<s>
In	O
signal	O
processing	O
,	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
comprises	O
those	O
techniques	O
that	O
study	O
a	O
signal	O
in	O
both	O
the	O
time	O
and	O
frequency	O
domains	O
simultaneously	O
,	O
using	O
various	O
time	O
–	O
frequency	O
representations	O
.	O
</s>
<s>
Rather	O
than	O
viewing	O
a	O
1-dimensional	O
signal	O
(	O
a	O
function	O
,	O
real	O
or	O
complex-valued	O
,	O
whose	O
domain	O
is	O
the	O
real	O
line	O
)	O
and	O
some	O
transform	O
(	O
another	O
function	O
whose	O
domain	O
is	O
the	O
real	O
line	O
,	O
obtained	O
from	O
the	O
original	O
via	O
some	O
transform	O
)	O
,	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
studies	O
a	O
two-dimensional	O
signal	O
–	O
a	O
function	O
whose	O
domain	O
is	O
the	O
two-dimensional	O
real	O
plane	O
,	O
obtained	O
from	O
the	O
signal	O
via	O
a	O
time	O
–	O
frequency	O
transform	O
.	O
</s>
<s>
A	O
simple	O
example	O
is	O
that	O
the	O
4-fold	O
periodicity	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
–	O
and	O
the	O
fact	O
that	O
two-fold	O
Fourier	B-Algorithm
transform	I-Algorithm
reverses	O
direction	O
–	O
can	O
be	O
interpreted	O
by	O
considering	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
as	O
a	O
90°	O
rotation	O
in	O
the	O
associated	O
time	O
–	O
frequency	O
plane	O
:	O
4	O
such	O
rotations	O
yield	O
the	O
identity	O
,	O
and	O
2	O
such	O
rotations	O
simply	O
reverse	O
direction	O
(	O
reflection	O
through	O
the	O
origin	O
)	O
.	O
</s>
<s>
The	O
practical	O
motivation	O
for	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
is	O
that	O
classical	O
Fourier	O
analysis	O
assumes	O
that	O
signals	O
are	O
infinite	O
in	O
time	O
or	O
periodic	O
,	O
while	O
many	O
signals	O
in	O
practice	O
are	O
of	O
short	O
duration	O
,	O
and	O
change	O
substantially	O
over	O
their	O
duration	O
.	O
</s>
<s>
This	O
is	O
poorly	O
represented	O
by	O
traditional	O
methods	O
,	O
which	O
motivates	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
most	O
basic	O
forms	O
of	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
is	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
STFT	B-Algorithm
)	O
,	O
but	O
more	O
sophisticated	O
techniques	O
have	O
been	O
developed	O
,	O
notably	O
wavelets	O
and	O
least-squares	B-General_Concept
spectral	I-General_Concept
analysis	I-General_Concept
methods	O
for	O
unevenly	O
spaced	O
data	O
.	O
</s>
<s>
In	O
signal	O
processing	O
,	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
is	O
a	O
body	O
of	O
techniques	O
and	O
methods	O
used	O
for	O
characterizing	O
and	O
manipulating	O
signals	O
whose	O
statistics	O
vary	O
in	O
time	O
,	O
such	O
as	O
transient	O
signals	O
.	O
</s>
<s>
Since	O
many	O
signals	O
of	O
interest	O
–	O
such	O
as	O
speech	O
,	O
music	O
,	O
images	O
,	O
and	O
medical	O
signals	O
–	O
have	O
changing	O
frequency	O
characteristics	O
,	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
has	O
broad	O
scope	O
of	O
applications	O
.	O
</s>
<s>
Whereas	O
the	O
technique	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
extended	O
to	O
obtain	O
the	O
frequency	O
spectrum	O
of	O
any	O
slowly	O
growing	O
locally	O
integrable	O
signal	O
,	O
this	O
approach	O
requires	O
a	O
complete	O
description	O
of	O
the	O
signal	O
's	O
behavior	O
over	O
all	O
time	O
.	O
</s>
<s>
Once	O
such	O
a	O
representation	O
has	O
been	O
generated	O
other	O
techniques	O
in	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
may	O
then	O
be	O
applied	O
to	O
the	O
signal	O
in	O
order	O
to	O
extract	O
information	O
from	O
the	O
signal	O
,	O
to	O
separate	O
the	O
signal	O
from	O
noise	O
or	O
interfering	O
signals	O
,	O
etc	O
.	O
</s>
<s>
Short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
including	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
)	O
,	O
</s>
<s>
Wavelet	B-Algorithm
transform	I-Algorithm
,	O
</s>
<s>
Bilinear	O
time	O
–	O
frequency	O
distribution	O
function	O
(	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
,	O
or	O
WDF	O
)	O
,	O
</s>
<s>
Modified	B-Algorithm
Wigner	I-Algorithm
distribution	I-Algorithm
function	I-Algorithm
,	O
Gabor	O
–	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
,	O
and	O
so	O
on	O
(	O
see	O
Gabor	B-Algorithm
–	I-Algorithm
Wigner	I-Algorithm
transform	I-Algorithm
)	O
.	O
</s>
<s>
The	O
high	O
clarity	O
of	O
the	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
(	O
WDF	O
)	O
obtained	O
for	O
some	O
signals	O
is	O
due	O
to	O
the	O
auto-correlation	O
function	O
inherent	O
in	O
its	O
formulation	O
;	O
however	O
,	O
the	O
latter	O
also	O
causes	O
the	O
cross-term	O
problem	O
.	O
</s>
<s>
Therefore	O
,	O
if	O
we	O
want	O
to	O
analyze	O
a	O
single-term	O
signal	O
,	O
using	O
the	O
WDF	O
may	O
be	O
the	O
best	O
approach	O
;	O
if	O
the	O
signal	O
is	O
composed	O
of	O
multiple	O
components	O
,	O
some	O
other	O
methods	O
like	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
,	O
Gabor-Wigner	O
distribution	O
or	O
Modified	O
B-Distribution	O
functions	O
may	O
be	O
better	O
choices	O
.	O
</s>
<s>
But	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
can	O
.	O
</s>
<s>
Stationary	B-Algorithm
random	I-Algorithm
process	I-Algorithm
:	O
the	O
statistical	O
properties	O
do	O
not	O
change	O
with	O
t	O
.	O
Its	O
auto-covariance	O
function	O
:	O
</s>
<s>
If	O
varies	O
with	O
and	O
is	O
nonzero	O
when	O
,	O
then	O
is	O
a	O
non-stationary	B-Algorithm
random	O
process	O
.	O
</s>
<s>
Decompose	O
by	O
the	O
AF	O
and	O
the	O
FRFT	O
Any	O
non-stationary	B-Algorithm
random	O
process	O
can	O
be	O
expressed	O
as	O
a	O
summation	O
of	O
the	O
fractional	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
or	O
chirp	O
multiplication	O
)	O
of	O
stationary	B-Algorithm
random	I-Algorithm
process	I-Algorithm
.	O
</s>
<s>
The	O
Linear	B-Algorithm
canonical	I-Algorithm
transform	I-Algorithm
(	O
LCT	O
)	O
is	O
really	O
helpful	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
LCTs	O
can	O
shift	O
the	O
time	O
–	O
frequency	O
distribution	O
to	O
any	O
location	O
,	O
dilate	O
it	O
in	O
the	O
horizontal	O
and	O
vertical	O
direction	O
without	O
changing	O
its	O
area	O
on	O
the	O
plane	O
,	O
shear	O
(	O
or	O
twist	O
)	O
it	O
,	O
and	O
rotate	O
it	O
(	O
Fractional	O
Fourier	B-Algorithm
transform	I-Algorithm
)	O
.	O
</s>
<s>
By	O
using	O
the	O
time	O
–	O
frequency	O
distribution	O
function	O
,	O
we	O
can	O
filter	O
in	O
the	O
Euclidean	O
time	O
–	O
frequency	O
domain	O
or	O
in	O
the	O
fractional	O
domain	O
by	O
employing	O
the	O
fractional	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Filter	O
design	O
in	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
always	O
deals	O
with	O
signals	O
composed	O
of	O
multiple	O
components	O
,	O
so	O
one	O
cannot	O
use	O
WDF	O
due	O
to	O
cross-term	O
.	O
</s>
<s>
The	O
Gabor	B-Algorithm
transform	I-Algorithm
,	O
Gabor	O
–	O
Wigner	B-Algorithm
distribution	I-Algorithm
function	I-Algorithm
,	O
or	O
Cohen	O
's	O
class	O
distribution	O
function	O
may	O
be	O
better	O
choices	O
.	O
</s>
<s>
Such	O
filtering	O
is	O
traditionally	O
done	O
in	O
the	O
time	O
domain	O
or	O
in	O
the	O
frequency	O
domain	O
;	O
however	O
,	O
this	O
may	O
not	O
be	O
possible	O
in	O
the	O
case	O
of	O
non-stationary	B-Algorithm
signals	O
that	O
are	O
multicomponent	O
as	O
such	O
components	O
could	O
overlap	O
in	O
both	O
the	O
time	O
domain	O
and	O
also	O
in	O
the	O
frequency	O
domain	O
;	O
as	O
a	O
consequence	O
,	O
the	O
only	O
possible	O
way	O
to	O
achieve	O
component	O
separation	O
and	O
therefore	O
a	O
signal	O
decomposition	O
is	O
to	O
implement	O
a	O
time	O
–	O
frequency	O
filter	O
.	O
</s>
<s>
By	O
the	O
Nyquist	O
–	O
Shannon	O
sampling	O
theorem	O
,	O
we	O
can	O
conclude	O
that	O
the	O
minimum	O
number	O
of	O
sampling	O
points	O
without	O
aliasing	B-Error_Name
is	O
equivalent	O
to	O
the	O
area	O
of	O
the	O
time	O
–	O
frequency	O
distribution	O
of	O
a	O
signal	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
using	O
Gabor	B-Algorithm
transform	I-Algorithm
causes	O
an	O
improvement	O
in	O
the	O
clarity	O
and	O
readability	O
of	O
the	O
representation	O
,	O
therefore	O
improving	O
its	O
interpretation	O
and	O
application	O
to	O
practical	O
problems	O
.	O
</s>
<s>
Consequently	O
,	O
when	O
the	O
signal	O
we	O
tend	O
to	O
sample	O
is	O
composed	O
of	O
single	O
component	O
,	O
we	O
use	O
the	O
WDF	O
;	O
however	O
,	O
if	O
the	O
signal	O
consists	O
of	O
more	O
than	O
one	O
component	O
,	O
using	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
,	O
Gabor-Wigner	O
distribution	O
function	O
,	O
or	O
other	O
reduced	O
interference	O
TFDs	O
may	O
achieve	O
better	O
results	O
.	O
</s>
<s>
Conventionally	O
,	O
the	O
operation	O
of	O
modulation	B-Application
and	O
multiplexing	B-Architecture
concentrates	O
in	O
time	O
or	O
in	O
frequency	O
,	O
separately	O
.	O
</s>
<s>
Light	O
is	O
an	O
electromagnetic	O
wave	O
,	O
so	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
applies	O
to	O
optics	O
in	O
the	O
same	O
way	O
as	O
for	O
general	O
electromagnetic	O
wave	O
propagation	O
.	O
</s>
<s>
As	O
acoustic	O
signals	O
are	O
used	O
as	O
speech	O
in	O
communication	O
between	O
the	O
human-sender	O
and	O
-receiver	O
,	O
their	O
undelayedly	O
transmission	O
in	O
technical	O
communication	O
systems	O
is	O
crucial	O
,	O
which	O
makes	O
the	O
use	O
of	O
simpler	O
TFDs	O
,	O
such	O
as	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
,	O
suitable	O
to	O
analyze	O
these	O
signals	O
in	O
real-time	O
by	O
reducing	O
computational	O
complexity	O
.	O
</s>
<s>
In	O
biomedicine	O
,	O
one	O
can	O
use	O
time	O
–	O
frequency	O
distribution	O
to	O
analyze	O
the	O
electromyography	O
(	O
EMG	O
)	O
,	O
electroencephalography	B-Application
(	O
EEG	B-Application
)	O
,	O
electrocardiogram	B-Application
(	O
ECG	B-Application
)	O
or	O
otoacoustic	O
emissions	O
(	O
OAEs	O
)	O
.	O
</s>
<s>
Early	O
work	O
in	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
can	O
be	O
seen	O
in	O
the	O
Haar	O
wavelets	O
(	O
1909	O
)	O
of	O
Alfréd	O
Haar	O
,	O
though	O
these	O
were	O
not	O
significantly	O
applied	O
to	O
signal	O
processing	O
.	O
</s>
<s>
More	O
substantial	O
work	O
was	O
undertaken	O
by	O
Dennis	O
Gabor	O
,	O
such	O
as	O
Gabor	O
atoms	O
(	O
1947	O
)	O
,	O
an	O
early	O
form	O
of	O
wavelets	O
,	O
and	O
the	O
Gabor	B-Algorithm
transform	I-Algorithm
,	O
a	O
modified	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Particularly	O
in	O
the	O
1930s	O
and	O
1940s	O
,	O
early	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
developed	O
in	O
concert	O
with	O
quantum	O
mechanics	O
(	O
Wigner	O
developed	O
the	O
Wigner	O
–	O
Ville	O
distribution	O
in	O
1932	O
in	O
quantum	O
mechanics	O
,	O
and	O
Gabor	O
was	O
influenced	O
by	O
quantum	O
mechanics	O
–	O
see	O
Gabor	O
atom	O
)	O
;	O
this	O
is	O
reflected	O
in	O
the	O
shared	O
mathematics	O
of	O
the	O
position-momentum	O
plane	O
and	O
the	O
time	O
–	O
frequency	O
plane	O
–	O
as	O
in	O
the	O
Heisenberg	O
uncertainty	O
principle	O
(	O
quantum	O
mechanics	O
)	O
and	O
the	O
Gabor	O
limit	O
(	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
)	O
,	O
ultimately	O
both	O
reflecting	O
a	O
symplectic	O
structure	O
.	O
</s>
<s>
An	O
early	O
practical	O
motivation	O
for	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
was	O
the	O
development	O
of	O
radar	O
–	O
see	O
ambiguity	B-Algorithm
function	I-Algorithm
.	O
</s>
