信号处理VMD 变分模态分解，示例+完整代码

首先， pip install vmdpy

下面两段代码，1为VMD函数的定义

                               2为一个模态分解的示例

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# -*- coding: utf-8 -*-
"""
Created on Wed Feb 20 19:24:58 2019

@author: Vinícius Rezende Carvalho
"""
import numpy as np

def  VMD(f, alpha, tau, K, DC, init, tol):
    """
    u,u_hat,omega = VMD(f, alpha, tau, K, DC, init, tol)
    Variational mode decomposition
    Python implementation by Vinícius Rezende Carvalho - [email protected]
    code based on Dominique Zosso's MATLAB code, available at:
    https://www.mathworks.com/matlabcentral/fileexchange/44765-variational-mode-decomposition
    Original paper:
    Dragomiretskiy, K. and Zosso, D. (2014) ‘Variational Mode Decomposition’, 
    IEEE Transactions on Signal Processing, 62(3), pp. 531–544. doi: 10.1109/TSP.2013.2288675.
    
    
    Input and Parameters:
    ---------------------
    f       - 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
    """
    
    if len(f)%2:
       f = f[:-1]

    # Period and sampling frequency of input signal
    fs = 1./len(f)
    
    ltemp = len(f)//2 
    fMirr =  np.append(np.flip(f[:ltemp],axis = 0),f)  
    fMirr = np.append(fMirr,np.flip(f[-ltemp:],axis = 0))

    # Time Domain 0 to T (of mirrored signal)
    T = len(fMirr)
    t = np.arange(1,T+1)/T  
    
    # Spectral Domain discretization
    freqs = t-0.5-(1/T)

    # Maximum number of iterations (if not converged yet, then it won't anyway)
    Niter = 500
    # For future generalizations: individual alpha for each mode
    Alpha = alpha*np.ones(K)
    
    # Construct and center f_hat
    f_hat = np.fft.fftshift((np.fft.fft(fMirr)))
    f_hat_plus = np.copy(f_hat) #copy f_hat
    f_hat_plus[:T//2] = 0

    # Initialization of omega_k
    omega_plus = np.zeros([Niter, K])


    if init == 1:
        for i in range(K):
            omega_plus[0,i] = (0.5/K)*(i)
    elif init == 2:
        omega_plus[0,:] = np.sort(np.exp(np.log(fs) + (np.log(0.5)-np.log(fs))*np.random.rand(1,K)))
    else:
        omega_plus[0,:] = 0
            
    # if DC mode imposed, set its omega to 0
    if DC:
        omega_plus[0,0] = 0
    
    # start with empty dual variables
    lambda_hat = np.zeros([Niter, len(freqs)], dtype = complex)
    
    # other inits
    uDiff = tol+np.spacing(1) # update step
    n = 0 # loop counter
    sum_uk = 0 # accumulator
    # matrix keeping track of every iterant // could be discarded for mem
    u_hat_plus = np.zeros([Niter, len(freqs), K],dtype=complex)    

    #*** Main loop for iterative updates***

    while ( uDiff > tol and  n < Niter-1 ): # not converged and below iterations limit
        # update first mode accumulator
        k = 0
        sum_uk = u_hat_plus[n,:,K-1] + sum_uk - u_hat_plus[n,:,0]
        
        # 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[k]*(freqs - omega_plus[n,k])**2)
        
        # update first omega if not held at 0
        if not(DC):
            omega_plus[n+1,k] = np.dot(freqs[T//2:T],(abs(u_hat_plus[n+1, T//2:T, k])**2))/np.sum(abs(u_hat_plus[n+1,T//2:T,k])**2)

        # update of any other mode
        for k in np.arange(1,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[k]*(freqs - omega_plus[n,k])**2)
            # center frequencies
            omega_plus[n+1,k] = np.dot(freqs[T//2:T],(abs(u_hat_plus[n+1, T//2:T, k])**2))/np.sum(abs(u_hat_plus[n+1,T//2:T,k])**2)
            
        # Dual ascent
        lambda_hat[n+1,:] = lambda_hat[n,:] + tau*(np.sum(u_hat_plus[n+1,:,:],axis = 1) - f_hat_plus)
        
        # loop counter
        n = n+1
        
        # converged yet?
        uDiff = np.spacing(1)
        for i in range(K):
            uDiff = uDiff + (1/T)*np.dot((u_hat_plus[n,:,i]-u_hat_plus[n-1,:,i]),np.conj((u_hat_plus[n,:,i]-u_hat_plus[n-1,:,i])))

        uDiff = np.abs(uDiff)        
            
    #Postprocessing and cleanup
    
    #discard empty space if converged early
    Niter = np.min([Niter,n])
    omega = omega_plus[:Niter,:]
    
    idxs = np.flip(np.arange(1,T//2+1),axis = 0)
    # Signal reconstruction
    u_hat = np.zeros([T, K],dtype = complex)
    u_hat[T//2:T,:] = u_hat_plus[Niter-1,T//2:T,:]
    u_hat[idxs,:] = np.conj(u_hat_plus[Niter-1,T//2:T,:])
    u_hat[0,:] = np.conj(u_hat[-1,:])    
    
    u = np.zeros([K,len(t)])
    for k in range(K):
        u[k,:] = np.real(np.fft.ifft(np.fft.ifftshift(u_hat[:,k])))
        
    # remove mirror part
    u = u[:,T//4:3*T//4]

    # recompute spectrum
    u_hat = np.zeros([u.shape[1],K],dtype = complex)
    for k in range(K):
        u_hat[:,k]=np.fft.fftshift(np.fft.fft(u[k,:]))

    return u, u_hat, omega

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import numpy as np
import matplotlib.pyplot as plt
from vmdpy import VMD
# Time Domain 0 to T
T = 1000
fs = 1/T
t = np.arange(1,T+1)/T
freqs = 2*np.pi*(t-0.5-fs)/(fs)
# center frequencies of components
f_1 = 2
f_2 = 24
f_3 = 288
# modes
v_1 = (np.cos(2*np.pi*f_1*t))
v_2 = 1/4*(np.cos(2*np.pi*f_2*t))
v_3 = 1/16*(np.cos(2*np.pi*f_3*t))
#
#% for visualization purposes
fsub = {1:v_1,2:v_2,3:v_3}
wsub = {1:2*np.pi*f_1,2:2*np.pi*f_2,3:2*np.pi*f_3}

# composite signal, including noise

f = v_1 + v_2 + v_3 + 0.1*np.random.randn(v_1.size)
f_hat = np.fft.fftshift((np.fft.fft(f)))

# some sample parameters for VMD
alpha = 2000       # moderate bandwidth constraint
tau = 0.            # noise-tolerance (no strict fidelity enforcement)
K = 3              # 3 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)
#%%
# Simple Visualization of decomposed modes
plt.figure()
plt.plot(u.T)
plt.title('Decomposed modes')
# For convenience here: Order omegas increasingly and reindex u/u_hat
sortIndex = np.argsort(omega[-1,:])
omega = omega[:,sortIndex]
u_hat = u_hat[:,sortIndex]
u = u[sortIndex,:]
linestyles = ['b', 'g', 'm', 'c', 'c', 'r', 'k']

fig1 = plt.figure()
plt.subplot(411)
plt.plot(t,f)
plt.xlim((0,1))
for key, value in fsub.items():
    plt.subplot(4,1,key+1)
    plt.plot(t,value)
fig1.suptitle('Original input signal and its components')


fig2 = plt.figure()
plt.loglog(freqs[T//2:], abs(f_hat[T//2:]))
plt.xlim(np.array([1,T/2])*np.pi*2)
ax = plt.gca()
ax.grid(which='major', axis='both', linestyle='--')
fig2.suptitle('Input signal spectrum')


fig3 = plt.figure()
for k in range(K):
    plt.semilogx(2*np.pi/fs*omega[:,k], np.arange(1,omega.shape[0]+1), linestyles[k])
fig3.suptitle('Evolution of center frequencies omega')


fig4 = plt.figure()
plt.loglog(freqs[T//2:], abs(f_hat[T//2:]), 'k:')
plt.xlim(np.array([1, T//2])*np.pi*2)
for k in range(K):
    plt.loglog(freqs[T//2:], abs(u_hat[T//2:,k]), linestyles[k])
fig4.suptitle('Spectral decomposition')
plt.legend(['Original','1st component','2nd component','3rd component'])


fig4 = plt.figure()

for k in range(K):
    plt.subplot(3,1,k+1)
    plt.plot(t,u[k,:], linestyles[k])
    plt.plot(t, fsub[k+1], 'k:')

    plt.xlim((0,1))
    plt.title('Reconstructed mode %d'%(k+1))
plt.show()
print('finish')