<s>
The	O
Gabor	B-Algorithm
transform	I-Algorithm
,	O
named	O
after	O
Dennis	O
Gabor	O
,	O
is	O
a	O
special	O
case	O
of	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
function	O
to	O
be	O
transformed	O
is	O
first	O
multiplied	O
by	O
a	O
Gaussian	O
function	O
,	O
which	O
can	O
be	O
regarded	O
as	O
a	O
window	B-Language
function	I-Language
,	O
and	O
the	O
resulting	O
function	O
is	O
then	O
transformed	O
with	O
a	O
Fourier	O
transform	O
to	O
derive	O
the	O
time-frequency	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
The	O
window	B-Language
function	I-Language
means	O
that	O
the	O
signal	O
near	O
the	O
time	O
being	O
analyzed	O
will	O
have	O
higher	O
weight	O
.	O
</s>
<s>
The	O
Gabor	B-Algorithm
transform	I-Algorithm
of	O
a	O
signal	O
x(t )	O
is	O
defined	O
by	O
this	O
formula	O
:	O
</s>
<s>
This	O
simplification	O
makes	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
practical	O
and	O
realizable	O
.	O
</s>
<s>
The	O
window	B-Language
function	I-Language
width	O
can	O
also	O
be	O
varied	O
to	O
optimize	O
the	O
time-frequency	O
resolution	O
tradeoff	O
for	O
a	O
particular	O
application	O
by	O
replacing	O
the	O
with	O
for	O
some	O
chosen	O
.	O
</s>
<s>
The	O
Gabor	B-Algorithm
transform	I-Algorithm
is	O
invertible	O
.	O
</s>
<s>
The	O
Gabor	B-Algorithm
transform	I-Algorithm
has	O
many	O
properties	O
like	O
those	O
of	O
the	O
Fourier	O
transform	O
.	O
</s>
<s>
The	O
main	O
application	O
of	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
is	O
used	O
in	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Through	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
by	O
applying	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
,	O
the	O
available	O
bandwidth	O
can	O
be	O
known	O
and	O
those	O
frequency	O
bands	O
can	O
be	O
used	O
for	O
other	O
applications	O
and	O
bandwidth	O
is	O
saved	O
.	O
</s>
<s>
The	O
right	O
side	O
picture	O
shows	O
the	O
input	O
signal	O
x(t )	O
and	O
the	O
output	O
of	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
As	O
in	O
short	B-Algorithm
time	I-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
the	O
resolution	O
in	O
time	O
and	O
frequency	O
domain	O
can	O
be	O
adjusted	O
by	O
choosing	O
different	O
window	B-Language
function	I-Language
width	O
.	O
</s>
<s>
In	O
Gabor	B-Algorithm
transform	I-Algorithm
cases	O
,	O
by	O
adding	O
variance	O
,	O
as	O
following	O
equation	O
:	O
</s>
<s>
So	O
the	O
Scaled	O
Gabor	B-Algorithm
transform	I-Algorithm
can	O
be	O
written	O
as	O
:	O
</s>
<s>
With	O
a	O
large	O
,	O
the	O
window	B-Language
function	I-Language
will	O
be	O
narrow	O
,	O
causing	O
higher	O
resolution	O
in	O
time	O
domain	O
but	O
lower	O
resolution	O
in	O
frequency	O
domain	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
time-frequency	B-Algorithm
analysis	I-Algorithm
based	O
on	O
the	O
resulting	O
complex-valued	O
extension	O
of	O
the	O
time-causal	O
limit	O
kernel	O
makes	O
it	O
possible	O
to	O
capture	O
essentially	O
similar	O
transformations	O
of	O
a	O
temporal	O
signal	O
as	O
the	O
Gabor	O
function	O
can	O
,	O
and	O
corresponding	O
to	O
the	O
Heisenberg	O
group	O
,	O
see	O
for	O
further	O
details	O
.	O
</s>
