<s>
The	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
STFT	B-Algorithm
)	O
,	O
is	O
a	O
Fourier-related	B-Algorithm
transform	I-Algorithm
used	O
to	O
determine	O
the	O
sinusoidal	O
frequency	O
and	O
phase	O
content	O
of	O
local	O
sections	O
of	O
a	O
signal	O
as	O
it	O
changes	O
over	O
time	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
procedure	O
for	O
computing	O
STFTs	B-Algorithm
is	O
to	O
divide	O
a	O
longer	O
time	O
signal	O
into	O
shorter	O
segments	O
of	O
equal	O
length	O
and	O
then	O
compute	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
separately	O
on	O
each	O
shorter	O
segment	O
.	O
</s>
<s>
One	O
then	O
usually	O
plots	O
the	O
changing	O
spectra	O
as	O
a	O
function	O
of	O
time	O
,	O
known	O
as	O
a	O
spectrogram	O
or	O
waterfall	O
plot	O
,	O
such	O
as	O
commonly	O
used	O
in	O
software	B-Architecture
defined	I-Architecture
radio	I-Architecture
(	O
SDR	O
)	O
based	O
spectrum	O
displays	O
.	O
</s>
<s>
Full	O
bandwidth	O
displays	O
covering	O
the	O
whole	O
range	O
of	O
an	O
SDR	O
commonly	O
use	O
fast	O
Fourier	B-Algorithm
transforms	I-Algorithm
(	O
FFTs	O
)	O
with	O
2^24	O
points	O
on	O
desktop	O
computers	O
.	O
</s>
<s>
Simply	O
,	O
in	O
the	O
continuous-time	O
case	O
,	O
the	O
function	O
to	O
be	O
transformed	O
is	O
multiplied	O
by	O
a	O
window	B-Language
function	I-Language
which	O
is	O
nonzero	O
for	O
only	O
a	O
short	O
period	O
of	O
time	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
a	O
one-dimensional	O
function	O
)	O
of	O
the	O
resulting	O
signal	O
is	O
taken	O
,	O
then	O
the	O
window	O
is	O
slid	O
along	O
the	O
time	O
axis	O
until	O
the	O
end	O
resulting	O
in	O
a	O
two-dimensional	O
representation	O
of	O
the	O
signal	O
.	O
</s>
<s>
where	O
is	O
the	O
window	B-Language
function	I-Language
,	O
commonly	O
a	O
Hann	O
window	O
or	O
Gaussian	O
window	O
centered	O
around	O
zero	O
,	O
and	O
is	O
the	O
signal	O
to	O
be	O
transformed	O
(	O
note	O
the	O
difference	O
between	O
the	O
window	B-Language
function	I-Language
and	O
the	O
frequency	O
)	O
.	O
</s>
<s>
is	O
essentially	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
,	O
a	O
complex	O
function	O
representing	O
the	O
phase	O
and	O
magnitude	O
of	O
the	O
signal	O
over	O
time	O
and	O
frequency	O
.	O
</s>
<s>
Often	O
phase	O
unwrapping	O
is	O
employed	O
along	O
either	O
or	O
both	O
the	O
time	O
axis	O
,	O
,	O
and	O
frequency	O
axis	O
,	O
,	O
to	O
suppress	O
any	O
jump	O
discontinuity	O
of	O
the	O
phase	O
result	O
of	O
the	O
STFT	B-Algorithm
.	O
</s>
<s>
Given	O
that	O
the	O
STFT	B-Algorithm
is	O
essentially	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
times	O
a	O
window	B-Language
function	I-Language
,	O
the	O
STFT	B-Algorithm
is	O
also	O
called	O
windowed	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
or	O
time-dependent	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
In	O
this	O
case	O
,	O
m	O
is	O
discrete	O
and	O
ω	O
is	O
continuous	O
,	O
but	O
in	O
most	O
typical	O
applications	O
the	O
STFT	B-Algorithm
is	O
performed	O
on	O
a	O
computer	O
using	O
the	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
so	O
both	O
variables	O
are	O
discrete	O
and	O
quantized	B-Algorithm
.	O
</s>
<s>
The	O
magnitude	O
squared	O
of	O
the	O
STFT	B-Algorithm
yields	O
the	O
spectrogram	O
representation	O
of	O
the	O
power	O
spectral	O
density	O
of	O
the	O
function	O
:	O
</s>
<s>
See	O
also	O
the	O
modified	O
discrete	O
cosine	O
transform	O
(	O
MDCT	O
)	O
,	O
which	O
is	O
also	O
a	O
Fourier-related	B-Algorithm
transform	I-Algorithm
that	O
uses	O
overlapping	O
windows	O
.	O
</s>
<s>
If	O
only	O
a	O
small	O
number	O
of	O
ω	O
are	O
desired	O
,	O
or	O
if	O
the	O
STFT	B-Algorithm
is	O
desired	O
to	O
be	O
evaluated	O
for	O
every	O
shift	O
m	O
of	O
the	O
window	O
,	O
then	O
the	O
STFT	B-Algorithm
may	O
be	O
more	O
efficiently	O
evaluated	O
using	O
a	O
sliding	B-Algorithm
DFT	I-Algorithm
algorithm	O
.	O
</s>
<s>
The	O
STFT	B-Algorithm
is	O
invertible	O
,	O
that	O
is	O
,	O
the	O
original	O
signal	O
can	O
be	O
recovered	O
from	O
the	O
transform	O
by	O
the	O
inverse	O
STFT	B-Algorithm
.	O
</s>
<s>
The	O
most	O
widely	O
accepted	O
way	O
of	O
inverting	O
the	O
STFT	B-Algorithm
is	O
by	O
using	O
the	O
overlap-add	B-Algorithm
(	O
OLA	O
)	O
method	O
,	O
which	O
also	O
allows	O
for	O
modifications	O
to	O
the	O
STFT	B-Algorithm
complex	O
spectrum	O
.	O
</s>
<s>
So	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
seen	O
as	O
a	O
sort	O
of	O
phase	O
coherent	O
sum	O
of	O
all	O
of	O
the	O
STFTs	B-Algorithm
of	O
x(t )	O
.	O
</s>
<s>
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
X(τ,ω )	O
for	O
τ	O
fixed	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
window	B-Language
function	I-Language
has	O
the	O
following	O
properties	O
:	O
</s>
<s>
One	O
of	O
the	O
pitfalls	O
of	O
the	O
STFT	B-Algorithm
is	O
that	O
it	O
has	O
a	O
fixed	O
resolution	O
.	O
</s>
<s>
This	O
is	O
one	O
of	O
the	O
reasons	O
for	O
the	O
creation	O
of	O
the	O
wavelet	B-Algorithm
transform	I-Algorithm
and	O
multiresolution	O
analysis	O
,	O
which	O
can	O
give	O
good	O
time	O
resolution	O
for	O
high-frequency	O
events	O
and	O
good	O
frequency	O
resolution	O
for	O
low-frequency	O
events	O
,	O
the	O
combination	O
best	O
suited	O
for	O
many	O
real	O
signals	O
.	O
</s>
<s>
The	O
boundary	O
of	O
the	O
uncertainty	O
principle	O
(	O
best	O
simultaneous	O
resolution	O
of	O
both	O
)	O
is	O
reached	O
with	O
a	O
Gaussian	O
window	B-Language
function	I-Language
(	O
or	O
mask	O
function	O
)	O
,	O
as	O
the	O
Gaussian	O
minimizes	O
the	O
Fourier	O
uncertainty	O
principle	O
.	O
</s>
<s>
This	O
is	O
called	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
(	O
and	O
with	O
modifications	O
for	O
multiresolution	O
becomes	O
the	O
Morlet	O
wavelet	B-Algorithm
transform	I-Algorithm
)	O
.	O
</s>
<s>
One	O
can	O
consider	O
the	O
STFT	B-Algorithm
for	O
varying	O
window	O
size	O
as	O
a	O
two-dimensional	O
domain	O
(	O
time	O
and	O
frequency	O
)	O
,	O
as	O
illustrated	O
in	O
the	O
example	O
below	O
,	O
which	O
can	O
be	O
calculated	O
by	O
varying	O
the	O
window	O
size	O
.	O
</s>
<s>
When	O
we	O
use	O
it	O
,	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
called	O
the	O
"	O
Gabor	B-Algorithm
transform	I-Algorithm
"	O
.	O
</s>
<s>
Taking	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
produces	O
N	O
complex	O
coefficients	O
.	O
</s>
<s>
The	O
Rayleigh	O
frequency	O
is	O
an	O
important	O
consideration	O
in	O
applications	O
of	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
STFT	B-Algorithm
)	O
,	O
as	O
well	O
as	O
any	O
other	O
method	O
of	O
harmonic	O
analysis	O
on	O
a	O
signal	O
of	O
finite	O
record-length	O
.	O
</s>
<s>
STFTs	B-Algorithm
as	O
well	O
as	O
standard	O
Fourier	B-Algorithm
transforms	I-Algorithm
and	O
other	O
tools	O
are	O
frequently	O
used	O
to	O
analyze	O
music	O
.	O
</s>
<s>
The	O
height	O
of	O
each	O
bar	O
(	O
augmented	O
by	O
color	O
)	O
represents	O
the	O
amplitude	B-Application
of	O
the	O
frequencies	O
within	O
that	O
band	O
.	O
</s>
