bg physiological signal processing 02 signals and systems2019psp mk 7778

Nguyễn Công Phương PHYSIOLOGICAL SIGNAL PROCESSING Signals and Systems 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 Signals and Systems Characterization and Representation of Discrete – Time Signals Characterization and Representation of Discrete – Time Systems s i tes.google.com/site/ncpdhbkhn Signals and Systems • • • Physiology: – Physiological Systems, – Physiological Signals Signal Processing: [Bruce, 2001] the manipulation of a signal for the purpose of: – Extracting information from the signal, – Extracting information about the relationships of two (or more) signals, – Producing an alternative representation of the signal Why signals are processed? [Bruce, 2001] – To remove unwanted signal components that are corrupting the signal of interest, – To extract information by rendering it in a more obvious or more useful form, – To predict future values of the signal in order to anticipate the behavior of its source s i tes.google.com/site/ncpdhbkhn Signals and Systems t (s) T s i tes.google.com/site/ncpdhbkhn n Signals and Systems Characterization and Representation of Discrete – Time Signals a) Types of Signals b) Discrete – Time Signals Characterization and Representation of Discrete – Time Systems s i tes.google.com/site/ncpdhbkhn Types of Signals (1) • Signal: a physical quantity varying as a function (of time, space, etc.) and carrying information Index L R i + C v e – +– Current as a function of time M T W T F S Closing value of the stock exchange index as a function of days s i tes.google.com/site/ncpdhbkhn Image as a function of x–y coordinates Types of Signals (2) Index i (A) t (s) M T W T F S s(t) s[n] Continuous – time signal Discrete – time signal Value of s(t) is defined for every value of time t Value of s(t) is defined only at discrete time s i tes.google.com/site/ncpdhbkhn Types of Signals (3) i (A) i (A) T t (s) s(t) n s[n ] = s(t ) t = nT = s(nT ) T : sampling period f s = : sampling frequency T s i tes.google.com/site/ncpdhbkhn Types of Signals (4) Deterministic signal The future value is predictable Random signal The future value is unpredictable s i tes.google.com/site/ncpdhbkhn 10 Block Diagrams & Signal – Flow Graphs (2) a y[n] = ax[n ] a y[n] = ax[n ] x[ n] Multiplier Gain branch x[ n] s i tes.google.com/site/ncpdhbkhn 39 Block Diagrams & Signal – Flow Graphs (3) x[ n] z −1 y[n] = x[n − 1] Unit delay Unit delay branch z −1 x[ n] y[n] = x[n − 1] s i tes.google.com/site/ncpdhbkhn 40 Block Diagrams & Signal – Flow Graphs (4) w[n] Splitter Pick-off node w[n] w[n] w[n] w[n] w[n] s i tes.google.com/site/ncpdhbkhn 41 Ex Block Diagrams & Signal – Flow Graphs (5) Find y[n]? w[n ] x[ n ] a y[n ] z −1 w[n ] = x[n ] + cw[n − 1] c b bx[n ] + cy[n ] b + ac y[n ] = aw[n ] + bw[n − 1] bx[n − 1] + cy[n − 1] → w[n − 1] = b + ac = a ( x[n ] + cw[n − 1]) + bw[n − 1] = ax[n] + (ac + b)w[n − 1] → y[n ] = ax[n] + (ac + b) → w[n ] = bx[n − 1] + cy[n − 1] = ax[n ] + bx[n − 1] + cy[n − 1] b + ac s i tes.google.com/site/ncpdhbkhn 42 Signals and Systems Characterization and Representation of Discrete – Time Signals Characterization and Representation of Discrete – Time Systems a) b) c) d) e) Properties of Discrete – Time Systems Impulse Response Convolution Block Diagrams & Signal – Flow Graphs Linear Constant – Coefficient Difference Equations f) Properties of LTI Systems s i tes.google.com/site/ncpdhbkhn 43 Linear Constant – Coefficient Difference Equations (LCCDE) N M k =1 k =1 y[n ] = −  a k y[n − k ] +  bk x[n − k ] • ak: feedback coefficients • bk: feedforward coefficients • If ak & bk are fixed, then the system is time – invariant • If ak & bk depend on n, then time – varying • N is the order of the system s i tes.google.com/site/ncpdhbkhn 44 Signals and Systems Characterization and Representation of Discrete – Time Signals Characterization and Representation of Discrete – Time Systems a) b) c) d) e) f) Properties of Discrete – Time Systems Impulse Response Convolution Block Diagrams & Signal – Flow Graphs Linear Constant – Coefficient Difference Equations Properties of LTI Systems i ii iii iv Properties of Convolution Causality and Stability Convolution of Periodic Sequences Response to Simple Test Sequences s i tes.google.com/site/ncpdhbkhn 45 Properties of Convolution (1) x[n] x[n] x[n] δ[n – n0] δ[n] x[n − n0 ] = δ [n − n0 ]* x[n ] x[ n ] = δ [ n ] * x[ n ] y[ n ] x[ n] x[n − n0 ] y[ n ] h[ n ] h[n] x[n] y[n] = x[n]* h[n] = h[n]* x[n] y[ n ] x[ n ] h1[n] y[ n ] x[ n] h2[n] h[n] = h1[n]*h2[n] ( x[n] * h1[n ]) * h2[n ] = x[n] * ( h1[n] * h2[n ]) s i tes.google.com/site/ncpdhbkhn 46 Properties of Convolution (2) y[n] x[n] h1[n] y[n] x[n] h2[n] h2[n] h1[n] ( x[n]* h1[n ]) * h2[n ] = ( x[n]* h2 [n]) * h1[n] h1[n] x[n] y[n] + y[ n ] x[ n] h[n] = h1[n]+h2[n] h2[n] x[n] * (h1[n ] + h2 [n]) = x[n] * h1[n] + x[n ]* h2 [n] s i tes.google.com/site/ncpdhbkhn 47 Causality and Stability • A linear time – invariant system with impulse response h[n] is causal if: h[ n] = for n