<s>
In	O
mathematics	O
and	O
signal	O
processing	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
a	O
specific	O
singular	O
integral	O
that	O
takes	O
a	O
function	O
,	O
of	O
a	O
real	O
variable	O
and	O
produces	O
another	O
function	O
of	O
a	O
real	O
variable	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
given	O
by	O
the	O
Cauchy	O
principal	O
value	O
of	O
the	O
convolution	B-Language
with	O
the	O
function	O
(	O
see	O
)	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
has	O
a	O
particularly	O
simple	O
representation	O
in	O
the	O
frequency	O
domain	O
:	O
It	O
imparts	O
a	O
phase	O
shift	O
of	O
±90°	O
(	O
radians	O
)	O
to	O
every	O
frequency	O
component	O
of	O
a	O
function	O
,	O
the	O
sign	O
of	O
the	O
shift	O
depending	O
on	O
the	O
sign	O
of	O
the	O
frequency	O
(	O
see	O
)	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
important	O
in	O
signal	O
processing	O
,	O
where	O
it	O
is	O
a	O
component	O
of	O
the	O
analytic	O
representation	O
of	O
a	O
real-valued	O
signal	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
was	O
first	O
introduced	O
by	O
David	O
Hilbert	O
in	O
this	O
setting	O
,	O
to	O
solve	O
a	O
special	O
case	O
of	O
the	O
Riemann	O
–	O
Hilbert	O
problem	O
for	O
analytic	B-Language
functions	I-Language
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
can	O
be	O
thought	O
of	O
as	O
the	O
convolution	B-Language
of	O
with	O
the	O
function	O
,	O
known	O
as	O
the	O
Cauchy	O
kernel	O
.	O
</s>
<s>
Because	O
is	O
not	O
integrable	O
across	O
,	O
the	O
integral	O
defining	O
the	O
convolution	B-Language
does	O
not	O
always	O
converge	O
.	O
</s>
<s>
Instead	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
defined	O
using	O
the	O
Cauchy	O
principal	O
value	O
(	O
denoted	O
here	O
by	O
)	O
.	O
</s>
<s>
This	O
is	O
precisely	O
the	O
convolution	B-Language
of	O
with	O
the	O
tempered	O
distribution	O
.	O
</s>
<s>
When	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
applied	O
twice	O
in	O
succession	O
to	O
a	O
function	O
,	O
the	O
result	O
is	O
:	O
</s>
<s>
This	O
fact	O
can	O
most	O
easily	O
be	O
seen	O
by	O
considering	O
the	O
effect	O
of	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
on	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
(	O
see	O
,	O
below	O
)	O
.	O
</s>
<s>
For	O
an	O
analytic	B-Language
function	I-Language
in	O
the	O
upper	O
half-plane	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
describes	O
the	O
relationship	O
between	O
the	O
real	O
part	O
and	O
the	O
imaginary	O
part	O
of	O
the	O
boundary	O
values	O
.	O
</s>
<s>
That	O
is	O
,	O
if	O
is	O
analytic	O
in	O
the	O
upper	O
half	O
complex	O
plane	O
,	O
and	O
,	O
then	O
up	O
to	O
an	O
additive	O
constant	O
,	O
provided	O
this	O
Hilbert	B-Algorithm
transform	I-Algorithm
exists	O
.	O
</s>
<s>
In	O
signal	O
processing	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
is	O
commonly	O
denoted	O
by	O
.	O
</s>
<s>
However	O
,	O
in	O
mathematics	O
,	O
this	O
notation	O
is	O
already	O
extensively	O
used	O
to	O
denote	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
.	O
</s>
<s>
Occasionally	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
may	O
be	O
denoted	O
by	O
.	O
</s>
<s>
Furthermore	O
,	O
many	O
sources	O
define	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
as	O
the	O
negative	O
of	O
the	O
one	O
defined	O
here	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
arose	O
in	O
Hilbert	O
's	O
1905	O
work	O
on	O
a	O
problem	O
Riemann	O
posed	O
concerning	O
analytic	B-Language
functions	I-Language
,	O
which	O
has	O
come	O
to	O
be	O
known	O
as	O
the	O
Riemann	O
–	O
Hilbert	O
problem	O
.	O
</s>
<s>
Hilbert	O
's	O
work	O
was	O
mainly	O
concerned	O
with	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
for	O
functions	O
defined	O
on	O
the	O
circle	O
.	O
</s>
<s>
Some	O
of	O
his	O
earlier	O
work	O
related	O
to	O
the	O
Discrete	O
Hilbert	B-Algorithm
Transform	I-Algorithm
dates	O
back	O
to	O
lectures	O
he	O
gave	O
in	O
Göttingen	O
.	O
</s>
<s>
Schur	O
improved	O
Hilbert	O
's	O
results	O
about	O
the	O
discrete	O
Hilbert	B-Algorithm
transform	I-Algorithm
and	O
extended	O
them	O
to	O
the	O
integral	O
case	O
.	O
</s>
<s>
In	O
1928	O
,	O
Marcel	O
Riesz	O
proved	O
that	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
can	O
be	O
defined	O
for	O
u	O
in	O
(	O
Lp	O
space	O
)	O
for	O
,	O
that	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
a	O
bounded	O
operator	O
on	O
for	O
,	O
and	O
that	O
similar	O
results	O
hold	O
for	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
on	O
the	O
circle	O
as	O
well	O
as	O
the	O
discrete	O
Hilbert	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
was	O
a	O
motivating	O
example	O
for	O
Antoni	O
Zygmund	O
and	O
Alberto	O
Calderón	O
during	O
their	O
study	O
of	O
singular	O
integrals	O
.	O
</s>
<s>
Various	O
generalizations	O
of	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
,	O
such	O
as	O
the	O
bilinear	O
and	O
trilinear	O
Hilbert	B-Algorithm
transforms	I-Algorithm
are	O
still	O
active	O
areas	O
of	O
research	O
today	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
a	O
multiplier	O
operator	O
.	O
</s>
<s>
where	O
denotes	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
When	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
applied	O
twice	O
,	O
the	O
phase	O
of	O
the	O
negative	O
and	O
positive	O
frequency	O
components	O
of	O
are	O
respectively	O
shifted	O
by	O
+180°	O
and	O
−180°	O
,	O
which	O
are	O
equivalent	O
amounts	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
the	O
sin	O
and	O
cos	O
functions	O
can	O
be	O
defined	O
by	O
taking	O
the	O
principal	O
value	O
of	O
the	O
integral	O
at	O
infinity	O
.	O
</s>
<s>
This	O
definition	O
agrees	O
with	O
the	O
result	O
of	O
defining	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
distributionally	O
.	O
</s>
<s>
An	O
extensive	O
table	O
of	O
Hilbert	B-Algorithm
transforms	I-Algorithm
is	O
available	O
.	O
</s>
<s>
Note	O
that	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
a	O
constant	O
is	O
zero	O
.	O
</s>
<s>
It	O
is	O
by	O
no	O
means	O
obvious	O
that	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
well-defined	O
at	O
all	O
,	O
as	O
the	O
improper	O
integral	O
defining	O
it	O
must	O
converge	O
in	O
a	O
suitable	O
sense	O
.	O
</s>
<s>
However	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
well-defined	O
for	O
a	O
broad	O
class	O
of	O
functions	O
,	O
namely	O
those	O
in	O
for	O
.	O
</s>
<s>
In	O
the	O
case	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
still	O
converges	O
pointwise	O
almost	O
everywhere	O
,	O
but	O
may	O
itself	O
fail	O
to	O
be	O
integrable	O
,	O
even	O
locally	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
an	O
function	O
does	O
converge	O
,	O
however	O
,	O
in	O
-weak	O
,	O
and	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
a	O
bounded	O
operator	O
from	O
to	O
.	O
</s>
<s>
(	O
In	O
particular	O
,	O
since	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
also	O
a	O
multiplier	O
operator	O
on	O
,	O
Marcinkiewicz	O
interpolation	O
and	O
a	O
duality	O
argument	O
furnishes	O
an	O
alternative	O
proof	O
that	O
is	O
bounded	O
on	O
.	O
)	O
</s>
<s>
The	O
same	O
best	O
constants	O
hold	O
for	O
the	O
periodic	O
Hilbert	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
an	O
anti-self	O
adjoint	O
operator	O
relative	O
to	O
the	O
duality	O
pairing	O
between	O
and	O
the	O
dual	O
space	O
where	O
and	O
are	O
Hölder	B-Algorithm
conjugates	I-Algorithm
and	O
.	O
</s>
<s>
Because	O
(""	O
is	O
the	O
identity	O
operator	O
)	O
on	O
the	O
real	O
Banach	O
space	O
of	O
real-valued	O
functions	O
in	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
defines	O
a	O
linear	O
complex	O
structure	O
on	O
this	O
Banach	O
space	O
.	O
</s>
<s>
In	O
particular	O
,	O
when	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
gives	O
the	O
Hilbert	O
space	O
of	O
real-valued	O
functions	O
in	O
the	O
structure	O
of	O
a	O
complex	O
Hilbert	O
space	O
.	O
</s>
<s>
The	O
(	O
complex	O
)	O
eigenstates	O
of	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
admit	O
representations	O
as	O
holomorphic	O
functions	O
in	O
the	O
upper	O
and	O
lower	O
half-planes	O
in	O
the	O
Hardy	O
space	O
by	O
the	O
Paley	O
–	O
Wiener	O
theorem	O
.	O
</s>
<s>
Formally	O
,	O
the	O
derivative	O
of	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
the	O
derivative	O
,	O
i.e.	O
</s>
<s>
For	O
most	O
operational	O
purposes	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
can	O
be	O
treated	O
as	O
a	O
convolution	B-Language
.	O
</s>
<s>
For	O
example	O
,	O
in	O
a	O
formal	O
sense	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
a	O
convolution	B-Language
is	O
the	O
convolution	B-Language
of	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
applied	O
on	O
only	O
one	O
of	O
either	O
of	O
the	O
factors	O
:	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
has	O
the	O
following	O
invariance	O
properties	O
on	O
.	O
</s>
<s>
Up	O
to	O
a	O
multiplicative	O
constant	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
the	O
only	O
bounded	O
operator	O
on	O
2	O
with	O
these	O
properties	O
.	O
</s>
<s>
In	O
fact	O
there	O
is	O
a	O
wider	O
set	O
of	O
operators	O
that	O
commute	O
with	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
and	O
its	O
conjugate	O
consist	O
of	O
exactly	O
those	O
functions	O
with	O
Fourier	B-Algorithm
transforms	I-Algorithm
vanishing	O
on	O
the	O
negative	O
and	O
positive	O
parts	O
of	O
the	O
real	O
axis	O
respectively	O
.	O
</s>
<s>
Since	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
equal	O
to	O
,	O
with	O
being	O
the	O
orthogonal	O
projection	O
from	O
onto	O
and	O
the	O
identity	O
operator	O
,	O
it	O
follows	O
that	O
and	O
its	O
orthogonal	O
complement	O
are	O
eigenspaces	O
of	O
for	O
the	O
eigenvalues	O
.	O
</s>
<s>
It	O
is	O
further	O
possible	O
to	O
extend	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
to	O
certain	O
spaces	O
of	O
distributions	O
.	O
</s>
<s>
Since	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
commutes	O
with	O
differentiation	O
,	O
and	O
is	O
a	O
bounded	O
operator	O
on	O
,	O
restricts	O
to	O
give	O
a	O
continuous	O
transform	O
on	O
the	O
inverse	O
limit	O
of	O
Sobolev	O
spaces	O
:	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
can	O
then	O
be	O
defined	O
on	O
the	O
dual	O
space	O
of	O
,	O
denoted	O
,	O
consisting	O
of	O
distributions	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
define	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
on	O
the	O
space	O
of	O
tempered	O
distributions	O
as	O
well	O
by	O
an	O
approach	O
due	O
to	O
Gel'fand	O
and	O
Shilov	O
,	O
but	O
considerably	O
more	O
care	O
is	O
needed	O
because	O
of	O
the	O
singularity	O
in	O
the	O
integral	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
transform	I-Algorithm
can	O
be	O
defined	O
for	O
functions	O
in	O
as	O
well	O
,	O
but	O
it	O
requires	O
some	O
modifications	O
and	O
caveats	O
.	O
</s>
<s>
Properly	O
understood	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
maps	O
to	O
the	O
Banach	O
space	O
of	O
bounded	B-Algorithm
mean	I-Algorithm
oscillation	I-Algorithm
(	O
BMO	O
)	O
classes	O
.	O
</s>
<s>
Interpreted	O
naïvely	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
a	O
bounded	O
function	O
is	O
clearly	O
ill-defined	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
resulting	O
integral	O
converges	O
pointwise	O
almost	O
everywhere	O
,	O
and	O
with	O
respect	O
to	O
the	O
BMO	O
norm	O
,	O
to	O
a	O
function	O
of	O
bounded	B-Algorithm
mean	I-Algorithm
oscillation	I-Algorithm
.	O
</s>
<s>
Under	O
these	O
circumstances	O
,	O
if	O
and	O
are	O
sufficiently	O
integrable	O
,	O
then	O
one	O
is	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
the	O
other	O
.	O
</s>
<s>
The	O
(	O
non-tangential	O
)	O
boundary	O
limit	O
of	O
as	O
is	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
.	O
</s>
<s>
Titchmarsh	O
's	O
theorem	O
(	O
named	O
for	O
E	O
.	O
C	O
.	O
Titchmarsh	O
who	O
included	O
it	O
in	O
his	O
1937	O
work	O
)	O
makes	O
precise	O
the	O
relationship	O
between	O
the	O
boundary	O
values	O
of	O
holomorphic	O
functions	O
in	O
the	O
upper	O
half-plane	O
and	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
It	O
gives	O
necessary	O
and	O
sufficient	O
conditions	O
for	O
a	O
complex-valued	O
square-integrable	B-Algorithm
function	I-Algorithm
on	O
the	O
real	O
line	O
to	O
be	O
the	O
boundary	O
value	O
of	O
a	O
function	O
in	O
the	O
Hardy	O
space	O
of	O
holomorphic	O
functions	O
in	O
the	O
upper	O
half-plane	O
.	O
</s>
<s>
The	O
theorem	O
states	O
that	O
the	O
following	O
conditions	O
for	O
a	O
complex-valued	O
square-integrable	B-Algorithm
function	I-Algorithm
are	O
equivalent	O
:	O
</s>
<s>
The	O
real	O
and	O
imaginary	O
parts	O
of	O
are	O
Hilbert	B-Algorithm
transforms	I-Algorithm
of	O
each	O
other	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
vanishes	O
for	O
.	O
</s>
<s>
where	O
is	O
a	O
real-valued	O
function	O
in	O
and	O
is	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
(	O
of	O
class	O
)	O
of	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
an	O
function	O
need	O
not	O
converge	O
in	O
the	O
mean	O
to	O
another	O
function	O
.	O
</s>
<s>
For	O
a	O
periodic	O
function	O
the	O
circular	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
defined	O
:	O
</s>
<s>
The	O
circular	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
used	O
in	O
giving	O
a	O
characterization	O
of	O
Hardy	O
space	O
and	O
in	O
the	O
study	O
of	O
the	O
conjugate	O
function	O
in	O
Fourier	O
series	O
.	O
</s>
<s>
is	O
known	O
as	O
the	O
Hilbert	B-Algorithm
kernel	I-Algorithm
since	O
it	O
was	O
in	O
this	O
form	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
was	O
originally	O
studied	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
kernel	I-Algorithm
(	O
for	O
the	O
circular	O
Hilbert	B-Algorithm
transform	I-Algorithm
)	O
can	O
be	O
obtained	O
by	O
making	O
the	O
Cauchy	O
kernel	O
periodic	O
.	O
</s>
<s>
Many	O
results	O
about	O
the	O
circular	O
Hilbert	B-Algorithm
transform	I-Algorithm
may	O
be	O
derived	O
from	O
the	O
corresponding	O
results	O
for	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
from	O
this	O
correspondence	O
.	O
</s>
<s>
A	O
Fourier	B-Algorithm
transform	I-Algorithm
property	O
indicates	O
that	O
this	O
complex	O
heterodyne	B-Algorithm
operation	O
can	O
shift	O
all	O
the	O
negative	O
frequency	O
components	O
of	O
above	O
0Hz	O
.	O
</s>
<s>
In	O
that	O
case	O
,	O
the	O
imaginary	O
part	O
of	O
the	O
result	O
is	O
a	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
the	O
real	O
part	O
.	O
</s>
<s>
This	O
is	O
an	O
indirect	O
way	O
to	O
produce	O
Hilbert	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
is	O
called	O
angle	B-Device
modulation	I-Device
,	O
which	O
includes	O
both	O
phase	B-Device
modulation	I-Device
and	O
frequency	B-Application
modulation	I-Application
.	O
</s>
<s>
the	O
result	O
is	O
single-sideband	B-Device
modulation	I-Device
:	O
</s>
<s>
The	O
function	O
presents	O
two	O
challenges	O
to	O
practical	O
implementation	O
as	O
a	O
convolution	B-Language
:	O
</s>
<s>
But	O
windowing	B-Language
the	O
length	O
also	O
reduces	O
the	O
effective	O
frequency	O
range	O
of	O
the	O
transform	O
.	O
</s>
<s>
See	O
also	O
quadrature	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
It	O
is	O
a	O
non-causal	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
For	O
a	O
discrete	O
function	O
,	O
with	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
)	O
,	O
and	O
discrete	O
Hilbert	B-Algorithm
transform	I-Algorithm
the	O
DTFT	O
of	O
in	O
the	O
region	O
is	O
given	O
by	O
:	O
</s>
<s>
The	O
inverse	O
DTFT	O
,	O
using	O
the	O
convolution	B-Language
theorem	O
,	O
is	O
:	O
</s>
<s>
When	O
the	O
convolution	B-Language
is	O
performed	O
numerically	O
,	O
an	O
FIR	O
approximation	O
is	O
substituted	O
for	O
,	O
as	O
shown	O
in	O
Figure	O
1	O
.	O
</s>
<s>
The	O
MATLAB	B-Language
function	O
,	O
,	O
convolves	O
a	O
u[n]	O
sequence	O
with	O
the	O
periodic	B-Algorithm
summation	I-Algorithm
:	O
</s>
<s>
The	O
convolution	B-Language
is	O
implemented	O
in	O
the	O
frequency	O
domain	O
as	O
the	O
product	O
of	O
the	O
array	O
with	O
samples	O
of	O
the	O
distribution	O
(	O
whose	O
real	O
and	O
imaginary	O
components	O
are	O
all	O
just	O
0	O
or	O
)	O
.	O
</s>
<s>
Given	O
an	O
FIR	O
approximation	O
for	O
denoted	O
by	O
substituting	O
for	O
the	O
samples	O
results	O
in	O
an	O
FIR	O
version	O
of	O
the	O
convolution	B-Language
.	O
</s>
<s>
When	O
the	O
input	O
is	O
a	O
segment	O
of	O
a	O
pure	O
cosine	O
,	O
the	O
resulting	O
convolution	B-Language
for	O
two	O
different	O
values	O
of	O
is	O
depicted	O
in	O
Figure	O
4	O
(	O
red	O
and	O
blue	O
plots	O
)	O
.	O
</s>
<s>
An	O
appreciation	O
for	O
the	O
edge	O
effects	O
is	O
important	O
when	O
a	O
method	O
called	O
overlap-save	B-Algorithm
is	O
used	O
to	O
perform	O
the	O
convolution	B-Language
on	O
a	O
long	O
sequence	O
.	O
</s>
<s>
In	O
the	O
example	O
,	O
a	O
sine	O
function	O
is	O
created	O
by	O
computing	O
the	O
Discrete	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
a	O
cosine	O
function	O
,	O
which	O
was	O
processed	O
in	O
four	O
overlapping	O
segments	O
,	O
and	O
pieced	O
back	O
together	O
.	O
</s>
<s>
Effectively	O
,	O
,	O
whereas	O
the	O
overlap-save	B-Algorithm
method	I-Algorithm
needs	O
.	O
</s>
<s>
The	O
number	O
theoretic	O
Hilbert	B-Algorithm
transform	I-Algorithm
is	O
an	O
extension	O
of	O
the	O
discrete	O
Hilbert	B-Algorithm
transform	I-Algorithm
to	O
integers	O
modulo	O
an	O
appropriate	O
prime	O
number	O
.	O
</s>
<s>
In	O
this	O
it	O
follows	O
the	O
generalization	O
of	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
to	O
number	O
theoretic	O
transforms	O
.	O
</s>
<s>
The	O
number	O
theoretic	O
Hilbert	B-Algorithm
transform	I-Algorithm
can	O
be	O
used	O
to	O
generate	O
sets	O
of	O
orthogonal	O
discrete	O
sequences	O
.	O
</s>
