S变换介绍（附代码）

1、S变换

        作为小波变换和短时傅里叶变换的继承和发展， S 变换采用高斯窗函数且窗宽与频率的倒数成正比，免去了窗函数的选择和改善了窗宽固定的缺陷，并且时频表示中各频率分量的相位谱与原始信号保持直接的联系，使其在 PQD 分析中可以采用更多的特征量，同时， S 变换提取的特征量对噪声不敏感。因此，在电能电能质量扰动、轴承故障诊断领域运用广泛。适用于非平稳信号的时频分析方法，其定义为：

            

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 2、S变换matlab代码

unction [st,t,f] = st(timeseries,minfreq,maxfreq,samplingrate,freqsamplingrate) 
% Returns the Stockwell Transform of the timeseries. 
% Code by Robert Glenn Stockwell. 
% DO NOT DISTRIBUTE 
% BETA TEST ONLY 
% Reference is "Localization of the Complex Spectrum: The S Transform" 
% from IEEE Transactions on Signal Processing, vol. 44., number 4, April 1996, pages 998-1001. 
% 
%-------Inputs Needed------------------------------------------------ 
%   
%   *****All frequencies in (cycles/(time unit))!****** 
%	"timeseries" - vector of data to be transformed 
%-------Optional Inputs ------------------------------------------------ 
% 
%"minfreq" is the minimum frequency in the ST result(Default=0) 
%"maxfreq" is the maximum frequency in the ST result (Default=Nyquist) 
%"samplingrate" is the time interval between samples (Default=1) 
%"freqsamplingrate" is the frequency-sampling interval you desire in the ST result (Default=1) 
%Passing a negative number will give the default ex.  [s,t,f] = st(data,-1,-1,2,2) 
%-------Outputs Returned------------------------------------------------ 
% 
% st     -a complex matrix containing the Stockwell transform.  
%			 The rows of STOutput are the frequencies and the  
%         columns are the time values ie each column is  
%         the "local spectrum" for that point in time 
%  t      - a vector containing the sampled times 
%  f      - a vector containing the sampled frequencies 
%--------Additional details----------------------- 
%   %  There are several parameters immediately below that 
%  the user may change. They are: 
%[verbose]    if true prints out informational messages throughout the function. 
%[removeedge] if true, removes a least squares fit parabola 
%                and puts a 5% hanning taper on the edges of the time series. 
%                This is usually a good idea. 
%[analytic_signal]  if the timeseries is real-valued 
%                      this takes the analytic signal and STs it. 
%                      This is almost always a good idea. 
%[factor]     the width factor of the localizing gaussian 
%                ie, a sinusoid of period 10 seconds has a  
%                gaussian window of width factor*10 seconds. 
%                I usually use factor=1, but sometimes factor = 3 
%                to get better frequency resolution. 
%   Copyright (c) by Bob Stockwell 
%   $Revision: 1.2 $  $Date: 1997/07/08  $ 
 
 
% This is the S transform wrapper that holds default values for the function. 
TRUE = 1; 
FALSE = 0; 
%%% DEFAULT PARAMETERS  [change these for your particular application] 
verbose = TRUE;           
removeedge= FALSE; 
analytic_signal =  FALSE; 
factor = 1; 
%%% END of DEFAULT PARAMETERS 
 
 
%%%START OF INPUT VARIABLE CHECK 
% First:  make sure it is a valid time_series  
%         If not, return the help message 
 
if verbose disp(' '),end  % i like a line left blank 
 
if nargin == 0  
   if verbose disp('No parameters inputted.'),end 
   st_help 
   t=0;,st=-1;,f=0; 
   return 
end 
 
% Change to column vector 
if size(timeseries,2) > size(timeseries,1) 
	timeseries=timeseries';	 
end 
 
% Make sure it is a 1-dimensional array 
if size(timeseries,2) > 1 
   error('Please enter a *vector* of data, not matrix') 
	return 
elseif (size(timeseries)==[1 1]) == 1 
	error('Please enter a *vector* of data, not a scalar') 
	return 
end 
 
% use defaults for input variables 
 
if nargin == 1 
   minfreq = 0; 
   maxfreq = fix(length(timeseries)/2); 
   samplingrate=1; 
   freqsamplingrate=1; 
elseif nargin==2 
   maxfreq = fix(length(timeseries)/2); 
   samplingrate=1; 
   freqsamplingrate=1; 
   [ minfreq,maxfreq,samplingrate,freqsamplingrate] =  check_input(minfreq,maxfreq,samplingrate,freqsamplingrate,verbose,timeseries); 
elseif nargin==3  
   samplingrate=1; 
   freqsamplingrate=1; 
   [ minfreq,maxfreq,samplingrate,freqsamplingrate] =  check_input(minfreq,maxfreq,samplingrate,freqsamplingrate,verbose,timeseries); 
elseif nargin==4    
   freqsamplingrate=1; 
   [ minfreq,maxfreq,samplingrate,freqsamplingrate] =  check_input(minfreq,maxfreq,samplingrate,freqsamplingrate,verbose,timeseries); 
elseif nargin == 5 
      [ minfreq,maxfreq,samplingrate,freqsamplingrate] =  check_input(minfreq,maxfreq,samplingrate,freqsamplingrate,verbose,timeseries); 
else       
   if verbose disp('Error in input arguments: using defaults'),end 
   minfreq = 0; 
   maxfreq = fix(length(timeseries)/2); 
   samplingrate=1; 
   freqsamplingrate=1; 
end 
if verbose  
   disp(sprintf('Minfreq = %d',minfreq)) 
   disp(sprintf('Maxfreq = %d',maxfreq)) 
   disp(sprintf('Sampling Rate (time   domain) = %d',samplingrate)) 
   disp(sprintf('Sampling Rate (freq.  domain) = %d',freqsamplingrate)) 
   disp(sprintf('The length of the timeseries is %d points',length(timeseries))) 
 
   disp(' ') 
end 
%END OF INPUT VARIABLE CHECK 
 
% If you want to "hardwire" minfreq & maxfreq & samplingrate & freqsamplingrate do it here 
 
% calculate the sampled time and frequency values from the two sampling rates 
t = (0:length(timeseries)-1)*samplingrate; 
spe_nelements =ceil((maxfreq - minfreq+1)/freqsamplingrate)   ; 
f = (minfreq + [0:spe_nelements-1]*freqsamplingrate)/(samplingrate*length(timeseries)); 
if verbose disp(sprintf('The number of frequency voices is %d',spe_nelements)),end 
 
 
% The actual S Transform function is here: 
st = strans(timeseries,minfreq,maxfreq,samplingrate,freqsamplingrate,verbose,removeedge,analytic_signal,factor);  
% this function is below, thus nicely encapsulated 
 
%WRITE switch statement on nargout 
% if 0 then plot amplitude spectrum 
if nargout==0  
   if verbose disp('Plotting pseudocolor image'),end 
   pcolor(t,f,abs(st)) 
end 
 
 
return 
 
 
%^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 
%^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 
%^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 
%^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 
%^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 
 
 
function st = strans(timeseries,minfreq,maxfreq,samplingrate,freqsamplingrate,verbose,removeedge,analytic_signal,factor);  
% Returns the Stockwell Transform, STOutput, of the time-series 
% Code by R.G. Stockwell. 
% Reference is "Localization of the Complex Spectrum: The S Transform" 
% from IEEE Transactions on Signal Processing, vol. 44., number 4, 
% April 1996, pages 998-1001. 
% 
%-------Inputs Returned------------------------------------------------ 
%         - are all taken care of in the wrapper function above 
% 
%-------Outputs Returned------------------------------------------------ 
% 
%	ST    -a complex matrix containing the Stockwell transform. 
%			 The rows of STOutput are the frequencies and the 
%			 columns are the time values 
% 
% 
%----------------------------------------------------------------------- 
 
% Compute the length of the data. 
n=length(timeseries); 
original = timeseries; 
if removeedge 
    if verbose disp('Removing trend with polynomial fit'),end 
 	 ind = [0:n-1]'; 
    r = polyfit(ind,timeseries,2); 
    fit = polyval(r,ind) ; 
	 timeseries = timeseries - fit; 
    if verbose disp('Removing edges with 5% hanning taper'),end 
    sh_len = floor(length(timeseries)/10); 
    wn = hanning(sh_len); 
    if(sh_len==0) 
       sh_len=length(timeseries); 
       wn = 1&[1:sh_len]; 
    end 
    % make sure wn is a column vector, because timeseries is 
   if size(wn,2) > size(wn,1) 
      wn=wn';	 
   end 
    
   timeseries(1:floor(sh_len/2),1) = timeseries(1:floor(sh_len/2),1).*wn(1:floor(sh_len/2),1); 
	timeseries(length(timeseries)-floor(sh_len/2):n,1) = timeseries(length(timeseries)-floor(sh_len/2):n,1).*wn(sh_len-floor(sh_len/2):sh_len,1); 
   
end 
 
% If vector is real, do the analytic signal  
 
if analytic_signal 
   if verbose disp('Calculating analytic signal (using Hilbert transform)'),end 
   % this version of the hilbert transform is different than hilbert.m 
   %  This is correct! 
   ts_spe = fft(real(timeseries)); 
   h = [1; 2*ones(fix((n-1)/2),1); ones(1-rem(n,2),1); zeros(fix((n-1)/2),1)]; 
   ts_spe(:) = ts_spe.*h(:); 
   timeseries = ifft(ts_spe); 
end   
 
% Compute FFT's 
tic;vector_fft=fft(timeseries);tim_est=toc; 
vector_fft=[vector_fft,vector_fft]; 
tim_est = tim_est*ceil((maxfreq - minfreq+1)/freqsamplingrate)   ; 
if verbose disp(sprintf('Estimated time is %f',tim_est)),end 
 
% Preallocate the STOutput matrix 
st=zeros(ceil((maxfreq - minfreq+1)/freqsamplingrate),n); 
% Compute the mean 
% Compute S-transform value for 1 ... ceil(n/2+1)-1 frequency points 
if verbose disp('Calculating S transform...'),end 
if minfreq == 0 
   st(1,:) = mean(timeseries)*(1&[1:1:n]); 
else 
  	st(1,:)=ifft(vector_fft(minfreq+1:minfreq+n).*g_window(n,minfreq,factor)); 
end 
 
%the actual calculation of the ST 
% Start loop to increment the frequency point 
for banana=freqsamplingrate:freqsamplingrate:(maxfreq-minfreq) 
   st(banana/freqsamplingrate+1,:)=ifft(vector_fft(minfreq+banana+1:minfreq+banana+n).*g_window(n,minfreq+banana,factor)); 
end   % a fruit loop!   aaaaa ha ha ha ha ha ha ha ha ha ha 
% End loop to increment the frequency point 
if verbose disp('Finished Calculation'),end 
 
%%% end strans function 
 
%------------------------------------------------------------------------ 
function gauss=g_window(length,freq,factor) 
 
% Function to compute the Gaussion window for  
% function Stransform. g_window is used by function 
% Stransform. Programmed by Eric Tittley 
% 
%-----Inputs Needed-------------------------- 
% 
%	length-the length of the Gaussian window 
% 
%	freq-the frequency at which to evaluate 
%		  the window. 
%	factor- the window-width factor 
% 
%-----Outputs Returned-------------------------- 
% 
%	gauss-The Gaussian window 
% 
 
vector(1,:)=[0:length-1]; 
vector(2,:)=[-length:-1]; 
vector=vector.^2;     
vector=vector*(-factor*2*pi^2/freq^2); 
% Compute the Gaussion window 
gauss=sum(exp(vector)); 
 
%----------------------------------------------------------------------- 
 
%^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^% 
function [ minfreq,maxfreq,samplingrate,freqsamplingrate] =  check_input(minfreq,maxfreq,samplingrate,freqsamplingrate,verbose,timeseries) 
% this checks numbers, and replaces them with defaults if invalid 
 
% if the parameters are passed as an array, put them into the appropriate variables 
s = size(minfreq); 
l = max(s); 
if l > 1   
   if verbose disp('Array of inputs accepted.'),end 
   temp=minfreq; 
   minfreq = temp(1);; 
   if l > 1  maxfreq = temp(2);,end; 
   if l > 2  samplingrate = temp(3);,end; 
   if l > 3  freqsamplingrate = temp(4);,end; 
   if l > 4   
      if verbose disp('Ignoring extra input parameters.'),end 
   end; 
 
end       
      
   if minfreq < 0 | minfreq > fix(length(timeseries)/2); 
      minfreq = 0; 
      if verbose disp('Minfreq < 0 or > Nyquist. Setting minfreq = 0.'),end 
   end 
   if maxfreq > length(timeseries)/2  | maxfreq < 0  
      maxfreq = fix(length(timeseries)/2); 
      if verbose disp(sprintf('Maxfreq < 0 or > Nyquist. Setting maxfreq = %d',maxfreq)),end 
   end 
      if minfreq > maxfreq  
      temporary = minfreq; 
      minfreq = maxfreq; 
      maxfreq = temporary; 
      clear temporary; 
      if verbose disp('Swapping maxfreq <=> minfreq.'),end 
   end 
   if samplingrate <0 
      samplingrate = abs(samplingrate); 
      if verbose disp('Samplingrate <0. Setting samplingrate to its absolute value.'),end 
   end 
   if freqsamplingrate < 0   % check 'what if freqsamplingrate > maxfreq - minfreq' case 
      freqsamplingrate = abs(freqsamplingrate); 
      if verbose disp('Frequency Samplingrate negative, taking absolute value'),end 
   end 
 
% bloody odd how you don't end a function 
 
%^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^% 
function st_help 
   disp(' ') 
	disp('st()  HELP COMMAND') 
	disp('st() returns  - 1 or an error message if it fails') 
	disp('USAGE::    [localspectra,timevector,freqvector] = st(timeseries)') 
  	disp('NOTE::   The function st() sets default parameters then calls the function strans()') 
   disp(' ')   
   disp('You can call strans() directly and pass the following parameters') 
   disp(' **** Warning!  These inputs are not checked if strans() is called directly!! ****') 
  	disp('USAGE::  localspectra = strans(timeseries,minfreq,maxfreq,samplingrate,freqsamplingrate,verbose,removeedge,analytic_signal,factor) ') 
      
   disp(' ') 
   disp('Default parameters (available in st.m)') 
	disp('VERBOSE          - prints out informational messages throughout the function.') 
	disp('REMOVEEDGE       - removes the edge with a 5% taper, and takes') 
   disp('FACTOR           -  the width factor of the localizing gaussian') 
   disp('                    ie, a sinusoid of period 10 seconds has a ') 
   disp('                    gaussian window of width factor*10 seconds.') 
   disp('                    I usually use factor=1, but sometimes factor = 3') 
   disp('                    to get better frequency resolution.') 
   disp(' ') 
   disp('Default input variables') 
   disp('MINFREQ           - the lowest frequency in the ST result(Default=0)') 
   disp('MAXFREQ           - the highest frequency in the ST result (Default=nyquist') 
   disp('SAMPLINGRATE      - the time interval between successive data points (Default = 1)') 
   disp('FREQSAMPLINGRATE  - the number of frequencies between samples in the ST results') 
	 
% end of st_help procedure    
 
 

 代码引用来源IEEE Transactions on Signal Processing, vol. 44., number 4, April 1996, pages 998-1001.