Download - Blackman Tuckey method
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