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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

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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

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*(*)

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