bg physiological signal processing 06 discrete filters2020mk 9652

Nguyễn Công Phương PHYSIOLOGICAL SIGNAL PROCESSING Discrete Filters 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 Discrete Filters Types of Filters Transfer Function and Frequency Response Group Delay and Inverse System Minimum and Linear Phase Filters Design of FIR Filters Design of IIR Filters Structure of Realization of Discrete Filters s i tes.google.com/site/ncpdhbkhn Types of Filters (1) http://reactivex.io/documentation/operators/filter.html s i tes.google.com/site/ncpdhbkhn Types of Filters (2) FILTER s i tes.google.com/site/ncpdhbkhn Types of Filters (3) H H ωc ω H (0) = 1; H (∞ ) = Lowpass H Bandpass ω1 ω2 ωc H (0) = 0; H (∞ ) = Highpass Bandstop ω H H (0) = 0; H (∞) = ω ω1 ω2 ω H (0) = 1; H (∞ ) = s i tes.google.com/site/ncpdhbkhn Discrete Filters Types of Filters Transfer Function and Frequency Response a) Transfer Function b) Frequency Response Group Delay and Inverse System Minimum and Linear Phase Filters Design of FIR Filters Design of IIR Filters Structure of Realization of Discrete Filters s i tes.google.com/site/ncpdhbkhn Transfer Function (1) x H y System H (e jω ) = Y (e jω ) X (e jω ) Y ( s) H ( s) = X ( s) y (t ) h(t ) = x (t ) Y ( z) H ( z) = X ( z) y[ n] h[ n] = x[ n] s i tes.google.com/site/ncpdhbkhn Transfer Function (2) x H y System Y (z) H ( z) = X ( z) H ( z ) = → Y ( z) = → z1, z2 , , zM (zero(s) of the system) H ( z ) = ∞ → X ( z) = → p1 , p2 , , pN (pole(s) of the system) s i tes.google.com/site/ncpdhbkhn Transfer Function (3) N M k =1 k =1 y[ n] = − ak y[n − k ] +  bk x[n − k ] M → H ( z) = b z k =1 N X ( z) x[n] −k k +  ak z − k = B( z ) A( z ) System Y ( z) H ( z) = X ( z) Y ( z) y[ n] k =1 M B( z ) → H (e ) = = A( z ) z =e jω jω b e k =1 N − jω k k +  ak e− jωk k =1 M = b0 ∏ (1 − zk z ) k =1 N −1 ( − p z ) ∏ k k =1 M −1 = b0 − jω ( − z e ∏ k ) k =1 N − jω ( − p e ) ∏ k k =1 z =e j ω s i tes.google.com/site/ncpdhbkhn 10 FIR System Structures (3) Cascade Form M H ( z ) =  h[n ]z n =0 −n K = G ∏ (1 + Bɶk z −1 + Bɶ k z −2 ) k =0 s i tes.google.com/site/ncpdhbkhn 246 FIR System Structures (1) M y[n ] =  bk x[n − k ] k=0 bn , n = 0, 1, , M h[n] =  0, otherwise M Y (z) M −n H (z) = =  bn z =  h[n ]z − n X ( z ) n =0 n =0 Direct form Cascade form Direct form for linear-phase FIR systems Frequency-sampling form s i tes.google.com/site/ncpdhbkhn 247 FIR System Structures (4) Direct Form for Linear-Phase FIR Systems M y[n ] =  h[k ] x[n − k ] k =0 M −1 M M M   =  h[k ]x[n − k ] + h   x n −  +  h[k ]x[n − k ]  k = M +1  2  k =0 = = M −1 M −1 M M   h [ k ] x [ n − k ] + h x n − +  h[ M − kk ] x[n − M + k ]       k =0 k =0 M −1 M M   h [ k ] x [ n − k ] + x [ n − M + k ] + h x n − ( )      k =0 s i tes.google.com/site/ncpdhbkhn 248 FIR System Structures (5) Direct Form for Linear-Phase FIR Systems M y[n ] =  h[k ]x[n − k ] = k =0 M −1 M M   h [ k ] x [ n − k ] + x [ n − M + k ] + h x n − ( )      k =0 s i tes.google.com/site/ncpdhbkhn 249 FIR System Structures (6) Direct Form for Linear-Phase FIR Systems Ex H ( z ) = − 10 z − + z −2 − 20 z −3 + 35 z −4 − 20 z − + z − − 10 z − + z −8 s i tes.google.com/site/ncpdhbkhn 250 FIR System Structures (7) Direct Form for Linear-Phase FIR Systems Ex H ( z ) = − 10 z − + z −2 − 20 z −3 + 35 z −4 − 20 z − + z − − 10 z − + z −8 s i tes.google.com/site/ncpdhbkhn 251 FIR System Structures (1) M y[n ] =  bk x[n − k ] k =0 bn , n = 0, 1, , M h[n] =  0, otherwise M Y (z) M −n H (z) = =  bn z =  h[n ]z − n X ( z ) n =0 n =0 Direct form Cascade form Direct form for linear-phase FIR systems Frequency-sampling form s i tes.google.com/site/ncpdhbkhn 252 FIR System Structures (8) Frequency-Sampling Form − z− N H (z) = N 1− z −N = N N −1 H [k ] k =0 j  − z −1e 2π k N , H [k ] = H ( z ) z = e j π k / N  H [0] H [ N / 2] K  + + H [ k ] H ( z )    k −1 −1 − z + z  k =1  cos(∠H [k ]) − z −1 cos(∠H [k ] − 2π k / N ) Hk (z) = − cos(2π k / N ) z −1 + z −2 s i tes.google.com/site/ncpdhbkhn 253 FIR System Structures (9) Frequency-Sampling Form s i tes.google.com/site/ncpdhbkhn 254 Discrete Filters Types of Filters Transfer Function and Frequency Response Group Delay and Inverse System Minimum and Linear Phase Filters Design of FIR Filters Design of IIR Filters Structure of Realization of Discrete Filters a) b) c) d) Block Diagrams & Signal Flow Graphs IIR System Structures FIR System Structures Lattice Structures i ii e) All-Zero Lattice Structure All-Pole Lattice Structure Structure Conversion, Simulation, & Verification s i tes.google.com/site/ncpdhbkhn 255 Lattice Structure s i tes.google.com/site/ncpdhbkhn 256 Structures for Discrete – Time Systems Block Diagrams & Signal Flow Graphs IIR System Structures FIR System Structures Lattice Structures Structure Conversion, Simulation, & Verification s i tes.google.com/site/ncpdhbkhn 257 Discrete Filters Types of Filters Transfer Function and Frequency Response Group Delay and Inverse System Minimum and Linear Phase Filters Design of FIR Filters Design of IIR Filters Structure of Realization of Discrete Filters a) b) c) d) Block Diagrams & Signal Flow Graphs IIR System Structures FIR System Structures Structure Conversion, Simulation, & Verification s i tes.google.com/site/ncpdhbkhn 258 Structure Conversion, Simulation, & Verification (1) s i tes.google.com/site/ncpdhbkhn 259 Structure Conversion, Simulation, & Verification (2) • Verification of the converted structure can be obtained in many different ways The simplest is to: – (a) convert the given structure to direct form I, – (b) use the filter function to implement this structure, – (c) excite the structure in (b) and the structure to be tested with the same input and compare the corresponding outputs • Typical test inputs include the unit step, sinusoidal sequences, or random sequences • Another way is to obtain a well-known response like an impulse or a step response from one structure and compare it with that from another structure s i tes.google.com/site/ncpdhbkhn 260