浅谈VMD---变分模态分解
很多场景下，我们需要将信号进行分解，为我们下一步操作提供方便，常用的分解方法可以有EMD族类，例如EMD、EEMD、FEEMD、CEEMDAN、ICEEMDAN等，当然也有小波分解、经验小波分解等，总之分解方式多种多样，根据样本的特点，选用不同的分解方式。这里简要介绍VMD分解。
Konstantin等人在2014年提出了一个完全非递归的变分模态分解（VMD）它可以实现分解模态的同时提取。该模型寻找一组模态和它们各自的中心频率，以便这些模态共同再现输入信号，同时每个模态在解调到基带后都是平滑的。算法的本质是将经典的维纳滤波器推广到多个自适应波段，使得其具有坚实的理论基础，并且容易理解。采用交替方向乘子法对变分模型进行有效优化，使得模型对采样噪声的鲁棒性更强。
VMD分解的具体过程可以理解为变分问题的最优解，可以相应转化为变分问题的构造和求解。
以上就是VMD（变分模态分解）的理论部分，大家不一定全弄明白，因为本人看了原文，也不能完全弄懂里面的数学关系。大体知道分解的过程包含哪几个步骤即可，知网上面关于这类分解的文章也很多，大家可以参考浏览学习下。
下面直接上代码。
function [u, u_hat, omega] = VMD(signal, alpha, tau, K, DC, init, tol)
% Variational Mode Decomposition
% Authors: Konstantin Dragomiretskiy and Dominique Zosso
% [email protected] --- 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代码很长，尽量看，能看懂多少看懂多少。这里不再讲解，因为这里都是对数学原理的复现，如果要弄懂原理，建议比照原文和代码相结合，逐行去看。如果只是利用这种分解方式，关心得出的结果，那么就没有必要大费周章了。
function [u, u_hat, omega] = VMD(signal, alpha, tau, K, DC, init, tol)函数的输入输出，这里要解释一下。
函数的输入部分：signal代表输入信号，alpha表示数据保真度约束的平衡参数 ，tau表示时间步长，K表示分解层数，DC表示如果将第一模式置于DC(0频率)，则为true。 init表示信号的初始化，tol表示收敛容错准则。通常除了K，也就是分解模态数之外，其他参数都有相应的经验值。绝大部分文献对VMD的探索也是对分解模态数的确定，顶多再加上tau的讨论。（博主后面的文章中也会进行相应的讨论。）
函数的输出部分：u表示分解模式的集合，u_hat表示模式的光谱范围，omega 表示估计模态的中心频率。
下面是调用VMD分解的主程序。主要步骤就是输入信号值，确定VMD的分解参数，画图。
tic
clc
clear all
load('IMF1_7.mat')
x=IMF1_7;
t=1:length(IMF1_7);
%--------- 对于VMD参数进行设置---------------
alpha = 2000; % moderate bandwidth constraint：适度的带宽约束/惩罚因子
tau = 0.0244; % noise-tolerance (no strict fidelity enforcement)：噪声容限（没有严格的保真度执行）
K = 7; % modes：分解的模态数
DC = 0; % no DC part imposed：无直流部分
init = 1; % initialize omegas uniformly ：omegas的均匀初始化
tol = 1e-6 ;
%--------------- Run actual VMD code:数据进行vmd分解---------------------------
[u, u_hat, omega] = VMD(x, alpha, tau, K, DC, init, tol);
figure;
imfn=u;
n=size(imfn,1); %size(X,1),返回矩阵X的行数；size(X,2),返回矩阵X的列数；N=size(X,2)，就是把矩阵X的列数赋值给N
for n1=1:n
subplot(n,1,n1);
plot(t,u(n1,:));%输出IMF分量，a(:,n)则表示矩阵a的第n列元素，u(n1,:)表示矩阵u的n1行元素
ylabel(['IMF' ,int2str(n1)],'fontsize',11);%int2str(i)是将数值i四舍五入后转变成字符，y轴命名
end
xlabel('样本序列','fontsize',14,'fontname','宋体');
%时间\itt/s
toc;
下图记为分解的结果。
以上就是对VMD分解的简单描述，下面的博文中将探讨如何对分解层数进行相应固定。