<s>
the	O
case	O
of	O
the	O
spectrogram	O
or	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
the	O
method	O
of	O
reassignment	O
sharpens	O
blurry	O
time-frequency	O
data	O
by	O
relocating	O
the	O
data	O
according	O
to	O
local	O
estimates	O
of	O
instantaneous	O
frequency	O
and	O
group	O
delay	O
.	O
</s>
<s>
One	O
of	O
the	O
best-known	O
time-frequency	O
representations	O
is	O
the	O
spectrogram	O
,	O
defined	O
as	O
the	O
squared	O
magnitude	O
of	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
A	O
time-frequency	O
representation	O
having	O
improved	O
resolution	O
,	O
relative	O
to	O
the	O
spectrogram	O
,	O
is	O
the	O
Wigner	O
–	O
Ville	O
distribution	O
,	O
which	O
may	O
be	O
interpreted	O
as	O
a	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
with	O
a	O
window	O
function	O
that	O
is	O
perfectly	O
matched	O
to	O
the	O
signal	O
.	O
</s>
<s>
where	O
is	O
a	O
(	O
real-valued	O
)	O
lowpass	O
kernel	O
function	O
,	O
like	O
the	O
window	O
function	O
in	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
time-frequency	O
coordinates	O
thus	O
computed	O
are	O
equal	O
to	O
the	O
local	O
group	O
delay	O
,	O
and	O
local	O
instantaneous	O
frequency	O
,	O
and	O
are	O
computed	O
from	O
the	O
phase	O
of	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
which	O
is	O
normally	O
ignored	O
when	O
constructing	O
the	O
spectrogram	O
.	O
</s>
<s>
cannot	O
be	O
applied	O
directly	O
to	O
the	O
discrete	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
data	O
,	O
because	O
partial	O
derivatives	O
cannot	O
be	O
computed	O
directly	O
on	O
data	O
that	O
is	O
discrete	O
in	O
time	O
and	O
frequency	O
,	O
and	O
it	O
has	O
been	O
suggested	O
that	O
this	O
difficulty	O
has	O
been	O
the	O
primary	O
barrier	O
to	O
wider	O
use	O
of	O
the	O
method	O
of	O
reassignment	O
.	O
</s>
<s>
where	O
is	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
computed	O
using	O
an	O
analysis	O
window	O
is	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
computed	O
using	O
a	O
time-weighted	O
analysis	O
window	O
and	O
is	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
computed	O
using	O
a	O
time-derivative	O
analysis	O
window	O
.	O
</s>
<s>
Since	O
these	O
algorithms	O
operate	O
only	O
on	O
short-time	O
spectral	O
data	O
evaluated	O
at	O
a	O
single	O
time	O
and	O
frequency	O
,	O
and	O
do	O
not	O
explicitly	O
compute	O
any	O
derivatives	O
,	O
this	O
gives	O
an	O
efficient	O
method	O
of	O
computing	O
the	O
reassigned	O
discrete	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
often	O
be	O
used	O
to	O
estimate	O
the	O
amplitudes	O
and	O
phases	O
of	O
the	O
individual	O
components	O
in	O
a	O
multi-component	O
signal	O
,	O
such	O
as	O
a	O
quasi-harmonic	O
musical	O
instrument	O
tone	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
time	O
and	O
frequency	O
reassignment	O
operations	O
can	O
be	O
used	O
to	O
sharpen	O
the	O
representation	O
by	O
attributing	O
the	O
spectral	O
energy	O
reported	O
by	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
to	O
the	O
point	O
that	O
is	O
the	O
local	O
center	O
of	O
gravity	O
of	O
the	O
complex	O
energy	O
distribution	O
.	O
</s>
<s>
For	O
a	O
signal	O
consisting	O
of	O
a	O
single	O
component	O
,	O
the	O
instantaneous	O
frequency	O
can	O
be	O
estimated	O
from	O
the	O
partial	O
derivatives	O
of	O
phase	O
of	O
any	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
channel	O
that	O
passes	O
the	O
component	O
.	O
</s>
<s>
If	O
this	O
property	O
is	O
not	O
met	O
,	O
then	O
the	O
desired	O
multi-component	O
decomposition	O
cannot	O
be	O
achieved	O
,	O
because	O
the	O
parameters	O
of	O
individual	O
components	O
cannot	O
be	O
estimated	O
from	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
If	O
the	O
components	O
of	O
a	O
signal	O
are	O
separable	O
in	O
frequency	O
with	O
respect	O
to	O
a	O
particular	O
short-time	O
spectral	O
analysis	O
window	O
,	O
then	O
the	O
output	O
of	O
each	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
filter	O
is	O
a	O
filtered	O
version	O
of	O
,	O
at	O
most	O
,	O
a	O
single	O
dominant	O
(	O
having	O
significant	O
energy	O
)	O
component	O
,	O
and	O
so	O
the	O
derivative	O
,	O
with	O
respect	O
to	O
time	O
,	O
of	O
the	O
phase	O
of	O
the	O
is	O
equal	O
to	O
the	O
derivative	O
with	O
respect	O
to	O
time	O
,	O
of	O
the	O
phase	O
of	O
the	O
dominant	O
component	O
at	O
Therefore	O
,	O
if	O
a	O
component	O
,	O
having	O
instantaneous	O
frequency	O
is	O
the	O
dominant	O
component	O
in	O
the	O
vicinity	O
of	O
then	O
the	O
instantaneous	O
frequency	O
of	O
that	O
component	O
can	O
be	O
computed	O
from	O
the	O
phase	O
of	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
evaluated	O
at	O
That	O
is	O
,	O
</s>
<s>
Just	O
as	O
each	O
bandpass	O
filter	O
in	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
filterbank	O
may	O
pass	O
at	O
most	O
a	O
single	O
complex	O
exponential	O
component	O
,	O
two	O
temporal	O
events	O
must	O
be	O
sufficiently	O
separated	O
in	O
time	O
that	O
they	O
do	O
not	O
lie	O
in	O
the	O
same	O
windowed	O
segment	O
of	O
the	O
input	O
signal	O
.	O
</s>
