投影片 1

Size: px

Start display at page:

Download "投影片 1"

徂桑
5 years ago
Views:

1 第三屆的時頻分析與地球科學研討會經驗模態分解法混波問題探討與解決方案國立臺灣師範大學機電科技系助理教授 Shuen-De Wu 吳順德博士

2 Outline IMFs EMD Mode Mixing Problem Iterative Gaussian Filter Conclusion --

3 EMD! Hot! Citation Number: 318

4 FFT v.s. Hilbert Transform

5 FFT v.s. Hilbert Transform -5-

6 FFT v.s. Hilbert Transform -6-

7 FFT v.s. Hilbert Transform Third Harmonics -7-

8 FFT v.s. Hilbert Transform -8-

9 Mono component signal or multi component signal? EMD prefer to interpret the above signals as monocomponent signals Positive amplitude and positive IF.

10 Signal Decomposition original signal IMFs residue Ref [1] -10-

11 Objectives of EMD EMD decomposes an arbitrary signal into several intrinsic mode functions IMFs: IMF can be considered as a mono-component signal. -11-

12 What is IMF? An intrinsic mode function IMF is a function that satisfies two conditions: [Huang 1998] 1 in the whole data set, the number of extrema and the number of zero crossings must either equal or differ at most by one; Guaranteed the IMFs are narrow band signals at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. -1-

13 What is IMF? Zhihua Yang and Lihua Yang 009 proves that Condition 1 of the IMF can really be deduced from Condition and then gives the new definition of IMF: An Intrinsic Mode Function IMF is a function that satisfies the condition that at any time instant, the mean value of the upper envelope as defined by the local maxima and the lower envelope as defined by the local minima is zero.

14 What is IMF? Robert C. Sharpley and Vesselin Vatchev006 A function ψ is an IMF if and only if it is a weak-imf whose spline envelopes satisfy the condition that the absolute value of the lower spline envelope is equal to the upper envelope and this common envelope is a quadratic polynomial. Furthermore, the common spline envelope is constant i.e., ψ is an FM signal if and only if Q = 1/P for the associated self-adjoint differential equation: d dt df t P t Q t f t dt 0

15 What is IMF? El Hadji S. Diop, R. Alexandre, and A. O. Boudraa [010] An IMF is the solution of the following parabolic PDE h h t h x,0 1 h x S x 0

16 What is IMF? Ingrid Daubechies, Jianfeng Lu1, Hau-Tieng Wu 010 A continuous function f : R C, f L C is said to be intrinsic-mode-type IMT with accuracy 0 if j t f Ate with A and having the following properties: A t C 1, t C ' ' inf t 0,sup t ' A t ' t '' t ' t

17 How to obtain IMFs? Sifting Process Cubic Spline interpolation [Huang 1998] emd.ppt Some issues: Location of Extrema Envelops Stop criterion Boundary Extension

18 Flow chart How to obtain IMFs? Original data x t Search maxima and minima [7] Local maxima envelopes Local minima envelopes U L Mean envelope mt t t Find cubic spline coefficients interpolation c t xt mt IMF IMF Y ct N residue r t xt ct N Trend Y IMFs +residue -18-

19 Modified the Location of Extrema Z. Xu, B. Huang and S. Xu [008] Relocation Extrema by using parabolic interpolation R.T. Rato, M.D. Ortigueira, A.G. Batista [008]

20 Averaging Upper and Lower Envelopes B-spline: Q. Chen, N. Huang [006] High Order spline Yannis Kopsinis and SteveMcLaughlin [008] Rational Spline Peel, M.C., G.G.S. Pegram and T.A. McMahon [008][009] parabolic parameter spline interpolation algorithm S.R. Qina,, Y.M. Zhong [006] Higher order extrema Li Lin, Ji Hongbing [009] Piecewise parabolic polynomial Zhengguang Xu, Benxiong Huang, Kewei Li [010] Raised Cosine interpolation A. Roy and J.F. Doherty [010] Trigonometric interpolation S. D. Hawley, L. E. Atlas,and H. J. Chizeck [010]

21 Averaging Upper and Lower Envelopes Rational Spline Peel, M.C., G.G.S. Pegram and T.A. McMahon [008]

22 Comuputing Mean Envelope Directly Constrained Optimization quadratic programming Sylvain Meignen and Valérie Perrier [007] Piecewise linear M. G. Frei and I. Osorio [007] Interpolate mean envelope by cubic spline using some special knots Z. Xu, B. Huang and F. Zhang [009] Local Integral Mean Hong Hong, Xinlong Wang, and Zhiyong Tao [009] PDE approaches Eric Deléchelle, Jacques Lemoine, and Oumar Niang[005] El Hadji S. Diop, R. Alexandre, and A. O. Boudraa [010] Window Averaging Blackman Window F. Bao, X. Wanga, Z. Tao, Q. Wang S. Du [010]

23 Stop Criterion Standard Deviation Haung [1998] S Number Haung [003] Three Threshold Value G. Rilling, P. Flandrin [003] Energy Difference Tracking Cheng Junsheng, Yu Dejie, Yang Yu [006] Bandwidth B. Xuan, Q. Xie and S. Peng [007]

24 NTNU MT Stop Criterion Based on Bandwidth B a B f B exp ] [ ] [ 1 H t j t a t x jh t x t x A d t x t x

25 Boundary Extension characteristic wave extending method Huang [1998] neural network extending Y. Deng, W. Wang [001] Mirror Extending J. Zhao and D. Huang [001] Data Extending Zeng and He [005] Similarity Search J. Wang, Y. Peng and X. Peng [007] Combinig End Mirror and least square polynomial extending Zhang Qingjie, Zhu Huayong, Shen Lincheng[010] Ratio extension Be able to decompose Chirp Signal QIN WU and SHERMAN RIEMENSCHNEIDER [010]

26 Mode Mixing [Huang 009] Mode mixing which is defined as a single Intrinsic Mode Function IMF consisting of signals of widely disparate scales, a signal of a similar scale residing in different IMF components. Mode mixing is caused by Signal intermittency [Huang 009] Limitation sifting process

27 Example IMF Times IMF Times 0 IMF IMF IMF IMF residue Times

28 Solution of Signal Intermittency Noise Assisted Ensemble EMD ZHAOHUA WU and NORDEN E. HUANG [009] COMPLEMENTARY ENSEMBLE EMD JIA-RONG YEH, JIANN-SHING SHIEH and NORDEN E. HUANG [010] Wavelet with Translation Invariance Qin Pinle, Lin Yan, and Chen Ming [008] Intermittency Test Algorithm Build in Function of Visual Signal developed by AnCad Iterative Gaussian Window Averaging

29 One or two frequencies? 0 cos 0t cos60t cos 0t cos30t cos 0t cost

30 One or two frequencies? 0 cos 0t cos70t cos 0t 0.*cos70t cos 0t 0.08* cos70t One or two frequency?

31 Is it a Trend? 1 10 x t t 0.3cos3 t 8 x ' t 1 0.9sin3t 0, t 6 No local extrema

32 One or two frequencies? Gabriel Rilling and Patrick Flandrin [008] x t cost acosft, f 0,1 af 1 the extrema rate is exactly the same as that of the HF component a f 1 the extrema rate is exactly the same as that of the LF component

33 One or two frequencies? considered as a single waveform does something else two components are well separated

34 One or two frequencies? f t A f t A cosf 1 cos 1 t f f 1 A1 A Range 1 : [1] They are as a single waveform. A A or Range : [1] Decomposition requires 1.75 several 1 A 1 sifting. 4 iterations A1 for close frequency harmonics. A A 1 Range 國立台灣師範大學 3 : [1] 機電科技學系 -34-

35 The Effect of Iteration The larger the iteration number of EMD is, The narrower the transition zone is.

36 The Effect of Iteration Number

37 The Effect of Iteration Number E. Huang [010] Both the empirical and the theoretical results point to the asymptotic state of infinite number of sifting iterations. The empirical result further established the processes how the final IMFs of constant amplitude are reached.

38 Differentiation Operator Method The simulation signal Y 1 0 t sin0t 0.15sin70t A A1 EMD Y A t sin 0t 0.058sin 80t A A A 1 A EMD 1 A

39 Differentiation Operator Method Differential operator Signal t A f t A cosf f 1 cos 1 t f A1 A f 1 amplitude ratio A X t 1 A 1 Differential signal f t f A f t f A sinf amplitude ratio f A A X t f f sin 1 t A 1 A 1 Differentijgasda sateate -39-

40 Differentiation Operator Method Differential signal Y' 1 0 t 0cos0t 10.5cos70t A A A A 1 Y ' t 0cos0t 4.86cos80t A A A. 4 1 A1-40-

41 Differentiation Operator Method Integral Integral a Signal 1 b Signal -41-

42 Differentiation Operator Method Flow Chart Original Signal Differential operator EMD Integral IMFs+residue Disadvantage: Integration of IMF is not always a IMF

43 Oblique EMD OEMD Inflection Points Zhijing Yang, Lihua Yanga, Chunmei Qing [010]

44 Practical Consideration of differentiation Forward difference scheme have phase lag 1 x' n x n x n 1 T Location of the detected infection point is wrong. Central difference scheme or high order central difference scheme is suggested. 1 x' n x n 1 x n x n 1 T Central difference is a zero phase filter no phase lag.

45 NTNU MT Iterative Gaussian Filter,0,0, x f x u t x ku u xx t * 4 exp exp 4 1, x f kt x kt dy y f kt y x kt t x u

46 Gaussian Window Time domain: t 1 - g,t e Frequency domain: g fft,t G, e

47 Frequency Response of Gaussian Filter Low pass filter Weighted averaging Poor Transition Zone -47-

48 Survey of Iterative Gaussian Filter [1996] Iterative Smoothed Residuals: A Low-Pass Filter for Smoothing With Controlled Shrinkage [008] Decomposition of one-dimensional waveform using iterative Gaussian diffusive filtering methods

49 原始訊號 Xt Iterative Gaussian filter t r 0 xt for signal decomposition S t r t r n n g,t r n-1 t- S n n-1 t t 1- g,t r t n-1 t S n 0 No Yes 高頻訊號 = r n 低頻訊號 = xt - r n -49-

50 N 次迭代高斯濾波器 The larger the iteration number is, The narrower the transition zone is. -50-

51 NTNU MT -51- Low pass iterative gaussian filter i n 1 i 1 i fft n 1 i 1 i n 3 1,t G,t i G,t i G i n -1, Fn,,t fn,,t i g i n -1,t fn, xt,t fn, S xt,t g,t g,t g,t g,t,t- 3g 3g S xt,t g,t g,t- g S xt,t g S

52 A Toy Example -5-

53 Number of Iterations:

54 Number of Iterations: 0-54-

55 Number of Iterations:

56 Number of Iterations: 100 Intermittency can be filtered out from the original signal -56-

57 實務上的問題遮罩長度會隨著其標準差增加, 造成在時域上迴旋積的效率低落 Note: Gaussian filter is a noncausal FIR filter Noncausal zero phase extremas do not shift after applying the operation. Solution: Using IIR filter to implement the gaussian window -57-

58 IIR Gaussian filter IIR Gaussian filter Ian T. Young, Senior Member, IEEE, Lucas J. van Vliet, and Michael van Ginkel [00]

59 Non Causal Gaussian filter -59-

60 NTNU MT Zero Phase Filter noncausal filter j j j e X e H e V j j j e U e H e W * j j e V e U * j j e W e Y * * * * * j j j j j j j j j j j e X e H e X e H e H e V e H e U e H e W e Y Fourier transforms: Implementation of a zero-phase filtering scheme: Combining the above equations, we obtain [5] S. k. Mitra, Digital signal processing, McGram.Hill international edition. Hz Time reversal Hz Time reversal x[n] v[n] V[-n]=u[n] w[n] w[-n]=y[n]

61 Conclusions Suggestion Signal Intermittency Using Window Averaging Method to Remove Improve the ability of signal decomposition for EMD Using differential operator OEMD Open Question EMD = iterative noncausal filter?

62 Some Criticisms for EMD Orthogonality of IMFs Necessary for Linear System But EMD is designed to deal with nonlinear system. Lack of mathematical foundation Instability of sifting process

63 What do you want? An universal tool A useful tool

64 敬請批評與指教 Thanks for your attention! -64-

Stochastic Processes (XI) Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University 508 YiFu Lou talk 06/

Stochastic Processes (XI) Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University hjzhang001@gmail.com 508 YiFu Lou talk 06/04/2010 - Page 1 Outline 508 YiFu Lou talk 06/04/2010