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Nguyễn Công Phương PHYSIOLOGICAL SIGNAL PROCESSING Signal Sampling and Reconstruction 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 Representation of Physiological Signals X Physiological Signal Processing s i tes.google.com/site/ncpdhbkhn Signal Sampling and Reconstruction • Periodic Sampling of Continuous – Time Signals • Frequency Analysis of Periodic Sampling • Reconstruction of Continuous – Time Signals from Samples • Discrete Processing of Continuous – Time Signals s i tes.google.com/site/ncpdhbkhn Signal Sampling and Reconstruction Analog input Sensor Analog Pre-processing ADC DSP DAC Analog Post-processing s i tes.google.com/site/ncpdhbkhn Analog output Periodic Sampling of Continuous – Time Signals x c (t ) Ideal analog – digital converter Fs = 1/T x[n ] = x c (t ) t = nT = xc (nT ), x[n ] = xc (nT ) −∞ 2Ω H −Ω s −ΩH X ( e jΩT ) Nyquist rate 2Ω H ΩH Nyquist frequency FH π T Fs Folding frequency s i tes.google.com/site/ncpdhbkhn 2π T Ω [rad/s] Fs Sampling frequency Frequency Analysis of Periodic Sampling (4) T Ω s > 2Ω H Guard band −Ω s Ω s = 2π Fs Guard band −ΩH TX ( e jΩT ), Ω ≤ Ω s / X c ( jΩ) = Ω > Ωs / 0, X ( e jΩT ) −Ω H ΩH Ω s − ΩH Ω s Ω = 2π F X c ( jΩ ) Ω H = 2π FH ΩH Ω = 2π F Let xc(t) be a continuous – time bandlimited signal with Fourier transform X c ( jΩ) = for Ω > Ω H Then xc(t) can be uniquely determined by its samples x[n] = xc(nT), where n = 0, ±1, ±2, …, if the sampling frequency Ωs satisfies the condition 2π Ωs = ≥ 2Ω H T s i tes.google.com/site/ncpdhbkhn 10 Reconstruction of Continuous – Time Signals from Samples (5) ∞ xc ( t ) Continuous – time Fourier Transform Pairs 2π ∞ −∞ X c ( j Ω)e jΩ td Ω ∞ x[ n]e X (e jΩ T ) = − jΩTn n =−∞ x[n ] Discrete – time Fourier Transform Pairs x [n] = 2π π /T π − /T TX (e jΩT )e jΩ Tn d Ω s i tes.google.com/site/ncpdhbkhn X ( e jΩT ) Normalized Frequency Sampling Reconstruction x c (t ) = X c ( jΩ) Frequency Discrete – Time −∞ Aliasing Lowpass – Filtering Continuous – Time X c ( j Ω) = x c (t )e − jΩt dt 16 Reconstruction of Continuous – Time Signals from Samples (6) x c (t ) X c ( j 2π F ) Ideal analog – digital converter Fs = 1/T x[n ] X (e j 2π FT ) Ideal digital – analog converter Fs = 1/T y r (t ) Yc ( j2π F ) -0 -0 -0 -0 s i tes.google.com/site/ncpdhbkhn 17 Reconstruction of Continuous – Time Signals from Samples (7) Spectrum of x (t) Ex c xc (t) = cos(2π F0t ) e j 2π F0t + e− j 2π F0t x c ( t) = X (e = T j 2π F0T ) X [ j 2π ( F − kF )] c GBL ( j2π F ) T , = 0, F0 F ≤ Fs / F > Fs / s F GBL ( j 2π F ) 2T F − s − Fs Fs Fs T ∞ k =−∞ − F0 − Fs F0 < − F0 Fs F0 F Fs Spectrum of xr(t) No aliasing − Fs X r ( j 2π F0 ) = Gr ( j 2π F ) X (e j 2π F0T ) − F0 s i tes.google.com/site/ncpdhbkhn F0 < F0 Fs Fs 18 F Reconstruction of Continuous – Time Signals from Samples (8) Spectrum of x (t) Ex c xc (t) = cos(2π F0t ) e j 2π F0t + e− j 2π F0t x c ( t) = X (e = T j 2π F0T ) X [ j 2π ( F c − kFs )] GBL ( j2π F ) T , = 0, F ≤ Fs / F > Fs / − GBL ( j 2π F ) 2T Fs F F0 Fs T ∞ k =−∞ − Fs − F0 Fs < F0 < Fs Fs − Fs −F0 F F0 Fs Spectrum of xr(t) Aliasing − Fs X r ( j 2π F0 ) = Gr ( j 2π F ) X (e j 2π F0T ) −( Fs − F0 ) F0 < Fs − F0 Fs Fs F xr (t ) = cos[ 2π ( Fs − F0 )t ] ≠ xc (t) s i tes.google.com/site/ncpdhbkhn 19 Ex Reconstruction of Continuous – Time Signals from Samples (9) xc (t) = cos(2π F0t ) F0 = Fs / + ∆F , → xc (t) = cos[2π ( Fs / + ∆F )t ] ∆F ≤ Fs / xr (t ) = cos[ 2π ( Fs − F0 )t ] Fapparent = Fs − F0 = Fs / − ∆F → xr (t ) = cos 2(π Fa t ) = cos[2π ( Fs / − ∆F )t ] s i tes.google.com/site/ncpdhbkhn 20 Reconstruction of Continuous – Time Signals from Samples (10) xc (t) = e −At xc(t) Ex 2A X c ( jΩ ) = A + Ω2 A>0 0.5 -10 -8 -6 -4 -2 t 10 -8 -6 -4 -2 Ω 10 -8 -6 -4 -2 Ω 10 x[ n] = xc (nT ) = e = (e − AT −A n T X c(jΩ) 0.4 0.2 -10 ) =a , n n a = e− AT jω X(e ) X (e ) = x[n]e − j ωn Sum Xc (jΩ ) shifted 2π to left & scaled 1/T n =−∞ − a2 = , − 2a cos(ω ) + a Ω ω= Fs -10 yr(t) jω ∞ X (jΩ ) scaled 1/T c 0.5 -10 Xc (jΩ ) shifted 2π to right & scaled 1/ T -8 -6 -4 s i tes.google.com/site/ncpdhbkhn -2 t 10 21 Signal Sampling and Reconstruction • Periodic Sampling of Continuous – Time Signals • Frequency Analysis of Periodic Sampling • Reconstruction of Continuous – Time Signals from Samples • Discrete Processing of Continuous – Time Signals s i tes.google.com/site/ncpdhbkhn 22 Discrete – Time Processing of Continuous – Time Signals (1) LTI xc ( t ) Ideal ADC x[n ] Discrete – Time y[ n] Ideal DAC System jω Fs = 1/T X c ( jΩ) Fs = 1/T X ( e jω ) Y ( e ) h[n] H(ejω) xc ( t ) T 2T 3T X c ( jΩ) Ω ∞ 2π X (e ) = X c j Ω − k T T k =−∞ T x[n ] = xc (t ) t =nT = xc (nT ) Yr ( jΩ) −2π FH 2π FH t yr (t ) jΩT t − 2π T s i tes.google.com/site/ncpdhbkhn −2π FH 2π FH 2π T Ω 23 Discrete – Time Processing of Continuous – Time Signals (2) LTI xc ( t ) Ideal ADC x[n ] Discrete – Time y[ n] Ideal DAC System jω Fs = 1/T X c ( jΩ) Fs = 1/T X ( e jω ) Y ( e ) h[n] H(ejω) T ⋯ n −2π ⋯ H (e j ω ) ωc ωH 2π ω 2π ω Y (e jω ) = H (e jω ) X (e jω ) T y[ n] = h[n] * x[n] ⋯ −ωH −ωc Yr ( jΩ) X (e jω ) x[n] ⋯ yr (t ) n −2π s i tes.google.com/site/ncpdhbkhn −ωc ωc 24 Discrete – Time Processing of Continuous – Time Signals (3) LTI xc ( t ) Ideal ADC x[n ] Discrete – Time y[ n] Ideal DAC System jω Fs = 1/T X c ( jΩ) Fs = 1/T X ( e jω ) Y ( e ) h[n] H(ejω) T 2T 3T Yr ( jΩ) Gr ( jΩ) T yr (nT ) = y[ n] yr (t ) Y (e jΩT ) T t 2π − T − π −Ω c Ω c π T T 2π T Ω Yr ( jΩ ) = G r ( jΩ )Y (e jΩT ) yr (t ) t s i tes.google.com/site/ncpdhbkhn −Ω c Ω c Ω 25 Discrete – Time Processing of Continuous – Time Signals (4) LTI xc ( t ) Ideal ADC x[n ] Discrete – Time y[ n] Ideal DAC System jω Fs = 1/T X c ( jΩ) Fs = 1/T X ( e jω ) Y ( e ) h[n] H(ejω) yr (t ) Yr ( jΩ) H (e jω ) X (e jΩT ∞ 2π ) = Xc j Ω − T k =−∞ T Y (e jω ) = H (e jω ) X (e jω ) Yr ( jΩ ) = Gr ( jΩ)Y (e jΩT ) k T Gr ( jΩ) Y (e jΩT ) T 2π − T → Yr ( j Ω) = G BL ( jΩ) H ( e j ΩT − ) T π −Ω c Ω c π T T 2π X j Ω − c T k =−∞ H (e jΩT ) X c ( jΩ), = 0, s i tes.google.com/site/ncpdhbkhn ∞ 2π T Ω k Ω ≤π /T Ω >π /T 26 Discrete – Time Processing of Continuous – Time Signals (5) LTI xc ( t ) Ideal ADC x[n ] Discrete – Time y[ n] Ideal DAC System jω Fs = 1/T X c ( jΩ) Fs = 1/T X ( e jω ) Y ( e ) h[n] H(ejω) H (e jΩT ) X c ( jΩ ), Yr ( jΩ) = 0, H (e jΩT ), H effective ( jΩ) = 0, yr (t ) Yr ( jΩ) Ω ≤π /T Ω >π /T Ω ≤π /T Ω >π /T → Yr ( jΩ) = H effective ( jΩ) X c ( jΩ) s i tes.google.com/site/ncpdhbkhn 27 Ex yc (t ) = Discrete – Time Processing of Continuous – Time Signals (6) dxc (t ) Y ( j Ω) → Yc ( jΩ) = jΩX ( jΩ) → H c ( j Ω) = = jΩ dt X ( j Ω) j Ω, H c ( j Ω) = 0, Ω ≤ ΩH otherwise Ωs =2Ω H hc (t ) → h[ n] = hc ( nT ) jω → H (e ) ω =ΩT ∞ 2π = H c jΩ − j k , T k =−∞ T T= π ΩH jω → H ( e ) = H c ( jω / T ) = , ω ≤ π T T jω → h[ n] = 2π 0, j ω jωn ω e d = cos(π n ) − π T nT , π s i tes.google.com/site/ncpdhbkhn n=0 n≠0 28 Ex Discrete – Time Processing of Continuous – Time Signals (7) Yc ( s) Ω2n H c (s) = = X c ( s) s2 + 2ζΩ n s + Ω 2n Ωn If < ζ < → hc (t) = e −ζΩnt sin Ω n − ζ t u(t) 1−ζ ( → h[n] = hc (nT ) = = → H ( z) = = ) ( Ωn ) e −ζΩnnT sin Ω n − ζ nT u(n) 1−ζ Ωn 1−ζ Ωn 1−ζ ( e−ζΩnT ∞ Ωn 1−ζ ( e n= × ) n ( ) sin ΩnT − ζ n u( n) − ζΩ nT ) sin (Ω T n n ( ) − ζ n z − n ) e− ζΩnT sin ΩnT − ζ z −1 ( ) − 2e− ζΩnT cos ΩnT − ζ z −1 + e−2ζΩn T z −2 s i tes.google.com/site/ncpdhbkhn 29 Ex Discrete – Time Processing of Continuous – Time Signals (8) Yc ( s) Ω2n H c (s) = = X c ( s) s2 + 2ζΩ n s + Ω 2n Ωn If < ζ < → hc ( t ) = e −ζΩnt sin Ω n − ζ t u( t ) 1−ζ ( h[ n] = H ( z) = → y[ n] = Ωn 1−ζ Ωn 1−ζ Ωn 1−ζ (e × − ζΩ nT ) ) sin (Ω T n n ) − ζ n u(n ) ( ) e− ζΩnT sin Ω nT − ζ z −1 ( ) − 2e− ζΩn T cos Ω nT − ζ z −1 + e−2ζΩn T z −2 ( ) e− ζΩn T sin Ω nT − ζ x[n − 1] ( ) + 2e −ζΩnT cos ΩnT − ζ y[ n − 1] − e−2ζΩnT y[ n − 2] s i tes.google.com/site/ncpdhbkhn 30 ... 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 Representation of Physiological Signals... Signals X Physiological Signal Processing s i tes.google.com/site/ncpdhbkhn Signal Sampling and Reconstruction • Periodic Sampling of Continuous – Time Signals • Frequency Analysis of Periodic Sampling. .. Continuous – Time Signals from Samples • Discrete Processing of Continuous – Time Signals s i tes.google.com/site/ncpdhbkhn Signal Sampling and Reconstruction Analog input Sensor Analog Pre-processing