写出以下matlab函数的实现：[u, u_hat, omega] = VMD(signal, alpha, tau, k, DC, init, tol, maxiter);

function [u, u_hat, omega] = VMD(signal, alpha, tau, k, DC, init, tol, maxiter)
% Inputs:
% signal: the signal to decompose
% alpha: balancing parameter
% tau: time-step
% k: number of modes
% DC: boolean indicating whether to include a DC component
% init: method to initialize VMD ('rand' or 'mode')
% tol: tolerance for convergence
% maxiter: maximum number of iterations
% Outputs:
% u: the decomposed modes
% u_hat: the reconstructed signal
% omega: the instantaneous frequencies

% Calculate size of signal
N = length(signal);

% Calculate Fourier frequencies
freqs = ((-N/2:N/2-1)/N)/tau;

% Initialize VMD
if strcmp(init, 'rand')
u = rand(k, N);
elseif strcmp(init, 'mode')
u = zeros(k, N);
for i = 1:k
u(i,:) = abs(hilbert(signal)).*exp(-1j*2*pi*(i-1)*(0:N-1)/N);
end
else
error('Invalid initialization method')
end

% Initialize variables
omega = zeros(k, N);
u_hat = zeros(1, N);
u_avg = mean(u, 1);
iter = 0;
err = tol + 1;

% Perform VMD
while err > tol && iter < maxiter
% Update each mode
for j = 1:k
% Calculate mean of other modes
u_other = u([1:j-1 j+1:k], :);
u_mean = mean(u_other, 1);

% Calculate omega
omega(j,:) = imag(hilbert(u(j,:) - u_mean))./(2*pi*freqs);
omega(j,:) = smooth(omega(j,:), 20);

% Calculate alpha_0
alpha_0 = alpha*(j-1)/k;

% Calculate u_hat
u_hat = u_other*exp(1j*2*pi*freqs.*repmat(omega(j,:), k-1, 1));
if DC
u_hat = [u_hat; repmat(u_avg, k-1, 1)];
end

% Calculate lambda
lambda = sqrt(alpha_0./(abs(u_hat).^2 + eps));

% Update mode
u(j,:) = lambda.*(signal - sum(u_hat, 1)) + u_mean;
end

% Calculate error
err = norm(signal - sum(u, 1))/norm(signal);

% Increment iteration count
iter = iter + 1;
end

% Reconstruct signal
u_hat = sum(u, 1);

end