新旧版MATLAB中的希尔伯特-黄变换(HHT)及其边际谱的求取问题

最近，由于科研工作需求，需要处理某一类波动信号，遂利用经验模态分解（EMD）、集合经验模态分解（EEMD）和希尔伯特-黄变换对上述信号进行处理，期望获取有用的本征模态函数（IMF）、频率信息和边际谱等。

在信号实施和处理时发现，网上大多代码均由较早时期的MATLAB完成，彼时还没有自带的HHT函数。而在MATLAB2018a及其以后的版本中，集成和自带了HHT函数。

请各位网友解答我的疑惑，不胜感激！！！

问题1：函数hhspectrum()返回的是否为归一化后的频率？为何？

函数instfreq中有如下注释
The result FNORMHAT lies between 0.0 and 0.5.

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问题2：函数toimage()的原理是什么？为什么以各个IMF分量的瞬时幅值和瞬时频率为输入得到的im是N*N的矩阵（N表示信号长度）？

[im,tt,ff] = TOIMAGE(A,f,t,splx,sply) transforms a spectrum made of 1D functions (e.g., output of "hhspectrum") in an 2D image

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问题3：老版MATLAB求取边际谱时为何乘以$\frac{1}{fs}$？

for k=1:size(E,1)
    bjp(k)=sum(E(k,:))*1/fs; 
end

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问题4：如何根据新版MATLAB求取边际谱？

[P,F,T,insf,inse]=hht(d(5:N_imf,:)',fs);

关于hht函数的解释，请参照MATLAB——Hilbert-Huang Transform （HHT）

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问题5：如何根据新版MATLAB绘制三维时频谱图？

hht(d(5:N_imf,:)',fs);

上述代码可以直接绘制二维希尔伯特时频谱图，横轴表示时间、纵轴表示频率，颜色表示能量
那么三维的时频谱图该如何利用[P,F,T,insf,inse]=hht(d(5:N_imf,:)’,fs);的输出结果进行绘制？

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下面首先展示老版MATLAB求取边际谱：

%% 获取源数据，signal1,signal2,signal3分别表示三种不同的信号类型
[date,time,signal1,signal2,signal3]=textread('BulgingData\aaa.txt','%s%s%f%f%f','headerlines',0);
timestring="05:25:00.00";
for i=1:length(date)
    if time(i)==timestring
        for j=1:2048*4
            globalsignal(j)=signal1(i+j-1);
        end
    end
end

globalsignal=globalsignal';

%% 定义采样频率、时间、采样点数
fs=25;
t = (0:length(globalsignal)-1)/fs;
N = length(globalsignal);

%% 对globalsignal进行EEMD分解
d=eemd(globalsignal,3,100);
N_imf=fix(log2(length(globalsignal)));
d=d';

%% 绘制分解结果
figure()
for i=1:length(d(:,1))
    subplot(length(d(:,1)),1,i)
    plot(t, d(i,:))
end

%% 边际谱求取过程
c=d(2:N_imf,:);
[m,n]=size(c);
% 对IMF分量求取瞬时频率与振幅：
% A：每个IMF的振幅向量; fa:每个IMF对应的瞬时频率(归一化后); t：时间序列号
[A,fa,tt]=hhspectrum(c);
% 将每个IMF信号合成求取Hilbert谱
% E：对应的振幅值; Cenf：每个网格对应的中心频率
% 时频图(用颜色表示第三维值的大小); 三维图（三维坐标系：时间，中心频率，振幅）        
[E,tt1]=toimage(A,fa,tt,length(tt)); % A,fa为固定参数, tt,length(tt)为可变参数

figure()
% 边际谱是hilbert谱对时间的积分，从积分的角度来讲，
% 对任意一阶频率把所有的时间上的幅值都加起来，反映这阶频率在所有时间上的幅值积累
for k=1:size(E,1)
    bjp(k)=sum(E(k,:))*1/fs; 
end
f=(1:N)/N*(fs/2);
plot(f(1:2000),bjp(1:2000));
xlabel('频率 / Hz');
ylabel('信号幅值');
ylim([0 80])
title('信号边际谱')%要求边际谱必须先对信号进行EMD分解

利用上述代码，可以求取信号的边际谱，代码中涉及的hhspectrum、toimage以及instfreq如下所示。

function [A,f,tt] = hhspectrum(x,t,l,aff)
% 
% [A,f,tt] = HHSPECTRUM(x,t,l,aff) computes the Hilbert-Huang spectrum
%
% inputs:
%   - x   : matrix with one signal per row
%   - t   : time instants
%   - l   : estimation parameter for instfreq (integer >=1 (1:default))
%   - aff : if 1, displays the computation evolution
%
% outputs:
%   - A   : instantaneous amplitudes 
%   - f   : instantaneous frequencies
%   - tt  : truncated time instants
%
% calls:
%   - hilbert  : computes the analytic signal
%   - instfreq : computes the instantaneous frequency
%   - disprog : displays the computation evolution
%
%Examples:
%
%s = randn(1,512);
%imf = emd(s);
%[A,f,tt] = hhspectrum(imf(1:end-1,:));
%
%s = randn(10,512);
%[A,f,tt] = hhspectrum(s,1:512,2,1);
%
% rem: need the Time-Frequency Toolbox (http://tftb.nongnu.org)
%
% See also
%  emd, toimage, disp_hhs
%
% G. Rilling, last modification 3.2007
% [email protected]
error(nargchk(1,4,nargin));

if nargin < 2
    
    t=1:size(x,2);
    
end

if nargin < 3
    
    l=1;
    
end

if nargin < 4
    
    aff = 0;
    
end

if min(size(x)) == 1
    if size(x,2) == 1
        x = x';
        if nargin < 2
            t = 1:size(x,2);
        end
    end
    Nmodes = 1;
else
    Nmodes = size(x,1);
end

lt=length(t);

tt=t((l+1):(lt-l));

for i=1:Nmodes
    
    an(i,:)=hilbert(x(i,:)')';
    f(i,:)=instfreq(an(i,:)',tt,l)';

    A=abs(an(:,l+1:end-l));
    
    if aff
        disprog(i,Nmodes,max(Nmodes,100))
    end
end
end

function [fnormhat,t]=instfreq(x,t,L,trace)
%INSTFREQ Instantaneous frequency estimation.
%	[FNORMHAT,T]=INSTFREQ(X,T,L,TRACE) computes the instantaneous
%	frequency of the analytic signal X at time instant(s) T, using the
%	trapezoidal integration rule.
%	The result FNORMHAT lies between 0.0 and 0.5.
%
%	X : Analytic signal to be analyzed.
%	T : Time instants	        (default : 2:length(X)-1).
%	L : If L=1, computes the (normalized) instantaneous frequency
%	    of the signal X defined as angle(X(T+1)*conj(X(T-1)) ;
%	    if L>1, computes a Maximum Likelihood estimation of the
%	    instantaneous frequency of the deterministic part of the signal
%	    blurried in a white gaussian noise.
%	    L must be an integer       	(default : 1).
%	TRACE : if nonzero, the progression of the algorithm is shown
%	                                (default : 0).
%	FNORMHAT : Output (normalized) instantaneous frequency.
%	T : Time instants.
%
%	Examples :
%	 x=fmsin(70,0.05,0.35,25); [instf,t]=instfreq(x); plot(t,instf)
%	 N=64; SNR=10.0; L=4; t=L+1:N-L; x=fmsin(N,0.05,0.35,40);
%	 sig=sigmerge(x,hilbert(randn(N,1)),SNR);
%	 plotifl(t,[instfreq(sig,t,L),instfreq(x,t)]); grid;
%	 title ('theoretical and estimated instantaneous frequencies');
%
%	See also  KAYTTH, SGRPDLAY.

%	F. Auger, March 1994, July 1995.
%	Copyright (c) 1996 by CNRS (France).
%
%	------------------- CONFIDENTIAL PROGRAM --------------------
%	This program can not be used without the authorization of its
%	author(s). For any comment or bug report, please send e-mail to
%	[email protected]

if (nargin == 0)
    error('At least one parameter required');
end
[xrow xcol] = size(x)
if (xcol~=1)
    error('X must have only one column');
end

if (nargin == 1)
    t=2:xrow-1; L=1; trace=0.0;
elseif (nargin == 2)
    L = 1; trace=0.0;
elseif (nargin == 3)
    trace=0.0;
end

if L<1,
    error('L must be >=1');
end
[trow,tcol] = size(t);
if (trow~=1),
    error('T must have only one row');
end;

if (L==1),
    if any(t==1)|any(t==xrow),
        error('T can not be equal to 1 neither to the last element of X');
    else
        fnormhat=0.5*(angle(-x(t+1).*conj(x(t-1)))+pi)/(2*pi);
    end;
else
    H=kaytth(L);
    if any(t<=L)|any(t+L>xrow),
        error('The relation L2*pi, tetapred=tetapred-(2*pi); end;
            iter = 1;
            while (diff > 1e-6)&(iter<50),
                M4bis=M4 .* exp(-j*2.0*tetapred);
                teta = H * (unwrap(angle(M4bis))+2.0*tetapred);
                while teta<0.0 , teta=(2*pi)+teta; end;
                while teta>2*pi, teta=teta-(2*pi); end;
                diff=abs(teta-tetapred);
                tetapred=teta; iter=iter+1;
            end;
            fnormhat(icol,1)=teta/(2*pi);
        end;
    end;
end
end

%TOIMAGE  transforms a spectrum made of 1D functions in an 2D image
%
% [im,tt,ff] = TOIMAGE(A,f,t,splx,sply) transforms a spectrum made
% of 1D functions (e.g., output of "hhspectrum") in an 2D image
%
% inputs :   - A    : amplitudes of modes (1 mode per row of A)
%            - f    : instantaneous frequencies
%            - t    : time instants
%            - splx : number of columns of the output im (time resolution).
%                     If different from length(t), works only for uniform
%                     sampling.
%            - sply : number of rows of the output im (frequency resolution).
% outputs :  - im   : 2D image of the spectrum
%            - tt   : time instants in the image
%            - ff   : centers of the frequency bins
%
% Examples : [im,tt,ff] = toimage(A,f);[im,tt] = toimage(A,f,t);[im,tt,ff] = toimage(A,f,sply);
%            [im,tt,ff] = toimage(A,f,splx,sply);[im,tt,ff] = toimage(A,f,t,splx,sply);
%
%
% See also
%  emd, hhspectrum, disp_hhs
%
% G. Rilling, last modification 3.2007
% [email protected]

function [im,tt,ff] = toimage(A,f,varargin)


DEFSPL = 400;

error(nargchk(2,5,nargin));

switch nargin
    case 2
        t = 1:size(A,2);
        sply = DEFSPL;
        splx = length(t);
    case 3
        if isscalar(varargin{1})
            t = 1:size(A,2);
            splx = length(t);
            sply = varargin{1};
        else
            t = varargin{1};
            splx = length(t);
            sply = DEFSPL;
        end
    case 4
        if isscalar(varargin{1})
            t = 1:size(A,2);
            sply = varargin{1};
            splx = varargin{2};
        else
            t = varargin{1};
            sply = varargin{2};
            splx = length(t);
        end
    case 5
        t = varargin{1};
        splx = varargin{2};
        sply = varargin{3};
end
if isvector(A)
    A = A(:)';
    f = f(:)';
end


if issparse(A) || ~isreal(A) || length(size(A)) > 2
    error('A argument must be a real matrix')
end
if issparse(f) || ~isreal(f) || length(size(f)) > 2
    error('f argument must be a real matrix')
end
if any(size(f)~=size(A))
    error('A and f matrices must have the same size')
end
if issparse(t) || ~isreal(t) || ~isvector(t) || length(t)~=size(A,2)
    error('t argument must be a vector and its length must be the number of columns in A and f inputs')
end
if ~isscalar(splx) || ~isreal(splx) || splx ~= floor(splx) || splx <= 0
    error('splx argument must be a positive integer')
end
if ~isscalar(sply) || ~isreal(sply) || sply ~= floor(sply) || sply <= 0
    error('splx argument must be a positive integer')
end

if any(diff(diff(t))) && splx ~= length(t)
    warning('toimage:nonuniformtimeinsants','When splx differs from length(t), the function only works for equally spaced time instants. You may consider reformating your data (using e.g. interpolation) before using toimage.')
end

f = min(f,0.5);
f = max(f,0);

indf = round(2*f*(sply-1)+1);
indt = repmat(round(linspace(1,length(t),splx)),size(A,1),1);
im = accumarray([indf(:),indt(:)],A(:),[sply,splx]);

indt = indt(1,:);
tt = t(indt);
ff = (0:sply-1)*0.5/sply+1/(4*sply);

end