BLACKMAN –TUKEY BLACKMAN –TUKEY METHODMETHOD

BY:Sarbjeet Singh

M.E(ECE)

1ST YEAR,1ST SEM

NITTTR- Chandigarh

OBJECTIVEOBJECTIVEIntroductionProcedure of the methodComparision

INTRODUCTIONINTRODUCTION Used for power spectrum estimation Non parametric method Smoothen the periodgram

PROCEDUREPROCEDURE To calculate the autocorrelation function of the

data. To apply a suitable window function to the

data. To compute the FFT of the resulting data to

obtain the power density spectrum

METHODMETHOD The Blackman -Tukey estimate is

Where w(m) has length

and is zero for

m M

2 1M

12

( 1)

( ) ( ) ( )M

BT j fmxx xx

m M

P f r m w m e

CONTD.CONTD.Extending the limit on the sum to(-∞,∞)Hence

The Expected value of Blackman-Tukey power spectrum estimation is

where

1/2

1/2

( ) ( ) ( )BTxx xxP f P W f d

1/2

1/2

[ ( )] [ ( )] ( )BTxx xxE P f E P W f d

1/2

1/2

[ ( )] ( ) ( )xx xx BE p W d

CONTD.CONTD. Where is the Fourier transform of the

Bartlett window

We get

Hence

( )BW f

1/2 1/2

1/2 1/2

[ ( )] ( ) ( ) ( )BTxx xx BE P f W W f d d

1/2

1/2

[ ( )] ( ) ( )BTxx xxE P f W f d

CONTD.CONTD.The variance of the Blackman-Tukey power Spectrum

Estimate is

Therefore

2 2var[ ( )] {[ ( ) ]} { [ ( )]}BT BT BTxx xx xxP f E P f E P f

1/22 2

1/2

1var[ ( )] ( )[ ( ) ]BT

xx xxP f f W dN

12 2

( 1)

1( )[ ( )

M

xxm M

f w mN

PERFORMANCE COMPARISIONPERFORMANCE COMPARISION

Mean:

Variance:

Quality factor:

1/2

1/2

[ ( )] ( ) ( )BTxx xxE P f W f d

12 2

( 1)

1( )[ ( )

M

xxm M

f w mN

var[ ( )]BTxxP f

2{ [ ( )]}

var[ ( )]

BTxxBtxx

E P f

P f

CONTDCONTDFor rectangular & Bartlett window we have

(rectangular)

(triangular)

12

( 1)

1( ) 2

M

M

MW m

N N

2

3

M

N

1.5BT

NQ

M

PERFORMANCE COMPARISIONPERFORMANCE COMPARISION

Estimate Quality Factor

Bartlett 1.11NΔf

Welch(50% overlap) 1.39NΔf

Blackman-Tukey 2.34NΔf

COMPUTATIONAL PERFORMANCECOMPUTATIONAL PERFORMANCE

Estimate Number of computations

Bartlett

Welch(50% overlap)

Blackman-Tukey

2

0.9(log )

2

N

f

2

5.12(log )N

f

2

1.28(log )N

f

REFERENCESREFERENCESProakis & ManolakisJervis