<s>
In	O
mathematics	O
,	O
the	O
Hartley	B-Algorithm
transform	I-Algorithm
(	O
HT	O
)	O
is	O
an	O
integral	B-Algorithm
transform	I-Algorithm
closely	O
related	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FT	O
)	O
,	O
but	O
which	O
transforms	O
real-valued	O
functions	O
to	O
real-valued	O
functions	O
.	O
</s>
<s>
It	O
was	O
proposed	O
as	O
an	O
alternative	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
by	O
Ralph	O
V	O
.	O
L	O
.	O
Hartley	O
in	O
1942	O
,	O
and	O
is	O
one	O
of	O
many	O
known	O
Fourier-related	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
Compared	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
the	O
Hartley	B-Algorithm
transform	I-Algorithm
has	O
the	O
advantages	O
of	O
transforming	O
real	O
functions	O
to	O
real	O
functions	O
(	O
as	O
opposed	O
to	O
requiring	O
complex	O
numbers	O
)	O
and	O
of	O
being	O
its	O
own	O
inverse	O
.	O
</s>
<s>
The	O
discrete	O
version	O
of	O
the	O
transform	O
,	O
the	O
discrete	O
Hartley	B-Algorithm
transform	I-Algorithm
(	O
DHT	O
)	O
,	O
was	O
introduced	O
by	O
Ronald	O
N	O
.	O
Bracewell	O
in	O
1983	O
.	O
</s>
<s>
The	O
two-dimensional	O
Hartley	B-Algorithm
transform	I-Algorithm
can	O
be	O
computed	O
by	O
an	O
analog	O
optical	O
process	O
similar	O
to	O
an	O
optical	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
OFT	O
)	O
,	O
with	O
the	O
proposed	O
advantage	O
that	O
only	O
its	O
amplitude	O
and	O
sign	O
need	O
to	O
be	O
determined	O
rather	O
than	O
its	O
complex	O
phase	O
.	O
</s>
<s>
However	O
,	O
optical	O
Hartley	B-Algorithm
transforms	I-Algorithm
do	O
not	O
seem	O
to	O
have	O
seen	O
widespread	O
use	O
.	O
</s>
<s>
The	O
Hartley	B-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
is	O
defined	O
by	O
:	O
</s>
<s>
is	O
the	O
cosine-and-sine	O
(	O
cas	O
)	O
or	O
Hartley	O
kernel	B-Algorithm
.	O
</s>
<s>
The	O
Hartley	B-Algorithm
transform	I-Algorithm
has	O
the	O
convenient	O
property	O
of	O
being	O
its	O
own	O
inverse	O
(	O
an	O
involution	B-Algorithm
)	O
:	O
</s>
<s>
The	O
above	O
is	O
in	O
accord	O
with	O
Hartley	O
's	O
original	O
definition	O
,	O
but	O
(	O
as	O
with	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
)	O
various	O
minor	O
details	O
are	O
matters	O
of	O
convention	O
and	O
can	O
be	O
changed	O
without	O
altering	O
the	O
essential	O
properties	O
:	O
</s>
<s>
One	O
can	O
use	O
instead	O
of	O
as	O
the	O
kernel	B-Algorithm
.	O
</s>
<s>
in	O
the	O
choice	O
of	O
the	O
kernel	B-Algorithm
.	O
</s>
<s>
In	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
we	O
have	O
the	O
exponential	O
kernel	B-Algorithm
,	O
</s>
<s>
The	O
two	O
transforms	O
are	O
closely	O
related	O
,	O
however	O
,	O
and	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
assuming	O
it	O
uses	O
the	O
same	O
normalization	O
convention	O
)	O
can	O
be	O
computed	O
from	O
the	O
Hartley	B-Algorithm
transform	I-Algorithm
via	O
:	O
</s>
<s>
That	O
is	O
,	O
the	O
real	O
and	O
imaginary	O
parts	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
are	O
simply	O
given	O
by	O
the	O
even	O
and	O
odd	O
parts	O
of	O
the	O
Hartley	B-Algorithm
transform	I-Algorithm
,	O
respectively	O
.	O
</s>
<s>
Conversely	O
,	O
for	O
real-valued	O
functions	O
the	O
Hartley	B-Algorithm
transform	I-Algorithm
is	O
given	O
from	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
's	O
real	O
and	O
imaginary	O
parts	O
:	O
</s>
<s>
The	O
Hartley	B-Algorithm
transform	I-Algorithm
is	O
a	O
real	O
linear	B-Architecture
operator	I-Architecture
,	O
and	O
is	O
symmetric	B-Algorithm
(	O
and	O
Hermitian	O
)	O
.	O
</s>
<s>
From	O
the	O
symmetric	B-Algorithm
and	O
self-inverse	B-Algorithm
properties	O
,	O
it	O
follows	O
that	O
the	O
transform	O
is	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
(	O
indeed	O
,	O
orthogonal	B-Algorithm
)	O
.	O
</s>
<s>
Similar	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
the	O
Hartley	B-Algorithm
transform	I-Algorithm
of	O
an	O
even/odd	O
function	O
is	O
even/odd	O
,	O
respectively	O
.	O
</s>
<s>
The	O
properties	O
of	O
the	O
Hartley	O
kernel	B-Algorithm
,	O
for	O
which	O
Hartley	O
introduced	O
the	O
name	O
cas	O
for	O
the	O
function	O
(	O
from	O
cosine	O
and	O
sine	O
)	O
in	O
1942	O
,	O
follow	O
directly	O
from	O
trigonometry	O
,	O
and	O
its	O
definition	O
as	O
a	O
phase-shifted	O
trigonometric	O
function	O
For	O
example	O
,	O
it	O
has	O
an	O
angle-addition	O
identity	O
of	O
:	O
</s>
