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📋📋📋本文目录如下：🎁🎁🎁

目录

💥1 概述

📚2 运行结果

2.1 算例1

2.2 算例2 

2.3 算例3

🎉3 参考文献

🌈4 Matlab代码实现

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💥1 概述

变分模态分解（VMD）是近年来引入的一种自适应数据分析方法，在各个领域引起了广泛的关注。然而，VMD是基于信号模型窄带特性的假设而制定的。为了分析宽带非线性线性调频信号（NCS），我们提出了一种称为变分非线性线性调频模式分解（VNCMD）的替代方法。VNCMD的开发基于这样一个事实，即宽带NCS可以通过使用解调技术转换为窄带信号。因此，我们的分解问题被表述为最优解调问题，通过乘法的交替方向方法（ADMM）有效地求解。我们的方法可以看作是一个时频（TF）滤波器组，它同时提取所有信号模式。提供了一些模拟和真实的数据示例，展示了VNCMD在分析包含接近甚至交叉模式的NCS方面的有效性。

📚2 运行结果

2.1 算例1

 

 

 

2.2 算例2 

 

2.3 算例3

 

部分代码：

%% initialize
[K,N] = size(eIF);%K is the number of the components，N is thenumber of the samples
t = (0:N-1)/fs;%time
e = ones(N,1);
e2 = -2*e;
% e2(1) = -1;e2(end) = -1;
oper = spdiags([e e2 e], 0:2, N-2, N);% oper = spdiags([e e2 e], -1:1, N, N);%the modified second-order difference matrix
opedoub = oper'*oper;%
sinm = zeros(K,N);cosm = zeros(K,N);%
xm = zeros(K,N);ym = zeros(K,N);%denote the two demodulated quadrature signals
iternum = 300; %the maximum allowable iterations
IFsetiter = zeros(K,N,iternum+1); IFsetiter(:,:,1) = eIF; %the collection of the obtained IF time series of all the signal modes at each iteration
ssetiter = zeros(K,N,iternum+1); %the collection of the obtained signal modes at each iteration
lamuda = zeros(1,N);%Lagrangian multiplier
for i = 1:K
    sinm(i,:) = sin(2*pi*(cumtrapz(t,eIF(i,:))));
    cosm(i,:) = cos(2*pi*(cumtrapz(t,eIF(i,:))));
    Bm = spdiags(sinm(i,:)', 0, N, N);Bdoubm = spdiags((sinm(i,:).^2)', 0, N, N);%Bdoubm = Bm'*Bm
    Am = spdiags(cosm(i,:)', 0, N, N);Adoubm = spdiags((cosm(i,:).^2)', 0, N, N);%Adoubm = Am'*Am
    xm(i,:) = (2/alpha*opedoub + Adoubm)\(Am'*s(:));
    ym(i,:) = (2/alpha*opedoub + Bdoubm)\(Bm'*s(:));
    ssetiter(i,:,1) = xm(i,:).*cosm(i,:) + ym(i,:).*sinm(i,:);%
end
%% iterations 
iter = 1;% iteration counter
sDif = tol + 1;%
sum_x = sum(xm.*cosm,1);%cumulative sum
sum_y = sum(ym.*sinm,1);%cumulative sum
while ( sDif > tol &&  iter <= iternum ) % 
   
    betathr = 10^(iter/36-10);%gradually increase the parameter beta during the iterations
    if betathr>beta
        betathr = beta; 
    end
    
    u = projec(s - sum_x - sum_y - lamuda/alpha,var);%projection operation; u denotes the noise variable; if let var=0, the output u will be zeros. 
    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%  update each mode  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
    for i = 1:K
        
       % lamuda = zeros(1,N);% if one wants to drop the Lagrangian multiplier, just set it to zeros, i.e., delete the first symbol % in this line.
       
%%%%%%%%%%%%% update the two matrices A and B %%%%%%%%%%%%%%%%%%%%%%%%%%%         
       Bm = spdiags(sinm(i,:)', 0, N, N);Bdoubm = spdiags((sinm(i,:).^2)', 0, N, N);
       Am = spdiags(cosm(i,:)', 0, N, N);Adoubm = spdiags((cosm(i,:).^2)', 0, N, N);
%%%%%%%%%%%%% x-update %%%%%%%%%%%%%%%%%%%%%%%%%%%
       sum_x = sum_x - xm(i,:).*cosm(i,:);% remove the relevant component from the sum
       xm(i,:) = (2/alpha*opedoub + Adoubm)\(Am'* (s - sum_x - sum_y - u - lamuda/alpha)');%
       interx = xm(i,:).*cosm(i,:);% temp variable
       sum_x = sum_x + interx;% update the sum
%%%%%%%%%%%%% y-update %%%%%%%%%%%%%%%%%%%%%%%%%%%
       sum_y = sum_y - ym(i,:).*sinm(i,:);% remove the relevant component from the sum
       ym(i,:) = (2/alpha*opedoub + Bdoubm)\(Bm'* (s - sum_x - sum_y - u - lamuda/alpha)');
%%%%%%%%%%%%%  update the IFs  %%%%%%%%%%%%%%%%%%%%%%%%       
       ybar = Differ(ym(i,:),1/fs); xbar = Differ(xm(i,:),1/fs);%compute the derivative of the functions
       deltaIF = (xm(i,:).*ybar - ym(i,:).*xbar)./(xm(i,:).^2 + ym(i,:).^2)/2/pi;% obtain the frequency increment by arctangent demodulation
       deltaIF = (2/betathr*opedoub + speye(N))\deltaIF';% smooth the frequency increment by low pass filtering
       eIF(i,:) = eIF(i,:) - 0.5*deltaIF';% update the IF
%%%%%%%%%%%%%  update cos and sin functions  %%%%%%%%%%%%%%%%%%%%%%%%          
       sinm(i,:) = sin(2*pi*(cumtrapz(t,eIF(i,:))));
       cosm(i,:) = cos(2*pi*(cumtrapz(t,eIF(i,:))));
%%%%%%%%%%%%% update sums %%%%%%%%%%%%%%%%%       
       sum_x = sum_x - interx + xm(i,:).*cosm(i,:); %
       sum_y = sum_y + ym(i,:).*sinm(i,:);%
       ssetiter(i,:,iter+1) = xm(i,:).*cosm(i,:) + ym(i,:).*sinm(i,:);%
    end
    IFsetiter(:,:,iter+1) = eIF;
    
%%%%%%%%%%%%% update Lagrangian multiplier %%%%%%%%%%%%%%%%%     
lamuda = lamuda + alpha*(u + sum_x + sum_y -s);

%%%%%%%%%%%%%%%%%%%%%%%%%%% restart scheme %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
if norm(u + sum_x + sum_y -s)>norm(s) %
   lamuda = zeros(1,length(t));
   for i = 1:K
    Bm = spdiags(sinm(i,:)', 0, N, N);Bdoubm = spdiags((sinm(i,:).^2)', 0, N, N);%
    Am = spdiags(cosm(i,:)', 0, N, N);Adoubm = spdiags((cosm(i,:).^2)', 0, N, N);%
    xm(i,:) = (2/alpha*opedoub + Adoubm)\(Am'*s(:));
    ym(i,:) = (2/alpha*opedoub + Bdoubm)\(Bm'*s(:));
    ssetiter(i,:,iter+1) = xm(i,:).*cosm(i,:) + ym(i,:).*sinm(i,:);
   end
   sum_x = sum(xm.*cosm,1);%
   sum_y = sum(ym.*sinm,1);%
end

%%%%%%%%%%%%%  compute the convergence index %%%%%%%%%%%%%%%%%%  
    sDif = 0;
    for i = 1:K
        sDif = sDif + (norm(ssetiter(i,:,iter+1) - ssetiter(i,:,iter))/norm(ssetiter(i,:,iter))).^2;
    end
    iter = iter + 1;
end
    IFmset = IFsetiter(:,:,1:iter);
    smset = ssetiter(:,:,1:iter);
    IA = sqrt(xm.^2 + ym.^2);
end
   

🎉3 参考文献

部分理论来源于网络，如有侵权请联系删除。

[1]Chen S, Dong X, Peng Z, et al, Nonlinear Chirp Mode Decomposition: A Variational Method, IEEE Transactions on Signal Processing, 2017.

[2]Chen S, Dong X, Xing G, et al, Separation of Overlapped Non-Stationary Signals by Ridge Path Regrouping and Intrinsic Chirp Component Decomposition, IEEE Sensors Journal, 2017.

[3]S. Chen, Z. Peng, Y. Yang, et al, Intrinsic chirp component decomposition by using Fourier Series representation, Signal Processing, 2017, 137: 319-327.

🌈4 Matlab代码实现