bg physiological signal processing 08 time frequency2020c mk 9934

Nguyễn Công Phương PHYSIOLOGICAL SIGNAL PROCESSING Time – Frequency Representations Contents I Introduction II Introduction to Electrophysiology III Signals and Systems IV Fourier Analysis V Signal Sampling and Reconstruction VI The z-Transform VII.Discrete Filters VIII.Random Signals IX Time-Frequency Representations X Physiological Signal Processing s i tes.google.com/site/ncpdhbkhn Time – Frequency Representations The Spectrogram The Periodogram Spectral Analysis in Multiresolution s i tes.google.com/site/ncpdhbkhn The Spectrogram L−1 X [ k , n] =  w[ m] x[n + m]e− j ( 2π k / N )m m= s i tes.google.com/site/ncpdhbkhn Time – Frequency Representations The Spectrogram The Periodogram Spectral Analysis in Multiresolution s i tes.google.com/site/ncpdhbkhn The Periodogram I (ω ) = N −1  rˆ[ℓ]e − jω ℓ ℓ =− ( N −1) = N rˆ[0] = N N −1 − jωn x [ n ] e  n =0 N −1 x [ n] =  2π n= ≃ 2π  2π I  k =0  K K −1 s i tes.google.com/site/ncpdhbkhn π  π I (ω )dω −  2π k =  K K  2π I  k =0  K K −1  k  Time – Frequency Representations The Spectrogram The Periodogram a) Statistical Properties of the Periodogram b) The Modified Periodogram c) The Blackman – Tukey Method: Smoothing a Single Periodogram d) The Bartlett – Welch Method: Averaging Multiple Periodograms Spectral Analysis in Multiresolution s i tes.google.com/site/ncpdhbkhn I( ) I( ) I( ) I( ) Statistical Properties of the Periodogram (1) s i tes.google.com/site/ncpdhbkhn Time – Frequency Representations The Spectrogram The Periodogram a) Statistical Properties of the Periodogram i ii Mean of the Periodogram Variance & Covariance of the Periodogram b) The Modified Periodogram c) The Blackman – Tukey Method: Smoothing a Single Periodogram d) The Bartlett – Welch Method: Averaging Multiple Periodograms Spectral Analysis in Multiresolution s i tes.google.com/site/ncpdhbkhn Mean of the Periodogram (1) I (ω ) = N −1  rˆ[ℓ]e− jω ℓ ℓ =− ( N −1) → E[I (ω )] = N −1  E(rˆ[ℓ ])e− jω ℓ ℓ =− ( N −1) E(rˆ[ℓ ]) = − ℓ N r[ℓ ]  ℓ → E[ I (ω )] =  1 −  r [ℓ]e − jω ℓ N ℓ =−( N −1)  N −1 S (ω ) = ∞ − jωℓ r [ ℓ ] e  ℓ =−∞ → lim E[ I (ω )] = S (ω ) N →∞ s i tes.google.com/site/ncpdhbkhn 10 CWT (2) ∞ W y (τ , s ) =  y (τ )  t −τ ψ *  s s −∞ y (t ) = Cψ ∞ ∞   W (τ , s ) y −∞ −∞ Cψ = 2π ∞  −∞ Ψ (ω ) ω   dt   t −τ  ψ  s  s  ψ s,τ (t ) =  t −τ ψ s  s  ds  dτ  s dω SC y (τ , s ) = Wy (τ , s) s i tes.google.com/site/ncpdhbkhn 37 CWT (3) Ex ≤ t ≤ 0.5  ψ (t ) =  −1, 0.5 ≤ t ≤  0, otherwise  Ψ (ω ) = je − jω / sin (ω / 4) (ω / 4) 2π | | The Haar wavelet  1, s i tes.google.com/site/ncpdhbkhn 38 CWT (4) Ex (t/0.5) [(t - 0.5)/0.5] (t - 1) (t) The Haar wavelet ψ s,τ (t ) =  t −τ  ψ  s  s  s i tes.google.com/site/ncpdhbkhn 39 CWT (5) Ex The Mexican hat d − x2 / 2 − x2 / ψ (t ) = − e = (1 − x ) e dx 2π | | Ψ (ω ) = ωe −ω /2 s i tes.google.com/site/ncpdhbkhn 40 CWT (6) Ex The Mexican hat s i tes.google.com/site/ncpdhbkhn ψ s,τ (t ) =  t −τ  ψ  s  s  41 CWT (7) Ex s i tes.google.com/site/ncpdhbkhn 42 CWT (8) Ex s i tes.google.com/site/ncpdhbkhn 43 CWT (9) Ex s i tes.google.com/site/ncpdhbkhn 44 CWT (10) Ex s i tes.google.com/site/ncpdhbkhn 45 Time – Frequency Representations The Spectrogram The Periodogram Spectral Analysis in Multiresolution a) The Short – Time Fourier Transform (STFT) b) The Continuous Wavelet Transform (CWT) c) The Discrete Wavelet Transform (DWT) s i tes.google.com/site/ncpdhbkhn 46 DWT (1) http://ataspinar.com/2018/12/21/a-guide-for-usingthe-wavelet-transform-in-machine-learning / s i tes.google.com/site/ncpdhbkhn 47 c DWT (2) Wϕ [ j0 , k ] = N Wψ [ j, k ] = N N −1  x[n]ϕ n= [ n], (for all k ) [ n], (for all k and all j > j0 ) j0 , k N −1  x[n]ψ n= j, k x[ n] = Wϕ [ j0 , k ]ϕ j0 ,k [ n]  N k ∞ + Wψ [ j , k ]ψ j ,k [ n],   N j = j0 k s i tes.google.com/site/ncpdhbkhn ( n = 0,1, , N − 1] 48 DWT (3) Ex ϕ0,0 [ n] = {1,1,1,1}; ψ 0,0 [n] = {1,1, −1, −1}; ψ 1,0 [n] = { 2, − 2, 0,0}; ψ 1,1[ n] = {0, 0, 2, − 2}; x[n] = {1, 2,3, 4} Wϕ [0,0] =  x[n]ϕ0,0 [ n] = (1×1 + ×1 + ×1 + ×1) = n= Wψ [0, 0] =  x[n]ψ 0,0 [n] = [1×1 + ×1 + 3( −1) + 4( −1)] = −2 n =0 Wψ [1,0] =  x[ n]ψ 0,0 [ n] = [1× + 2(− 2) + × + × 0] = −0.7071 n =0 Wψ [1,1] =  x[n ]ψ 0,0 [n ] = [1 × + × + × + 4( − 2)] = −0.7071 n= s i tes.google.com/site/ncpdhbkhn 49 DWT (4) Ex ϕ0,0 [ n] = {1,1,1,1}; ψ 0,0 [n ] = {1,1, −1, −1}; ψ 1,0 [n ] = { 2, − 2, 0,0}; ψ 1,1[ n] = {0, 0, 2, − 2}; x[n ] = {1, 2,3, 4} Wϕ [0,0] = 5; Wψ [0, 0] = −2; Wψ [1, 0] = −0.7071; Wψ [1,1] = −0.7071   1 2   1 1 −1 − 0   1   −1     −2   =     − 0,7071     − 0,7071 −      1    1     −2   2 1 −  =    1 −1   −0, 7071          −0, 7071   − 2 1 −1 s i tes.google.com/site/ncpdhbkhn 50 DWT (5) Ex s i tes.google.com/site/ncpdhbkhn 51