<s>
The	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
(	O
WDF	O
)	O
is	O
used	O
in	O
signal	O
processing	O
as	O
a	O
transform	O
in	O
time-frequency	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Given	O
the	O
shared	O
algebraic	O
structure	O
between	O
position-momentum	O
and	O
time-frequency	O
conjugate	O
pairs	O
,	O
it	O
also	O
usefully	O
serves	O
in	O
signal	O
processing	O
,	O
as	O
a	O
transform	O
in	O
time-frequency	B-Algorithm
analysis	I-Algorithm
,	O
the	O
subject	O
of	O
this	O
article	O
.	O
</s>
<s>
Compared	O
to	O
a	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
such	O
as	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
,	O
the	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
provides	O
the	O
highest	O
possible	O
temporal	O
vs	O
frequency	O
resolution	O
which	O
is	O
mathematically	O
possible	O
within	O
the	O
limitations	O
of	O
the	O
uncertainty	O
principle	O
.	O
</s>
<s>
Subsequent	O
modifications	O
have	O
been	O
proposed	O
which	O
preserve	O
the	O
sharpness	O
of	O
the	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
but	O
largely	O
suppress	O
cross	O
terms	O
.	O
</s>
<s>
There	O
are	O
several	O
different	O
definitions	O
for	O
the	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
definition	O
given	O
here	O
is	O
specific	O
to	O
time-frequency	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
The	O
motivation	O
for	O
the	O
Wigner	O
function	O
is	O
that	O
it	O
reduces	O
to	O
the	O
spectral	B-Algorithm
density	I-Algorithm
function	I-Algorithm
at	O
all	O
times	O
for	O
stationary	O
processes	O
,	O
yet	O
it	O
is	O
fully	O
equivalent	O
to	O
the	O
non-stationary	O
autocorrelation	O
function	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
Wigner	O
function	O
tells	O
us	O
(	O
roughly	O
)	O
how	O
the	O
spectral	B-Algorithm
density	I-Algorithm
changes	O
in	O
time	O
.	O
</s>
<s>
Here	O
are	O
some	O
examples	O
illustrating	O
how	O
the	O
WDF	O
is	O
used	O
in	O
time-frequency	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
When	O
the	O
input	O
signal	O
is	O
a	O
linear	O
chirp	B-Application
function	I-Application
,	O
the	O
instantaneous	O
frequency	O
is	O
a	O
linear	O
function	O
.	O
</s>
<s>
The	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
is	O
best	O
suited	O
for	O
time-frequency	B-Algorithm
analysis	I-Algorithm
when	O
the	O
input	O
signal	O
's	O
phase	O
is	O
2nd	O
order	O
or	O
lower	O
.	O
</s>
<s>
The	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
is	O
not	O
a	O
linear	O
transform	O
.	O
</s>
<s>
A	O
cross	O
term	O
(	O
"	O
time	O
beats	O
"	O
)	O
occurs	O
when	O
there	O
is	O
more	O
than	O
one	O
component	O
in	O
the	O
input	O
signal	O
,	O
analogous	O
in	O
time	O
to	O
frequency	B-Application
beats	I-Application
.	O
</s>
<s>
By	O
contrast	O
,	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
does	O
not	O
have	O
this	O
feature	O
.	O
</s>
<s>
The	O
following	O
are	O
some	O
examples	O
that	O
exhibit	O
the	O
cross-term	O
feature	O
of	O
the	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
reduce	O
the	O
cross-term	O
difficulty	O
,	O
several	O
approaches	O
have	O
been	O
proposed	O
in	O
the	O
literature	O
,	O
some	O
of	O
them	O
leading	O
to	O
new	O
transforms	O
as	O
the	O
modified	B-Algorithm
Wigner	I-Algorithm
distribution	I-Algorithm
function	I-Algorithm
,	O
the	O
Gabor	B-Algorithm
–	I-Algorithm
Wigner	I-Algorithm
transform	I-Algorithm
,	O
the	O
Choi-Williams	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
and	O
Cohen	O
's	O
class	O
distribution	O
.	O
</s>
<s>
The	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
has	O
several	O
evident	O
properties	O
listed	O
in	O
the	O
following	O
table	O
.	O
</s>
<s>
When	O
a	O
signal	O
is	O
not	O
time	O
limited	O
,	O
its	O
Wigner	B-Algorithm
Distribution	I-Algorithm
Function	I-Algorithm
is	O
hard	O
to	O
implement	O
.	O
</s>
