time vs frequesnvy
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Transcript of time vs frequesnvy
8/20/2019 time vs frequesnvy
http://slidepdf.com/reader/full/time-vs-frequesnvy 1/3
40
and
the
TF
signai
is
essentially
stationary
over
the
window's
time
span
[80].
The
shorter
duration
of
the
analysis
window is
what
constitutes
the
short-time
nature
of
the STFT.
The
expression
for
the
discrete-time
sTFT
at
any time
n is
given
by,
s(",
f):
,,I""
s(m)w(n
-
m)e-i2nr'n
Therefore,
the
discrete
sTFT
is
obtained
by
frequency
sampling
as,
Var(S(ju))
*
Var(s(t))
>-
C
where
Var(.)
denotes
the
variance,
and C
is
a
constant.
(21)
S(n,k):
^9(t,
f)
lf=fi3:nT
Q2)
Where
N
is
the
total
number
of data
points
in
the
window and
is
the
frequency
sampling
factor.
Substituting
Equati
on
(22)
into
Equation
(21)
we
obtain
the following
discrete
STFT,
S(n,k)
:
t
s(m)w(n
-
m)e=ilP-
An
important
tradeoff
in
short-time
spectral
analysis
is time
versus
frequency
resolution.
Good
time
resolution
requires
short
duration
windows
u;(t)
whereas
good.
frequency
resolution
necessitates
long
duration
windows
[85].
This
tradeoff
is
what
is
known
as Heisenberg
uncertainty
principle.
(23)
(24)
Wigner
Distribution
(WD)
A
time-frequency
characteristic
of
a signal
that
overcomes
the above
mentioned
uncertainty
is the
Wigner
distribution
(WD).
It
is
a
bilinear
transformation
which
8/20/2019 time vs frequesnvy
http://slidepdf.com/reader/full/time-vs-frequesnvy 2/3
4T
maps
a
one
dimensional
(1-D)
time-frequency
signal
into
a two-dimensional
(2-
D)
time-frequency
characteri
zation.
This
was
originally
developed
in
quantum
mechanics
by
Wigner
in
1932
[81].
This
signal
transformation
has
many
desirable
properties
that
make
it an ideal
tool
for
TF signal
analysis
[S2].
It
also
plays
the
role
of
a
generalization
of
spectral
density
function
for
nonstationary
random
process.
The
Wigner
distribution
function
is
given
by,
where
the
overbar
symbol
denotes
complex
conjugation.
The
WD presumes
the
time-varying
nature
or
nonstationarities
of
the
signal.
and
provides
information
concerning
the
instantaneous
frequency
of signals.
It
does
not
suffer
from the
time
versus
frequency
tradeoff problems
of
short-time
fourier
transform
technique
[82].
The
Wigner
distribution
also
presumes
the
temporal
and
spectral
support
of
the
signal
as well
as
time
and
frequency
domain
shifts.
A
relation
exists
between
the
STFT
and
WD
where
the
squared
magnitude
of
the
STFT
is
equal
to
the
convolution
of
the
WD
of
the
signal
with
the
WD of
the
window
tu(t)
[g3]
(the
window
of
STFT or
WD
as
they
are
same),
that
is,
w(t,u)
:
l:s(r
*
)eA
-
|)"-i,,
a,
(25)
lS"(f,
r)l'
:
*
I I
W"(r,rDrv-(t
-
r,w
-
ridrdrt.
(26)
where
W"
and
W'-
ate
the
Wigner
distribution
of
the windowed
signal
and
the
window
respectively.
The
rectangular
windowed
discrete-time
Wigner
distribution
(WD)
of the
signal
s(n)
is
formally
defined
by,
L
W(n,
u)
:
t
w(m)s(n
*
m)s(n
-
m)e-izum
m=-L
(27)
Thus
it is
seen
from the
above
equation
that the
WD is
an
explicit
function
of
n,
the
windows
time
center
and
o,
the
frequency
variable.
The
segment window
8/20/2019 time vs frequesnvy
http://slidepdf.com/reader/full/time-vs-frequesnvy 3/3
*(*)
is
of
n'idth
2L
+
I
in
the
interval
of
[n
_
L,n
*I].
If
the
assumed
to
be
statisticalry
stationary
then
the
above
equation
as
the spectral
behavior
in
the
signal's
correlation domain
[g4].
The
wigner
distribution
is
a
real
valued
periodic
function
of
frequency
with
a
period
of
normalized
digital
frequency
0.5.
For
a
symmetric
window,
the
discrete
Wigner
distribution
algorithm
is
given
by
replacing,
bV
T.
L
W"(n,n,
:
:,
u(m)s(n
*
m)s(n
-
m)e='**
A.\
windowed
signal
is
can
be
considered
(28)
L
:
-tll(0)l'@)l'+
2n
I
w@)s(n
+
m)s(n
_
m)e#,
rn:0
0<&<ir/-1
where
*
:
ff;&
:
0,
r,2,...,rtr
-
1
and
span
over
one
period.
ft
designates
the
rear
part
of
the
operator,
*
is
digitar
frequency,
$
i,
th"
normalized
digitar
frequency,
and
n
is
the
data
window
center
time.
From
the
above
equation
it
is
apparent
that
FFT
algorithm
can
be
applied
to
efltciently
compute
the
discrete
WD
sampled
values.
Let
n
be
the
set
of
window
center
times
as
it
moves
along
the
entire
data
length,
where
n
€
{nttv2t,..,nU}
and
M
is
the
number
of
steps
of
the
overlapped
moving
window
up
to
the
ultimate
position
of
the
data
length.
The
value
of
M
can
be
controlled
by
controliing
the
window
width
L
or
by controlling the
step
size
of
the
window
as
it
moves
along
the
entire
data
length'
The
step
size
can
be
somewhere
between
r
to
2Ldepending
upon
the
percentage
of
data
is
allowed
to
overlap
in
successive
computation.
These
two paramet'et
L
and
M
must
be
chosen
based
on
both
signal
representation
within
the
window
and
computational
time.
After
evaluation
of
the
wD,
an
NxM
wD