bg physiological signal processing 03 fourier analysis2020mk 788

Nguyễn Công Phương PHYSIOLOGICAL SIGNAL PROCESSING Fourier Analysis Contents I Introduction II Introduction to Electrophysiology III Signals and Systems IV Fourier Analysis V Signal Sampling and Reconstruction VI z-Transform VII.Discrete Filters VIII.Random Signals IX Time-Frequency Representations X Physiological Signal Processing s i tes.google.com/site/ncpdhbkhn Fourier Analysis Sinusoidal Signals and their Properties Fourier Representation of Continuous – Time Signals Fourier Representation of Discrete – Time Signals Summary of Fourier Series and Fourier Transforms Properties of the Discrete – Time Fourier Transform Computational Fourier Analysis The Discrete Fourier Transform Fourier Analysis of Signals Using the DFT Fast Fourier Transform s i tes.google.com/site/ncpdhbkhn Fourier Analysis http://tex.stackexchange.com/questions/127375/replicate-the-fourier-transform-time-frequency-domains-correspondence-illustrati s i tes.google.com/site/ncpdhbkhn Fourier Analysis Sinusoidal Signals and their Properties a) Continuous – Time Sinusoids b) Discrete – Time Sinusoids c) Frequency Variables and Units Fourier Representation of Continuous – Time Signals Fourier Representation of Discrete – Time Signals Summary of Fourier Series and Fourier Transforms Properties of the Discrete – Time Fourier Transform Computational Fourier Analysis The Discrete Fourier Transform Fourier Analysis of Signals Using the DFT Fast Fourier Transform s i tes.google.com/site/ncpdhbkhn Continuous – Time Sinusoids (1) x (t ) = A cos( 2π F0t + θ ), − ∞ < t < ∞ Ω = 2π F0 T0 = 2π = F0 Ω0 e± jϕ = cos ϕ ± j sin ϕ A cos θ • • • • • T0 x( t ) A t A: the amplitude θ: phase (radians, rad) F0: frequency (Hertz, Hz) Ω0: angular frequency (rad/s) T0: period (s) A jθ jΩ0 t − jθ − jΩ 0t A cos(2π F0t + θ ) = (e e + e e ) s i tes.google.com/site/ncpdhbkhn Continuous – Time Sinusoids (2) x1(t ) = cos 2π F1t T1 t x2 (t) = cos 2π F2t T2 s i tes.google.com/site/ncpdhbkhn t Continuous – Time Sinusoids (3) http://tex.stackexchange.com/questions/1273 75/replicate-the-fourier-transform-timefrequency-domains-correspondence-illustrati s1 (t) = e jΩ0t = e j 2π F0t The fundamental/first harmonic s2 (t) = e j 2Ω 0t = e j 2π F0t The second harmonic ⋮ sk (t) = e jkΩ 0t = e j 2π kF0t The kth harmonic s i tes.google.com/site/ncpdhbkhn Continuous – Time Sinusoids (4) x1(t) = A0 cos(2π F0t) + A1 cos(2π 3F0t) + A2 cos(2π 5F0t) t t x2 (t) = B0 cos(2π F0t) + B1 cos(2π 2.83F0t) + B2 cos(2π 7.14 F0t) s i tes.google.com/site/ncpdhbkhn Fourier Analysis Sinusoidal Signals and their Properties a) Continuous – Time Sinusoids b) Discrete – Time Sinusoids c) Frequency Variables and Units Fourier Representation of Continuous – Time Signals Fourier Representation of Discrete – Time Signals Summary of Fourier Series and Fourier Transforms Properties of the Discrete – Time Fourier Transform Computational Fourier Analysis The Discrete Fourier Transform Fourier Analysis of Signals Using the DFT Fast Fourier Transform s i tes.google.com/site/ncpdhbkhn 10 “Good” Windows and the Uncertainty Principle (6) s i tes.google.com/site/ncpdhbkhn 108 “Good” Windows and the Uncertainty Principle (7) 1.2 0.5 Rectangular Bartlett 0.45 0.4 0.35 Amplitude Amplitude 0.8 0.6 0.4 0.3 0.25 0.2 0.15 0.1 0.2 0.05 100 200 300 400 500 F (Hz) 600 700 800 900 1000 100 200 300 400 500 F (Hz) 600 700 800 900 1000 Ex x(t ) = cos(2π Ft ) + cos(2π 3Ft ) + cos(2π 4Ft ) 0.5 0.55 Hann 0.45 0.5 0.4 0.45 0.4 Amplitude 0.35 Amplitude Hamming 0.3 0.25 0.2 0.35 0.3 0.25 0.2 0.15 0.15 0.1 0.1 0.05 0.05 100 200 300 400 500 F (Hz) 600 700 800 900 1000 100 s i tes.google.com/site/ncpdhbkhn 200 300 400 500 F (Hz) 600 700 800 900 109 1000 Fourier Analysis Sinusoidal Signals and their Properties Fourier Representation of Continuous – Time Signals Fourier Representation of Discrete – Time Signals Summary of Fourier Series and Fourier Transforms Properties of the Discrete – Time Fourier Transform Computational Fourier Analysis The Discrete Fourier Transform Fourier Analysis of Signals Using the DFT Fast Fourier Transform a) Direct Computation of the DFT b) The FFT Idea Using a Matrix Approach c) Decimation – in – Time FFT Algorithms s i tes.google.com/site/ncpdhbkhn 110 Direct Computation of the Discrete – Fourier Transform N −1 X [ k ] =  x[n]WNkn , k = 0, 1, , N − n =0 x[n] = N N −1 − kn X [ k ] W , n = 0,1, , N −  N k =0   2π   2π   X R[ k ] =   xR [n]cos  kn  + xI [n ]sin  kn    N  N  k =0  N −1   2π   2π  X I [k ] =   xR[ n]sin  kn  − xI [n]cos  kn   N   N  k =0  N −1 The total cost of computing all X[k] coefficients is approximately N2 operations!!! Fast Fourier Transform s i tes.google.com/site/ncpdhbkhn 111 The Fast Fourier Transform Idea Using a Matrix Approach (1)  X [0] 1  X [1]  1     X [2] 1    X [ ]   = 1  X [4] 1    X [ ]   1  X [6] 1     X [7] 1 1 1  1  = 1  1 1  1 W8 8 8 8 W W W W W W 1 1 1   x[0] W82 W83 W84 W85 W86 W87   x[1]    6 W8 W8 W8 W8 W8   x[ 2]   W86 W8 W84 W87 W82 W85   x[3] W84 W84 W84   x[ 4]   W82 W87 W84 W8 W86 W83   x[5] W84 W82 W86 W84 W82   x[6]   W8 W8 W8 W8 W8 W8   x[7] 1 W82 W84 W86 W8 W84 W84 W86 W84 W82 1 W82 W84 W86 W82 W83 W84 W85 W84 W84 W86 W86 W84 W82 W87 1   x[0] W83 W85 W87   x[2]   6 W8 W8 W8   x[4]   W8 W87 W85   x[6] W84 W84 W84   x[1]    W87 W8 W83   x[3] W82 W86 W82   x[5]   W85 W83 W8   x[7] s i tes.google.com/site/ncpdhbkhn a c  b   e   ae + bf  = d   f   ce + df  b d  a   f   bf + ae  =    c   e   df + ce  112 The Fast Fourier Transform Idea Using a Matrix Approach (2) 1 W82 W84 W86 W8 W84 W84 W86 W84 W82 1 W82 W84 W86 W82 W83 W84 W85 8 W W 8 W W84 W 8 W W 1   x[0] W83 W85 W87   x[ 2]   6 W8 W8 W8   x[ 4]   W8 W87 W85   x[6] W84 W84 W84   x[1]    W87 W8 W83   x[3] W82 W86 W82   x[5]   W8 W8 W8   x[7] 1 W82 W84 W86 W8 W83 W85 W84 W82 W86 W82 W86 W82 W84 W86 W84 W84 W82 1 W84 W86 W83 W8 W87 −1 −W8 −1 −W83 −1 −W85 W84 W84 W82 −W82 −W83 −W86 −W8 −W82 −W87 0.8 0.6 W81 0.2 W84 W80 -0.2 -0.4 -0.6 -0.8 -1   x[0] W87   x[2]   W8   x[4]   W85   x[6] −1   x[1]    −W87   x[3]  −W86   x[5]    −W85   x[7] s i tes.google.com/site/ncpdhbkhn W83 0.4 Imagi nary Part  X [0] 1  X [1]  1     X [2] 1    X [ ]   = 1  X [4] 1    X [ ]   1  X [6] 1     X [7] 1 1 1  1  = 1  1 1  1 W82 W85 -1 W87 -0.5 Rea l Part W86 0.5 W84 = −1 W85 = −W81 W86 = −W82 W87 = −W83 113 The Fast Fourier Transform Idea Using a Matrix Approach (3)  X [0] 1  X [1]  1     X [2] 1    X [ ]   = 1  X [4] 1    X [ ]   1  X [6] 1     X [7] 1 1 1 W4 =  1  1 1 W82 W84 W86 W8 W83 W85 W84 W82 W86 W82 W83 W8 W87 −1 −W8 −1 −W83 −1 −W85 W86 W82 W84 W86 W4 W42 W84 W84 W82 1 W84 W86 W84 W42 W43 W42 −W86 −W82 W82 −W8 −W87  1 0 W W43  ; D =  W42  0 W82   W4  0 W84 −W82 −W83   x[0] W87   x[2]   W8   x[4]   W85   x[6] −1   x[1]    −W87   x[3] −W86   x[5]  5  −W8   x[7]   x[0]  x[1]   x [2 ]  x[3]  ; X   ; XOdd =   Even =   x [4 ]  x[5]      W83   x[6]  x[7] XTop = [ X [0] X [1] X [2] X [3]] ; X Bottom = [ X [4] X [5] X [6] X [7]] T T  XTop   W4 → =  X Bottom   W4 s i tes.google.com/site/ncpdhbkhn D8 W4   x Even  −D8 W4   xOdd  114 The Fast Fourier Transform Idea Using a Matrix Approach (4)  XTop   W4  = X  Bottom   W4 D8 W4  x Even  −D8 W4   xOdd  X T = X E + DN X O → X B = X E − DN X O N − point DFT : X E = WN / x E & XO = WN / 2x O N – point Input Sequence n = Even N/2 – point Sequence N – point FFT n = Odd xE [n ] N/2 – point Sequence xO [ n] N/2 – point DFT N/2 – point DFT X E [k ] X O [k ] Merge N/2 – point DFT N – point DFT Coefficients X [k ] s i tes.google.com/site/ncpdhbkhn 115 The Fast Fourier Transform Idea Using a Matrix Approach (5)  X [0] 1  X [1]  1     X [2] 1    X [ ]   = 1  X [4] 1    X [ ]   1  X [6] 1     X [7] 1  X [0] 1  X [2] 1     X [4] 1   1 X [ ]  =  X [1]  1    X [ ]   1  X [5] 1    X [ ]   1 1 W82 W84 W86 W8 W83 W85 W84 W84 W86 W84 W82 1 W82 W84 W86 W82 W83 −1 −W8 W86 W8 −1 −W83 W82 W87 −1 −W85 W84 W84 W86 W84 W82 −W82 −W83 −W86 −W8 −W82 −W87 1 W82 W84 W86 W84 W84 W86 W84 W82 1 1 W82 W84 W86 W84 W84 −1 −1 −1 −1 −W8 −W83 −W85 −W87 −W82 −W86 −W82 −W86 W8 W83 W85 W87 W82 W83 W86 W8 W82 W87 W86 W85   x[0] W87   x[2]   W8   x[4]   W85   x[6] −1   x[1]    −W87   x[3] −W86   x[5]  5  −W8   x[7]   x[0]   x [ ]     x [2 ]   x [ ] W4   x T    →  X E  =  W4      −W83   x[4]  X O   W4D − W4D   x B    −W8   x[5]  −W87   x[6]   −W85   x[7] W86 W84 W82 s i tes.google.com/site/ncpdhbkhn 116 Fourier Analysis Sinusoidal Signals and their Properties Fourier Representation of Continuous – Time Signals Fourier Representation of Discrete – Time Signals Summary of Fourier Series and Fourier Transforms Properties of the Discrete – Time Fourier Transform Computational Fourier Analysis The Discrete Fourier Transform Fourier Analysis of Signals Using the DFT Fast Fourier Transform a) Direct Computation of the DFT b) The FFT Idea Using a Matrix Approach c) Decimation – in – Time FFT Algorithms s i tes.google.com/site/ncpdhbkhn 117 Algebraic Derivation (1) N −1 X [k ] =  x[n]WNkn , k = 0,1, , N − n= N −1 N −1 m =0 m= =  x[2m ]WNk ( 2m ) +  x[2m + 1]WNk (2 m+1) N −1 N −1 m =0 m= =  x[2m ]WNk (2 m ) + WNk  x[2m + 1]WNk (2 m ) a[ n] = x[ 2n], n = 0,1, 2, , N / − b[ n] = x[ 2n + 1], n = 0, 1, 2, , N / − WN2 = WN / → X [k ] = A[k ] + WNk B[k ], A[k ] = N / −1  m =0 B[ k ] = N / −1  m =0 s i tes.google.com/site/ncpdhbkhn k = 0,1, , N − a[m ]WNkm/ , k = 0,1, , N / − b[m]WNkm/ , k = 0, 1, , N / − 118 Algebraic Derivation (2) X [ k ] = A[k ] + WNk B[ k ], k = 0, 1, , N / − N k+ N N N    → X  k +  = A  k +  + WN B  k +  2 2 2    W  N  k+    N = −WNk N  → X k +  = A[k ] − WNk B[k ] 2  X [k ] = A[k ] + WNk B[k ], N  X  k +  = A[k ] − WNk B[k ], 2  N −1 N k = 0, 1, , − k = 0, 1, , s i tes.google.com/site/ncpdhbkhn 119 Algebraic Derivation (3) Ex X [k ] = DFT8{x[0], x[1], x[2], x[3], x[4], x[5], x[6], x[7]}, ≤ k ≤ A[k ] = DFT4{x[0], x[2], x[4], x[6]}, ≤ k ≤ ր ց ր ց X [k ] B[k ] = DFT4{x[1], x[3], x[5], x[7]}, 0≤k≤3 ր ց C [k ] = DFT2{x[0], x[4]}, k = 0,1 D[k ] = DFT2{x[2], x[6]}, k = 0,1 E [ k ] = DFT2{x[1], x [5]}, k = 0,1 F [ k ] = DFT2 {x [3], x[ 7]}, k = 0,1 A[k ] = C[k ] + W82k D[k ], k = 0,1 A[k + 2] = C[k ] − W82 k D[k ], k = 0,1 B[ k ] = E [k ] + W82k F [ k ], k = 0,1  X [k ] = A[ k ] + W8k B[k ], k = 0, 1, 2, → k  X [ k + 2] = A[ k ] − W8 B[ k ], k = 0,1, 2, B[ k + 2] = E [k ] − W82 k F [ k ], k = 0, s i tes.google.com/site/ncpdhbkhn 120 Algebraic Derivation (4) Ex X [ k ] = DFT8{x[ 0], x[1], x[ 2], x[3], x[ 4], x[ 5], x[ 6], x[ 7]}, ≤ k ≤ A[ k ] = C[ k ] + W82 k D[ k ], k = 0,1 A[k + 2] = C[ k ] − W82k D[ k ], k = 0,1  X m [ p] = X m−1[ p] + WNr X m −1[ q] → r  X m [ q] = X m −1[ p ] − WN X m −1[ q] X m −1[ p] X m[ p ] WNr X m −1[ q] −1 s i tes.google.com/site/ncpdhbkhn X m [ q] 121 Algebraic Derivation (5) Ex X [ k ] = DFT8{x[ 0], x[1], x[ 2], x[3], x[ 4], x[ 5], x[ 6], x[ 7]}, ≤ k ≤ x[ 0] x[ 0] x[ 0] x[0] x[1] x[ 2] x[ 4] x[4] x[ 2] x[ 4] x[ 2] x[ 2] x[ 3] x[ 6] x[ 6] x[ 6] x[ 4] x[1] x[1] x[1] x[ 5] x[ 3] x[ 5] x[ 5] x[ 6] x[ 5] x[ 3] x[ 3] x[ 7] x[ 7] x[ 7] x[ 7] Algebraic Merge two – point DFTs Merge two – point DFTs Merge two – point DFTs Merge two – point DFTs s i tes.google.com/site/ncpdhbkhn X [ 0] Merge two – point DFTs X [1] X [ 2] X [ 3] Merge two – point DFTs X [ 4] Merge two – point DFTs X [ 5] X [ 6] X [ 7] Merging 122 ... tes.google.com/site/ncpdhbkhn Fourier Analysis Sinusoidal Signals and their Properties Fourier Representation of Continuous – Time Signals Fourier Representation of Discrete – Time Signals Summary of Fourier Series and Fourier. .. Time Signals a) Fourier Series for Continuous – Time Periodic Signals b) Fourier Transform for Continuous – Time Aperiodic Signals Fourier Representation of Discrete – Time Signals Summary of Fourier. .. 50 Fourier Analysis Sinusoidal Signals and their Properties Fourier Representation of Continuous – Time Signals Fourier Representation of Discrete – Time Signals Summary of Fourier Series and Fourier