变分模态分解（VMD）原理-附代码

1 VMD算法原理

VMD的思想认为待分解信号是由不同IMF的子信号组成的。VMD为避免信号分解过程中出现模态混叠，在计算IMF时舍弃了传统信号分解算法所使用的递归求解的思想，VMD采用的是完全非递归的模态分解。与传统信号分解算法相比，VMD拥有非递归求解和自主选择模态个数的优点。该算法可将第j条线路的暂态零序电流信号分解为K个中心角频率为的本征模态函数，其中K为人为指定的模态分量个数。不同于EMD，VMD将每个IMF定义为调幅调频函数，可表示为：

                              （1）

VMD算法可分为变分问题的构造和求解两部分：

(1)变分问题的构造：

                                                    （2）

①对暂态零序电流信号进行Hilbert变换，获得K个模态分量的解析信号，并得到单边频谱：

                       (3)

式中为冲激函数。

②将各模态的频谱调制到基频带上，得到：

③计算式(3)梯度的平方范数，并估计每个模态信号的带宽，构造变分问题如下：

            (4)

(2)变分问题的求解

将上述约束性转化为非约束性变分问题，在式(4)中引入二次惩罚因子和拉格朗日乘法算子，扩展的拉格朗日表达式为：

           (5)

采用乘法算子交替方向法(Alternate Direction Method of Multipliers，ADMM)求解

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 constraint2000或20000
% tau     - time-step of the dual ascent ( pick 0 for noise-slack ) 0
% K       - the number of modes to be recovered
% DC      - true if the first mode is put and kept at DC (0-freq) 0
% 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