信号处理——EMD、VMD的一点小思考

作者：桂。

时间：2017-03-06  20:57:22

链接：http://www.cnblogs.com/xingshansi/p/6511916.html 

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前言

本文为Hilbert变换一篇的内容补充，主要内容为：

　　1）EMD原理介绍

　　2）代码分析

　　3）一种权衡的小trick

　　4）问题补充

内容主要为自己的学习总结，并多有借鉴他人，最后一并给出链接。

一、EMD原理介绍

　　A-EMD的意义

很多人都知道EMD(Empirical Mode Decomposition)可以将信号分解不同频率特性，并且结合Hilbert求解包络以及瞬时频率。EMD、Hilbert、瞬时频率三者有无内在联系？答案是：有。

按照Hilbert变换一篇的介绍，

$f(t) = \frac{{d\Phi (t)}}{{d(t)}}$

然而，这样求解瞬时频率在某些情况下有问题，可能出现$f(t)$为负的情况：我1秒手指动5下，频率是5Hz；反过来，频率为8Hz时，手指1秒动8下，可如果频率为-5Hz呢？负频率没有意义。

考虑信号

$x(t) = {x_1}(t) + {x_2}(t) = {A_1}{e^{j{\omega _1}t}} + {A_2}{e^{j{\omega _2}t}} = A(t){e^{j\varphi (t)}}$

为了简单起见，假设$A_1$和$A_2$恒定，且$\omega_1$和$\omega_2$是正的。信号$x(t)$的频谱应由两个在$\omega_1$和$\omega_2$的$\delta$函数组成，即

$X(\omega ) = {A_1}\delta (\omega  - {\omega _1}) + {A_2}\delta (\omega  - {\omega _2})$

因为假设$\omega_1$和$\omega_2$是正的，所以该信号解析。求得相位

$\Phi (t) = \frac{{{A_1}\sin {\omega _1}t + {A_{\rm{2}}}\sin {\omega _{\rm{2}}}t}}{{{A_1}\cos {\omega _1}t + {A_{\rm{2}}}\cos {\omega _{\rm{2}}}t}}$

分别取两组参数，对$t$求导，得到对应参数下的瞬时频率：

参数：

$\omega_1 = 10Hz$和$\omega_2 = 20Hz$.

• 组1：{$A_1 = 0.2, A_2 = 1$};
• 组2：{$A_1 = 1.2, A_2 = 1$}

对于组2，瞬时频率出现了负值。

可见：

对任意信号进行Hilbert变换，可能出现无法解释、缺乏实际意义的频率分量。Norden E. Hung等人对瞬时频率进行研究后发现，只有满足特定条件的信号，其瞬时频率才具有物理意义，并将此类信号成为：IMF/基本模式分量。　

　　B-EMD基本原理

此处给一个原理图：

　　C-基本模式分量（IMF）

EMD分解的IMF其瞬时频率具有实际物理意义，原因有两点：

• 限定1：
  • 在整个数据序列中，极值点的数量$N_e$（包括极大值、极小值点）与过零点的数量必须相等，或最多相差1个，即$(N_e-1) \le N_e \ge (N_e+1)$.
• 限定2：
  • 在任意时间点$t_i$上，信号局部极大值确定的上包络线$f_{max}(t)$和局部极小值确定的下包络线$f_{min}(t)$的均值为0.

限定1即要求信号具有类似传统平稳高斯过程的分布；限定2要求局部均值为0，同时用局部最大、最小值的包络作为近似，从而信号局部对称，避免了不对称带来的瞬时频率波动。

　　D-VMD

关于VMD(Variational Mode Decomposition)，具体原理可以参考其论文，这里我们只要记住一点：其分解的各个基本分量——即各解析信号的瞬时频率具有实际的物理意义。

 

二、代码分析

首先给出信号分别用VMD、EMD的分解结果:

给出对应的代码：

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%--------------- Preparation clear all; close all; clc; % Time Domain 0 to T T = 1000; fs = 1/T; t = (1:T)/T; freqs = 2*pi*(t-0.5-1/T)/(fs); % center frequencies of components f_1 = 2; f_2 = 24; f_3 = 288; % modes v_1 = (cos(2*pi*f_1*t)); v_2 = 1/4*(cos(2*pi*f_2*t)); v_3 = 1/16*(cos(2*pi*f_3*t)); % for visualization purposes wsub{1} = 2*pi*f_1; wsub{2} = 2*pi*f_2; wsub{3} = 2*pi*f_3; % composite signal, including noise f = v_1 + v_2 + v_3 + 0.1*randn(size(v_1)); % some sample parameters for VMD alpha = 2000;        % moderate bandwidth constraint tau = 0;            % noise-tolerance (no strict fidelity enforcement) K = 4;              % 4 modes DC = 0;             % no DC part imposed init = 1;           % initialize omegas uniformly tol = 1e-7;   %--------------- Run actual VMD code [u, u_hat, omega] = VMD(f, alpha, tau, K, DC, init, tol); subplot(size(u,1)+1,2,1); plot(t,f,'k');grid on; title('VMD分解'); subplot(size(u,1)+1,2,2); plot(freqs,abs(fft(f)),'k');grid on; title('对应频谱'); for i = 2:size(u,1)+1     subplot(size(u,1)+1,2,i*2-1);     plot(t,u(i-1,:),'k');grid on;     subplot(size(u,1)+1,2,i*2);     plot(freqs,abs(fft(u(i-1,:))),'k');grid on; end   %---------------run EMD code imf = emd(f); figure; subplot(size(imf,1)+1,2,1); plot(t,f,'k');grid on; title('EMD分解'); subplot(size(imf,1)+1,2,2); plot(freqs,abs(fft(f)),'k');grid on; title('对应频谱'); for i = 2:size(imf,1)+1     subplot(size(imf,1)+1,2,i*2-1);     plot(t,imf(i-1,:),'k');grid on;     subplot(size(imf,1)+1,2,i*2);     plot(freqs,abs(fft(imf(i-1,:))),'k');grid on; end

　　附上两个子程序的code.

VMD:

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function [u, u_hat, omega] = VMD(signal, alpha, tau, K, DC, init, tol) % Variational Mode Decomposition % Authors: Konstantin Dragomiretskiy and Dominique Zosso % zosso@math.ucla.edu --- http://www.math.ucla.edu/~zosso % Initial release 2013-12-12 (c) 2013 % % Input and Parameters: % --------------------- % signal  - the time domain signal (1D) to be decomposed % alpha   - the balancing parameter of the data-fidelity constraint % tau     - time-step of the dual ascent ( pick 0 for noise-slack ) % K       - the number of modes to be recovered % DC      - true if the first mode is put and kept at DC (0-freq) % init    - 0 = all omegas start at 0 %                    1 = all omegas start uniformly distributed %                    2 = all omegas initialized randomly % tol     - tolerance of convergence criterion; typically around 1e-6 % % Output: % ------- % u       - the collection of decomposed modes % u_hat   - spectra of the modes % omega   - estimated mode center-frequencies % % When using this code, please do cite our paper: % ----------------------------------------------- % K. Dragomiretskiy, D. Zosso, Variational Mode Decomposition, IEEE Trans. % on Signal Processing (in press) % please check here for update reference: %          http://dx.doi.org/10.1109/TSP.2013.2288675       %---------- Preparations   % Period and sampling frequency of input signal save_T = length(signal); fs = 1/save_T;   % extend the signal by mirroring T = save_T; f_mirror(1:T/2) = signal(T/2:-1:1); f_mirror(T/2+1:3*T/2) = signal; f_mirror(3*T/2+1:2*T) = signal(T:-1:T/2+1); f = f_mirror;   % Time Domain 0 to T (of mirrored signal) T = length(f); t = (1:T)/T;   % Spectral Domain discretization freqs = t-0.5-1/T;   % Maximum number of iterations (if not converged yet, then it won't anyway) N = 500;   % For future generalizations: individual alpha for each mode Alpha = alpha*ones(1,K);   % Construct and center f_hat f_hat = fftshift((fft(f))); f_hat_plus = f_hat; f_hat_plus(1:T/2) = 0;   % matrix keeping track of every iterant // could be discarded for mem u_hat_plus = zeros(N, length(freqs), K);   % Initialization of omega_k omega_plus = zeros(N, K); switch init     case 1         for i = 1:K             omega_plus(1,i) = (0.5/K)*(i-1);         end     case 2         omega_plus(1,:) = sort(exp(log(fs) + (log(0.5)-log(fs))*rand(1,K)));     otherwise         omega_plus(1,:) = 0; end   % if DC mode imposed, set its omega to 0 if DC     omega_plus(1,1) = 0; end   % start with empty dual variables lambda_hat = zeros(N, length(freqs));   % other inits uDiff = tol+eps; % update step n = 1; % loop counter sum_uk = 0; % accumulator       % ----------- Main loop for iterative updates         while ( uDiff > tol &&  n < N ) % not converged and below iterations limit           % update first mode accumulator     k = 1;     sum_uk = u_hat_plus(n,:,K) + sum_uk - u_hat_plus(n,:,1);           % update spectrum of first mode through Wiener filter of residuals     u_hat_plus(n+1,:,k) = (f_hat_plus - sum_uk - lambda_hat(n,:)/2)./(1+Alpha(1,k)*(freqs - omega_plus(n,k)).^2);           % update first omega if not held at 0     if ~DC         omega_plus(n+1,k) = (freqs(T/2+1:T)*(abs(u_hat_plus(n+1, T/2+1:T, k)).^2)')/sum(abs(u_hat_plus(n+1,T/2+1:T,k)).^2);     end           % update of any other mode     for k=2:K                   % accumulator         sum_uk = u_hat_plus(n+1,:,k-1) + sum_uk - u_hat_plus(n,:,k);                   % mode spectrum         u_hat_plus(n+1,:,k) = (f_hat_plus - sum_uk - lambda_hat(n,:)/2)./(1+Alpha(1,k)*(freqs - omega_plus(n,k)).^2);                   % center frequencies         omega_plus(n+1,k) = (freqs(T/2+1:T)*(abs(u_hat_plus(n+1, T/2+1:T, k)).^2)')/sum(abs(u_hat_plus(n+1,T/2+1:T,k)).^2);               end           % Dual ascent     lambda_hat(n+1,:) = lambda_hat(n,:) + tau*(sum(u_hat_plus(n+1,:,:),3) - f_hat_plus);           % loop counter     n = n+1;           % converged yet?     uDiff = eps;     for i=1:K         uDiff = uDiff + 1/T*(u_hat_plus(n,:,i)-u_hat_plus(n-1,:,i))*conj((u_hat_plus(n,:,i)-u_hat_plus(n-1,:,i)))';     end     uDiff = abs(uDiff);       end     %------ Postprocessing and cleanup     % discard empty space if converged early N = min(N,n); omega = omega_plus(1:N,:);   % Signal reconstruction u_hat = zeros(T, K); u_hat((T/2+1):T,:) = squeeze(u_hat_plus(N,(T/2+1):T,:)); u_hat((T/2+1):-1:2,:) = squeeze(conj(u_hat_plus(N,(T/2+1):T,:))); u_hat(1,:) = conj(u_hat(end,:));   u = zeros(K,length(t));   for k = 1:K     u(k,:)=real(ifft(ifftshift(u_hat(:,k)))); end   % remove mirror part u = u(:,T/4+1:3*T/4);   % recompute spectrum clear u_hat; for k = 1:K     u_hat(:,k)=fftshift(fft(u(k,:)))'; end   end

EMD:

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%EMD  computes Empirical Mode Decomposition % % %   Syntax % % % IMF = EMD(X) % IMF = EMD(X,...,'Option_name',Option_value,...) % IMF = EMD(X,OPTS) % [IMF,ORT,NB_ITERATIONS] = EMD(...) % % %   Description % % % IMF = EMD(X) where X is a real vector computes the Empirical Mode % Decomposition [1] of X, resulting in a matrix IMF containing 1 IMF per row, the % last one being the residue. The default stopping criterion is the one proposed % in [2]: % %   at each point, mean_amplitude < THRESHOLD2*envelope_amplitude %   & %   mean of boolean array {(mean_amplitude)/(envelope_amplitude) > THRESHOLD} < TOLERANCE %   & %   |#zeros-#extrema|<=1 % % where mean_amplitude = abs(envelope_max+envelope_min)/2 % and envelope_amplitude = abs(envelope_max-envelope_min)/2 % % IMF = EMD(X) where X is a complex vector computes Bivariate Empirical Mode % Decomposition [3] of X, resulting in a matrix IMF containing 1 IMF per row, the % last one being the residue. The default stopping criterion is similar to the % one proposed in [2]: % %   at each point, mean_amplitude < THRESHOLD2*envelope_amplitude %   & %   mean of boolean array {(mean_amplitude)/(envelope_amplitude) > THRESHOLD} < TOLERANCE % % where mean_amplitude and envelope_amplitude have definitions similar to the % real case % % IMF = EMD(X,...,'Option_name',Option_value,...) sets options Option_name to % the specified Option_value (see Options) % % IMF = EMD(X,OPTS) is equivalent to the above syntax provided OPTS is a struct % object with field names corresponding to option names and field values being the % associated values % % [IMF,ORT,NB_ITERATIONS] = EMD(...) returns an index of orthogonality %                       ________ %         _  |IMF(i,:).*IMF(j,:)| %   ORT = \ _____________________ %         / %         ?       || X ||?%        i~=j % % and the number of iterations to extract each mode in NB_ITERATIONS % % %   Options % % %  stopping criterion options: % % STOP: vector of stopping parameters [THRESHOLD,THRESHOLD2,TOLERANCE] % if the input vector's length is less than 3, only the first parameters are % set, the remaining ones taking default values. % default: [0.05,0.5,0.05] % % FIX (int): disable the default stopping criterion and do exactly <FIX> % number of sifting iterations for each mode % % FIX_H (int): disable the default stopping criterion and do <FIX_H> sifting % iterations with |#zeros-#extrema|<=1 to stop [4] % %  bivariate/complex EMD options: % % COMPLEX_VERSION: selects the algorithm used for complex EMD ([3]) % COMPLEX_VERSION = 1: "algorithm 1" % COMPLEX_VERSION = 2: "algorithm 2" (default) % % NDIRS: number of directions in which envelopes are computed (default 4) % rem: the actual number of directions (according to [3]) is 2*NDIRS % %  other options: % % T: sampling times (line vector) (default: 1:length(x)) % % MAXITERATIONS: maximum number of sifting iterations for the computation of each % mode (default: 2000) % % MAXMODES: maximum number of imfs extracted (default: Inf) % % DISPLAY: if equals to 1 shows sifting steps with pause % if equals to 2 shows sifting steps without pause (movie style) % rem: display is disabled when the input is complex % % INTERP: interpolation scheme: 'linear', 'cubic', 'pchip' or 'spline' (default) % see interp1 documentation for details % % MASK: masking signal used to improve the decomposition according to [5] % % %   Examples % % %X = rand(1,512); % %IMF = emd(X); % %IMF = emd(X,'STOP',[0.1,0.5,0.05],'MAXITERATIONS',100); % %T=linspace(0,20,1e3); %X = 2*exp(i*T)+exp(3*i*T)+.5*T; %IMF = emd(X,'T',T); % %OPTIONS.DISLPAY = 1; %OPTIONS.FIX = 10; %OPTIONS.MAXMODES = 3; %[IMF,ORT,NBITS] = emd(X,OPTIONS); % % %   References % % % [1] N. E. Huang et al., "The empirical mode decomposition and the % Hilbert spectrum for non-linear and non stationary time series analysis", % Proc. Royal Soc. London A, Vol. 454, pp. 903-995, 1998 % % [2] G. Rilling, P. Flandrin and P. Gon鏰lves % "On Empirical Mode Decomposition and its algorithms", % IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing % NSIP-03, Grado (I), June 2003 % % [3] G. Rilling, P. Flandrin, P. Gon鏰lves and J. M. Lilly., % "Bivariate Empirical Mode Decomposition", % Signal Processing Letters (submitted) % % [4] N. E. Huang et al., "A confidence limit for the Empirical Mode % Decomposition and Hilbert spectral analysis", % Proc. Royal Soc. London A, Vol. 459, pp. 2317-2345, 2003 % % [5] R. Deering and J. F. Kaiser, "The use of a masking signal to improve % empirical mode decomposition", ICASSP 2005 % % % See also %  emd_visu (visualization), %  emdc, emdc_fix (fast implementations of EMD), %  cemdc, cemdc_fix, cemdc2, cemdc2_fix (fast implementations of bivariate EMD), %  hhspectrum (Hilbert-Huang spectrum) % % % G. Rilling, last modification: 3.2007 % gabriel.rilling@ens-lyon.fr     function [imf,ort,nbits] = emd(varargin)   [x,t,sd,sd2,tol,MODE_COMPLEX,ndirs,display_sifting,sdt,sd2t,r,imf,k,nbit,NbIt,MAXITERATIONS,FIXE,FIXE_H,MAXMODES,INTERP,mask] = init(varargin{:});   if display_sifting   fig_h = figure; end     %main loop : requires at least 3 extrema to proceed while ~stop_EMD(r,MODE_COMPLEX,ndirs) && (k < MAXMODES+1 || MAXMODES == 0) && ~any(mask)     % current mode   m = r;     % mode at previous iteration   mp = m;     %computation of mean and stopping criterion   if FIXE     [stop_sift,moyenne] = stop_sifting_fixe(t,m,INTERP,MODE_COMPLEX,ndirs);   elseif FIXE_H     stop_count = 0;     [stop_sift,moyenne] = stop_sifting_fixe_h(t,m,INTERP,stop_count,FIXE_H,MODE_COMPLEX,ndirs);   else     [stop_sift,moyenne] = stop_sifting(m,t,sd,sd2,tol,INTERP,MODE_COMPLEX,ndirs);   end     % in case the current mode is so small that machine precision can cause   % spurious extrema to appear   if (max(abs(m))) < (1e-10)*(max(abs(x)))     if ~stop_sift       warning('emd:warning','forced stop of EMD : too small amplitude')     else       disp('forced stop of EMD : too small amplitude')     end     break   end       % sifting loop   while ~stop_sift && nbit<MAXITERATIONS       if(~MODE_COMPLEX && nbit>MAXITERATIONS/5 && mod(nbit,floor(MAXITERATIONS/10))==0 && ~FIXE && nbit > 100)       disp(['mode ',int2str(k),', iteration ',int2str(nbit)])       if exist('s','var')         disp(['stop parameter mean value : ',num2str(s)])       end       [im,iM] = extr(m);       disp([int2str(sum(m(im) > 0)),' minima > 0; ',int2str(sum(m(iM) < 0)),' maxima < 0.'])     end       %sifting     m = m - moyenne;       %computation of mean and stopping criterion     if FIXE       [stop_sift,moyenne] = stop_sifting_fixe(t,m,INTERP,MODE_COMPLEX,ndirs);     elseif FIXE_H       [stop_sift,moyenne,stop_count] = stop_sifting_fixe_h(t,m,INTERP,stop_count,FIXE_H,MODE_COMPLEX,ndirs);     else       [stop_sift,moyenne,s] = stop_sifting(m,t,sd,sd2,tol,INTERP,MODE_COMPLEX,ndirs);     end       % display     if display_sifting && ~MODE_COMPLEX       NBSYM = 2;       [indmin,indmax] = extr(mp);       [tmin,tmax,mmin,mmax] = boundary_conditions(indmin,indmax,t,mp,mp,NBSYM);       envminp = interp1(tmin,mmin,t,INTERP);       envmaxp = interp1(tmax,mmax,t,INTERP);       envmoyp = (envminp+envmaxp)/2;       if FIXE || FIXE_H         display_emd_fixe(t,m,mp,r,envminp,envmaxp,envmoyp,nbit,k,display_sifting)       else         sxp=2*(abs(envmoyp))./(abs(envmaxp-envminp));         sp = mean(sxp);         display_emd(t,m,mp,r,envminp,envmaxp,envmoyp,s,sp,sxp,sdt,sd2t,nbit,k,display_sifting,stop_sift)       end     end       mp = m;     nbit=nbit+1;     NbIt=NbIt+1;       if(nbit==(MAXITERATIONS-1) && ~FIXE && nbit > 100)       if exist('s','var')         warning('emd:warning',['forced stop of sifting : too many iterations... mode ',int2str(k),'. stop parameter mean value : ',num2str(s)])       else         warning('emd:warning',['forced stop of sifting : too many iterations... mode ',int2str(k),'.'])       end     end     end % sifting loop   imf(k,:) = m;   if display_sifting     disp(['mode ',int2str(k),' stored'])   end   nbits(k) = nbit;   k = k+1;       r = r - m;   nbit=0;     end %main loop   if any(r) && ~any(mask)   imf(k,:) = r; end   ort = io(x,imf);   if display_sifting   close end end   %--------------------------------------------------------------------------------------------------- % tests if there are enough (3) extrema to continue the decomposition function stop = stop_EMD(r,MODE_COMPLEX,ndirs) if MODE_COMPLEX   for k = 1:ndirs     phi = (k-1)*pi/ndirs;     [indmin,indmax] = extr(real(exp(i*phi)*r));     ner(k) = length(indmin) + length(indmax);   end   stop = any(ner < 3); else   [indmin,indmax] = extr(r);   ner = length(indmin) + length(indmax);   stop = ner < 3; end end   %--------------------------------------------------------------------------------------------------- % computes the mean of the envelopes and the mode amplitude estimate function [envmoy,nem,nzm,amp] = mean_and_amplitude(m,t,INTERP,MODE_COMPLEX,ndirs) NBSYM = 2; if MODE_COMPLEX   switch MODE_COMPLEX     case 1       for k = 1:ndirs         phi = (k-1)*pi/ndirs;         y = real(exp(-i*phi)*m);         [indmin,indmax,indzer] = extr(y);         nem(k) = length(indmin)+length(indmax);         nzm(k) = length(indzer);         [tmin,tmax,zmin,zmax] = boundary_conditions(indmin,indmax,t,y,m,NBSYM);         envmin(k,:) = interp1(tmin,zmin,t,INTERP);         envmax(k,:) = interp1(tmax,zmax,t,INTERP);       end       envmoy = mean((envmin+envmax)/2,1);       if nargout > 3         amp = mean(abs(envmax-envmin),1)/2;       end     case 2       for k = 1:ndirs         phi = (k-1)*pi/ndirs;         y = real(exp(-i*phi)*m);         [indmin,indmax,indzer] = extr(y);         nem(k) = length(indmin)+length(indmax);         nzm(k) = length(indzer);         [tmin,tmax,zmin,zmax] = boundary_conditions(indmin,indmax,t,y,y,NBSYM);         envmin(k,:) = exp(i*phi)*interp1(tmin,zmin,t,INTERP);         envmax(k,:) = exp(i*phi)*interp1(tmax,zmax,t,INTERP);       end       envmoy = mean((envmin+envmax),1);       if nargout > 3         amp = mean(abs(envmax-envmin),1)/2;       end   end else   [indmin,indmax,indzer] = extr(m);   nem = length(indmin)+length(indmax);   nzm = length(indzer);   [tmin,tmax,mmin,mmax] = boundary_conditions(indmin,indmax,t,m,m,NBSYM);   envmin = interp1(tmin,mmin,t,INTERP);   envmax = interp1(tmax,mmax,t,INTERP);   envmoy = (envmin+envmax)/2;   if nargout > 3     amp = mean(abs(envmax-envmin),1)/2;   end end end   %------------------------------------------------------------------------------- % default stopping criterion function [stop,envmoy,s] = stop_sifting(m,t,sd,sd2,tol,INTERP,MODE_COMPLEX,ndirs) try   [envmoy,nem,nzm,amp] = mean_and_amplitude(m,t,INTERP,MODE_COMPLEX,ndirs);   sx = abs(envmoy)./amp;   s = mean(sx);   stop = ~((mean(sx > sd) > tol | any(sx > sd2)) & (all(nem > 2)));   if ~MODE_COMPLEX     stop = stop && ~(abs(nzm-nem)>1);   end catch   stop = 1;   envmoy = zeros(1,length(m));   s = NaN; end end   %------------------------------------------------------------------------------- % stopping criterion corresponding to option FIX function [stop,moyenne]= stop_sifting_fixe(t,m,INTERP,MODE_COMPLEX,ndirs) try   moyenne = mean_and_amplitude(m,t,INTERP,MODE_COMPLEX,ndirs);   stop = 0; catch   moyenne = zeros(1,length(m));   stop = 1; end end   %------------------------------------------------------------------------------- % stopping criterion corresponding to option FIX_H function [stop,moyenne,stop_count]= stop_sifting_fixe_h(t,m,INTERP,stop_count,FIXE_H,MODE_COMPLEX,ndirs) try   [moyenne,nem,nzm] = mean_and_amplitude(m,t,INTERP,MODE_COMPLEX,ndirs);   if (all(abs(nzm-nem)>1))     stop = 0;     stop_count = 0;   else     stop_count = stop_count+1;     stop = (stop_count == FIXE_H);   end catch   moyenne = zeros(1,length(m));   stop = 1; end end   %------------------------------------------------------------------------------- % displays the progression of the decomposition with the default stopping criterion function display_emd(t,m,mp,r,envmin,envmax,envmoy,s,sb,sx,sdt,sd2t,nbit,k,display_sifting,stop_sift) subplot(4,1,1) plot(t,mp);hold on; plot(t,envmax,'--k');plot(t,envmin,'--k');plot(t,envmoy,'r'); title(['IMF ',int2str(k),';   iteration ',int2str(nbit),' before sifting']); set(gca,'XTick',[]) hold  off subplot(4,1,2) plot(t,sx) hold on plot(t,sdt,'--r') plot(t,sd2t,':k') title('stop parameter') set(gca,'XTick',[]) hold off subplot(4,1,3) plot(t,m) title(['IMF ',int2str(k),';   iteration ',int2str(nbit),' after sifting']); set(gca,'XTick',[]) subplot(4,1,4); plot(t,r-m) title('residue'); disp(['stop parameter mean value : ',num2str(sb),' before sifting and ',num2str(s),' after']) if stop_sift   disp('last iteration for this mode') end if display_sifting == 2   pause(0.01) else   pause end end   %--------------------------------------------------------------------------------------------------- % displays the progression of the decomposition with the FIX and FIX_H stopping criteria function display_emd_fixe(t,m,mp,r,envmin,envmax,envmoy,nbit,k,display_sifting) subplot(3,1,1) plot(t,mp);hold on; plot(t,envmax,'--k');plot(t,envmin,'--k');plot(t,envmoy,'r'); title(['IMF ',int2str(k),';   iteration ',int2str(nbit),' before sifting']); set(gca,'XTick',[]) hold  off subplot(3,1,2) plot(t,m) title(['IMF ',int2str(k),';   iteration ',int2str(nbit),' after sifting']); set(gca,'XTick',[]) subplot(3,1,3); plot(t,r-m) title('residue'); if display_sifting == 2   pause(0.01) else   pause end end   %--------------------------------------------------------------------------------------- % defines new extrema points to extend the interpolations at the edges of the % signal (mainly mirror symmetry) function [tmin,tmax,zmin,zmax] = boundary_conditions(indmin,indmax,t,x,z,nbsym)           lx = length(x);           if (length(indmin) + length(indmax) < 3)         error('not enough extrema')     end       % boundary conditions for interpolations :       if indmax(1) < indmin(1)         if x(1) > x(indmin(1))             lmax = fliplr(indmax(2:min(end,nbsym+1)));             lmin = fliplr(indmin(1:min(end,nbsym)));             lsym = indmax(1);         else             lmax = fliplr(indmax(1:min(end,nbsym)));             lmin = [fliplr(indmin(1:min(end,nbsym-1))),1];             lsym = 1;         end     else           if x(1) < x(indmax(1))             lmax = fliplr(indmax(1:min(end,nbsym)));             lmin = fliplr(indmin(2:min(end,nbsym+1)));             lsym = indmin(1);         else             lmax = [fliplr(indmax(1:min(end,nbsym-1))),1];             lmin = fliplr(indmin(1:min(end,nbsym)));             lsym = 1;         end     end           if indmax(end) < indmin(end)         if x(end) < x(indmax(end))             rmax = fliplr(indmax(max(end-nbsym+1,1):end));             rmin = fliplr(indmin(max(end-nbsym,1):end-1));             rsym = indmin(end);         else             rmax = [lx,fliplr(indmax(max(end-nbsym+2,1):end))];             rmin = fliplr(indmin(max(end-nbsym+1,1):end));             rsym = lx;         end     else         if x(end) > x(indmin(end))             rmax = fliplr(indmax(max(end-nbsym,1):end-1));             rmin = fliplr(indmin(max(end-nbsym+1,1):end));             rsym = indmax(end);         else             rmax = fliplr(indmax(max(end-nbsym+1,1):end));             rmin = [lx,fliplr(indmin(max(end-nbsym+2,1):end))];             rsym = lx;         end     end           tlmin = 2*t(lsym)-t(lmin);     tlmax = 2*t(lsym)-t(lmax);     trmin = 2*t(rsym)-t(rmin);     trmax = 2*t(rsym)-t(rmax);           % in case symmetrized parts do not extend enough     if tlmin(1) > t(1) || tlmax(1) > t(1)         if lsym == indmax(1)             lmax = fliplr(indmax(1:min(end,nbsym)));         else             lmin = fliplr(indmin(1:min(end,nbsym)));         end         if lsym == 1             error('bug')         end         lsym = 1;         tlmin = 2*t(lsym)-t(lmin);         tlmax = 2*t(lsym)-t(lmax);     end             if trmin(end) < t(lx) || trmax(end) < t(lx)         if rsym == indmax(end)             rmax = fliplr(indmax(max(end-nbsym+1,1):end));         else             rmin = fliplr(indmin(max(end-nbsym+1,1):end));         end     if rsym == lx         error('bug')     end         rsym = lx;         trmin = 2*t(rsym)-t(rmin);         trmax = 2*t(rsym)-t(rmax);     end                 zlmax =z(lmax);     zlmin =z(lmin);     zrmax =z(rmax);     zrmin =z(rmin);            tmin = [tlmin t(indmin) trmin];     tmax = [tlmax t(indmax) trmax];     zmin = [zlmin z(indmin) zrmin];     zmax = [zlmax z(indmax) zrmax]; end       %--------------------------------------------------------------------------------------------------- %extracts the indices of extrema function [indmin, indmax, indzer] = extr(x,t)   if(nargin==1)   t=1:length(x); end   m = length(x);   if nargout > 2   x1=x(1:m-1);   x2=x(2:m);   indzer = find(x1.*x2<0);     if any(x == 0)     iz = find( x==0 );     indz = [];     if any(diff(iz)==1)       zer = x == 0;       dz = diff([0 zer 0]);       debz = find(dz == 1);       finz = find(dz == -1)-1;       indz = round((debz+finz)/2);     else       indz = iz;     end     indzer = sort([indzer indz]);   end end   d = diff(x);   n = length(d); d1 = d(1:n-1); d2 = d(2:n); indmin = find(d1.*d2<0 & d1<0)+1; indmax = find(d1.*d2<0 & d1>0)+1;     % when two or more successive points have the same value we consider only one extremum in the middle of the constant area % (only works if the signal is uniformly sampled)   if any(d==0)     imax = [];   imin = [];     bad = (d==0);   dd = diff([0 bad 0]);   debs = find(dd == 1);   fins = find(dd == -1);   if debs(1) == 1     if length(debs) > 1       debs = debs(2:end);       fins = fins(2:end);     else       debs = [];       fins = [];     end   end   if length(debs) > 0     if fins(end) == m       if length(debs) > 1         debs = debs(1:(end-1));         fins = fins(1:(end-1));         else         debs = [];         fins = [];       end     end   end   lc = length(debs);   if lc > 0     for k = 1:lc       if d(debs(k)-1) > 0         if d(fins(k)) < 0           imax = [imax round((fins(k)+debs(k))/2)];         end       else         if d(fins(k)) > 0           imin = [imin round((fins(k)+debs(k))/2)];         end       end     end   end     if length(imax) > 0     indmax = sort([indmax imax]);   end     if length(imin) > 0     indmin = sort([indmin imin]);   end   end end   %---------------------------------------------------------------------------------------------------   function ort = io(x,imf) % ort = IO(x,imf) computes the index of orthogonality % % inputs : - x    : analyzed signal %          - imf  : empirical mode decomposition   n = size(imf,1);   s = 0;   for i = 1:n   for j =1:n     if i~=j       s = s + abs(sum(imf(i,:).*conj(imf(j,:)))/sum(x.^2));     end   end end   ort = 0.5*s; end %---------------------------------------------------------------------------------------------------   function [x,t,sd,sd2,tol,MODE_COMPLEX,ndirs,display_sifting,sdt,sd2t,r,imf,k,nbit,NbIt,MAXITERATIONS,FIXE,FIXE_H,MAXMODES,INTERP,mask] = init(varargin)   x = varargin{1}; if nargin == 2   if isstruct(varargin{2})     inopts = varargin{2};   else     error('when using 2 arguments the first one is the analyzed signal X and the second one is a struct object describing the options')   end elseif nargin > 2   try     inopts = struct(varargin{2:end});   catch     error('bad argument syntax')   end end   % default for stopping defstop = [0.05,0.5,0.05];   opt_fields = {'t','stop','display','maxiterations','fix','maxmodes','interp','fix_h','mask','ndirs','complex_version'};   defopts.stop = defstop; defopts.display = 0; defopts.t = 1:max(size(x)); defopts.maxiterations = 2000; defopts.fix = 0; defopts.maxmodes = 0; defopts.interp = 'spline'; defopts.fix_h = 0; defopts.mask = 0; defopts.ndirs = 4; defopts.complex_version = 2;   opts = defopts;       if(nargin==1)   inopts = defopts; elseif nargin == 0   error('not enough arguments') end     names = fieldnames(inopts); for nom = names'   if ~any(strcmpi(char(nom), opt_fields))     error(['bad option field name: ',char(nom)])   end   if ~isempty(eval(['inopts.',char(nom)])) % empty values are discarded     eval(['opts.',lower(char(nom)),' = inopts.',char(nom),';'])   end end   t = opts.t; stop = opts.stop; display_sifting = opts.display; MAXITERATIONS = opts.maxiterations; FIXE = opts.fix; MAXMODES = opts.maxmodes; INTERP = opts.interp; FIXE_H = opts.fix_h; mask = opts.mask; ndirs = opts.ndirs; complex_version = opts.complex_version;   if ~isvector(x)   error('X must have only one row or one column') end   if size(x,1) > 1   x = x.'; end   if ~isvector(t)   error('option field T must have only one row or one column') end   if ~isreal(t)   error('time instants T must be a real vector') end   if size(t,1) > 1   t = t'; end   if (length(t)~=length(x))   error('X and option field T must have the same length') end   if ~isvector(stop) || length(stop) > 3   error('option field STOP must have only one row or one column of max three elements') end   if ~all(isfinite(x))   error('data elements must be finite') end   if size(stop,1) > 1   stop = stop'; end   L = length(stop); if L < 3   stop(3)=defstop(3); end   if L < 2   stop(2)=defstop(2); end     if ~ischar(INTERP) || ~any(strcmpi(INTERP,{'linear','cubic','spline'}))   error('INTERP field must be ''linear'', ''cubic'', ''pchip'' or ''spline''') end   %special procedure when a masking signal is specified if any(mask)   if ~isvector(mask) || length(mask) ~= length(x)     error('masking signal must have the same dimension as the analyzed signal X')   end     if size(mask,1) > 1     mask = mask.';   end   opts.mask = 0;   imf1 = emd(x+mask,opts);   imf2 = emd(x-mask,opts);   if size(imf1,1) ~= size(imf2,1)     warning('emd:warning',['the two sets of IMFs have different sizes: ',int2str(size(imf1,1)),' and ',int2str(size(imf2,1)),' IMFs.'])   end   S1 = size(imf1,1);   S2 = size(imf2,1);   if S1 ~= S2     if S1 < S2       tmp = imf1;       imf1 = imf2;       imf2 = tmp;     end     imf2(max(S1,S2),1) = 0;   end   imf = (imf1+imf2)/2;   end     sd = stop(1); sd2 = stop(2); tol = stop(3);   lx = length(x);   sdt = sd*ones(1,lx); sd2t = sd2*ones(1,lx);   if FIXE   MAXITERATIONS = FIXE;   if FIXE_H     error('cannot use both ''FIX'' and ''FIX_H'' modes')   end end   MODE_COMPLEX = ~isreal(x)*complex_version; if MODE_COMPLEX && complex_version ~= 1 && complex_version ~= 2   error('COMPLEX_VERSION parameter must equal 1 or 2') end     % number of extrema and zero-crossings in residual ner = lx; nzr = lx;   r = x;   if ~any(mask) % if a masking signal is specified "imf" already exists at this stage   imf = []; end k = 1;   % iterations counter for extraction of 1 mode nbit=0;   % total iterations counter NbIt=0; end %---------------------------------------------------------------------------------------------------

关于EMD，有对应的工具箱。VMD也有扩展的二维分解，此处不再展开。

 

三、一种权衡的小trick

关于瞬时频率的原理以及代码，参考另一篇博文。

比较来看：

• EMD分解的IMF分量个数不能人为设定，而VMD(Variational Mode Decomposition)则可以;
• 但VMD也有弊端：分解过多，则信号断断续续，没有多少规律可言。

能不能取长补短呢？

自己之前做了一个小code，放在这里，供大家交流使用（此理论为自己首创，版权所有，拿去也不介意！(●'◡'●)）。
给定一个信号,下图是EMD分解结果，分解出了5个分量。

再来一个VMD（设定分量个数为3）的分解结果：

比较两个结果，可以发现：VMD的低频分量，更容易表达出经济波动的大趋势，而EMD则不易观察该特性。
或许有人会说：几个EMD分量叠加一下，也会有该效果，但如果不观察分解的数据，如何确定几个分量相加呢？更何况EMD总的IMF个数也是未知！

VMD的优势观察到了，但如何确定分量个数呢？
再来一个效果图：

这里分析了VMD分量从1~9，9种情况下某特征的曲线，可以观察到：个数增加到一定数量，曲线有了明显的下弯曲现象（该特性容易借助曲率，进行量化分析，不再展开），这个临界的个数就是分解的合适数量，此处：K=3，因为到4就有了明显的下弯曲。

可见通过该特征，即可理论上得出最优K。下面讲一讲这个某特征为何物？
上一段代码：

 

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for st=1:9     K=st+1;     [u, u_hat, omega] = VMD(data, length(data), 0, K, 0, 1, 1e-5);     u=flipud(u);     resf=zeros(1,K);     for i=1:K         testdata=u(i,:);         hilbert(testdata');          z=hilbert(testdata');                   % 希尔伯特变换         a=abs(z);                               % 包络线         fnor=instfreq(z);                       % 瞬时频率         resf(i)=mean(fnor);          end     subplot(3,3,st)     plot(resf,'k');title(['个数为',num2str(st)]);grid on; end

 

　　没错，该特征就是：分量瞬时频率的均值。如果分解个数过大，则分量会出现断断絮絮地现象，特别是在高频，这样一来，即使是高频，平均瞬时频率反而低一些，这也是下弯曲的根本原因。

这个小trick就介绍到这里。

 

四、问题补充

HHT算法中，有两处存在端点效应，VMD是否也有呢？这一点没有再去验证。另外，关于Hilbert的端点效应，在另一篇博文已经给出。

 

参考：

宋知用：《MATLAB在语音信号分析和合成中的应用》

了凡春秋： http://blog.sina.com.cn/s/blog_6163bdeb0102e2cd.html

VMD-code:https://cn.mathworks.com/matlabcentral/fileexchange/44765-variational-mode-decomposition

EMD原理图：http://blog.sciencenet.cn/blog-244606-256958.html