O
therefore	O
strongly	O
localized	O
in	O
the	O
frequency	O
spectrum	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
each	O
basis	O
function	O
corresponds	O
to	O
a	O
single	O
frequency	O
component	O
.	O
</s>
<s>
Applied	O
to	O
functions	O
of	O
continuous	O
arguments	O
,	O
Fourier-related	B-Algorithm
transforms	I-Algorithm
include	O
:	O
</s>
<s>
Fourier	B-Algorithm
transform	I-Algorithm
,	O
with	O
special	O
cases	O
:	O
</s>
<s>
When	O
the	O
input	O
function/waveform	O
is	O
periodic	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
output	O
is	O
a	O
Dirac	O
comb	O
function	O
,	O
modulated	O
by	O
a	O
discrete	O
sequence	O
of	O
finite-valued	O
coefficients	O
that	O
are	O
complex-valued	O
in	O
general	O
.	O
</s>
<s>
The	O
term	O
Fourier	O
series	O
actually	O
refers	O
to	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
which	O
is	O
a	O
sum	O
of	O
sinusoids	O
at	O
discrete	O
frequencies	O
,	O
weighted	O
by	O
the	O
Fourier	O
series	O
coefficients	O
.	O
</s>
<s>
When	O
the	O
non-zero	O
portion	O
of	O
the	O
input	O
function	O
has	O
finite	O
duration	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
continuous	O
and	O
finite-valued	O
.	O
</s>
<s>
Sine	O
and	O
cosine	O
transforms	O
:	O
When	O
the	O
input	O
function	O
has	O
odd	O
or	O
even	O
symmetry	O
around	O
the	O
origin	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
reduces	O
to	O
a	O
sine	O
or	O
cosine	O
transform	O
.	O
</s>
<s>
Hankel	B-Algorithm
transform	I-Algorithm
:	O
related	O
to	O
the	O
Fourier	B-Algorithm
Transform	I-Algorithm
of	O
radial	O
functions	O
.	O
</s>
<s>
Discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
:	O
Equivalent	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
"	O
continuous	O
"	O
function	O
that	O
is	O
constructed	O
from	O
the	O
discrete	O
input	O
function	O
by	O
using	O
the	O
sample	O
values	O
to	O
modulate	O
a	O
Dirac	O
comb	O
.	O
</s>
<s>
When	O
the	O
sample	O
values	O
are	O
derived	O
by	O
sampling	O
a	O
function	O
on	O
the	O
real	O
line	O
,	O
ƒ(x )	O
,	O
the	O
DTFT	O
is	O
equivalent	O
to	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
ƒ	O
.	O
</s>
<s>
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	O
)	O
:	O
</s>
<s>
Discrete	O
sine	O
and	O
cosine	O
transforms	O
:	O
'	O
When	O
the	O
input	O
sequence	O
has	O
odd	O
or	O
even	O
symmetry	O
around	O
the	O
origin	O
,	O
the	O
DTFT	O
reduces	O
to	O
a	O
discrete	O
sine	O
transform	O
(	O
DST	O
)	O
or	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
DCT	B-General_Concept
)	O
.	O
</s>
<s>
Regressive	B-Algorithm
discrete	I-Algorithm
Fourier	I-Algorithm
series	I-Algorithm
,	O
in	O
which	O
the	O
period	O
is	O
determined	O
by	O
the	O
data	O
rather	O
than	O
fixed	O
in	O
advance	O
.	O
</s>
<s>
Discrete	B-Algorithm
Chebyshev	I-Algorithm
transforms	I-Algorithm
(	O
on	O
the	O
'	O
roots	O
 '	O
grid	O
and	O
the	O
'	O
extrema	O
 '	O
grid	O
of	O
the	O
Chebyshev	O
polynomials	O
of	O
the	O
first	O
kind	O
)	O
.	O
</s>
<s>
Discrete-space	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DSFT	O
)	O
is	O
the	O
generalization	O
of	O
the	O
DTFT	O
from	O
1D	O
signals	O
to	O
2D	O
signals	O
.	O
</s>
<s>
Also	O
the	O
discretized	O
STFT	B-Algorithm
(	O
see	O
above	O
)	O
.	O
</s>
<s>
Hadamard	B-Algorithm
transform	I-Algorithm
(	O
Walsh	O
function	O
)	O
.	O
</s>
<s>
Fourier	B-Algorithm
transform	I-Algorithm
on	O
finite	O
groups	O
.	O
</s>
<s>
Discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
general	O
)	O
.	O
</s>
<s>
The	O
use	O
of	O
all	O
of	O
these	O
transforms	O
is	O
greatly	O
facilitated	O
by	O
the	O
existence	O
of	O
efficient	O
algorithms	O
based	O
on	O
a	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
.	O
</s>
<s>
Tables	O
of	O
Integral	B-Algorithm
Transforms	I-Algorithm
at	O
EqWorld	O
:	O
The	O
World	O
of	O
Mathematical	O
Equations	O
.	O
</s>
<s>
A	O
.	O
N	O
.	O
Akansu	O
and	O
H	O
.	O
Agirman-Tosun	O
,	O
"	O
Generalized	O
Discrete	B-Algorithm
Fourier	I-Algorithm
Transform	I-Algorithm
With	O
Nonlinear	O
Phase	O
"	O
,	O
IEEE	O
Transactions	O
on	O
Signal	O
Processing''	O
,	O
vol	O
.	O
</s>
