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Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
Blackman Tuckey method
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Blackman Tuckey method

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  • 1. BLACKMAN –TUKEYMETHOD BY:Sarbjeet Singh M.E(ECE) 1ST YEAR,1ST SEM NITTTR- Chandigarh
  • 2. OBJECTIVEIntroductionProcedure of the methodComparision
  • 3. INTRODUCTION  Used for power spectrum estimation  Non parametric method  Smoothen the periodgram
  • 4. PROCEDURE 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
  • 5. METHOD  The Blackman -Tukey estimate is M−1 Pxx ( f ) = BT ∑ m =−( M −1) rxx (m) w(m)e − j 2π fm Where w(m) has length 2 M − 1 and is zero for m ≥ M
  • 6. CONTD.Extending the limit on the sum to(-∞,∞)Hence 1/ 2 Pxx ( f ) = ∫ Pxx (α)W ( f − α) d α BT −1/ 2The Expected value of Blackman-Tukey power spectrum estimation is 1/ 2 E[ Pxx ( f )] = ∫ E[ Pxx (α W ( f −α d α BT )] ) − 2 1/where 1/ 2 E[ p xx (α = )] − 2 1/ ∫ xx (θ)WB (α−θ) dθ
  • 7. CONTD. Where WB ( f ) is the Fourier transform of the Bartlett window We get 1/ 2 1/ 2 E[ Pxx ( f )] = ∫ ∫  xx (θ )WB (α − θ )W ( f − α)d αdθ BT −1/ 2 −1/ 2 Hence 1/ 2 E[ Pxx ( f )] ≈ ∫ xx (θ)W ( f −θ) dθ BT − 2 1/
  • 8. CONTD.The variance of the Blackman-Tukey power Spectrum Estimate isvar[ Pxx ( f )] = E{[ Pxx ( f ) 2 ]} −{E[ Pxx ( f )]}2 BT BT BTTherefore 1/ 2 1 var[ Pxx ( f )] ≈ 2 ( f )[ ∫ W 2 (θ) dθ] BT xx N − 2 1/ M−1 1 ≈ ( f )[ 2 xx N ∑ m= ( M − − 1) w2 ( m)
  • 9. PERFORMANCE COMPARISION 1/ 2 Mean: E[ Pxx ( f )] ≈ BT ∫ xx (θ )W ( f −θ ) dθ −1/ 2 Variance: M−1 1 ∑ BT var[ P ( f )] xx ≈ ( f )[ 2 xx w2 ( m ) N m= ( M − − 1)Quality factor: {E[ Pxx ( f )]}2 BT Bt var[ Pxx ( f )]
  • 10. CONTD For rectangular & Bartlett window we have M −1 1 M (rectangular) N ∑ −( M −1) W 2 ( m) = 2 N 2M = (triangular) 3N N QBT = 1.5 M
  • 11. PERFORMANCE COMPARISION Estimate Quality Factor Bartlett 1.11NΔf Welch(50% overlap) 1.39NΔf Blackman-Tukey 2.34NΔf
  • 12. COMPUTATIONAL PERFORMANCE Estimate Number of computationsBartlett N 0.9 (log 2 ) 2 ∆fWelch(50% overlap) 5.12 N (log 2 ) ∆fBlackman-Tukey 1.28 N (log 2 ) ∆f
  • 13. REFERENCESProakis & ManolakisJervis

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