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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