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pwelch.m

%% Copyright (C) 2006 Peter V. Lanspeary
%%
%% This program is free software; you can redistribute it and/or
%% modify it under the terms of the GNU General Public License
%% as published by the Free Software Foundation; either version 2,
%% or (at your option) any later version.
%%
%% This program is distributed in the hope that it will be useful,
%% but WITHOUT ANY WARRANTY; without even the implied warranty of
%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
%% GNU General Public License for more details.
%%
%% You should have received a copy of the GNU General Public License
%% along with this program; If not, see <http://www.gnu.org/licenses/>.

%% USAGE:
%%   [spectra,freq] = pwelch(x,window,overlap,Nfft,Fs,
%%                           range,plot_type,detrend,sloppy)
%%     Estimate power spectral density of data "x" by the Welch (1967)
%%     periodogram/FFT method.  All arguments except "x" are optional.
%%         The data is divided into segments.  If "window" is a vector, each
%%     segment has the same length as "window" and is multiplied by "window"
%%     before (optional) zero-padding and calculation of its periodogram. If
%%     "window" is a scalar, each segment has a length of "window" and a
%%     Hamming window is used.
%%         The spectral density is the mean of the periodograms, scaled so that
%%     area under the spectrum is the same as the mean square of the
%%     data.  This equivalence is supposed to be exact, but in practice there
%%     is a mismatch of up to 0.5% when comparing area under a periodogram
%%     with the mean square of the data.
%%
%%  [spectra,freq] = pwelch(x,y,window,overlap,Nfft,Fs,
%%                          range,plot_type,detrend,sloppy,results)
%%     Two-channel spectrum analyser.  Estimate power spectral density, cross-
%%     spectral density, transfer function and/or coherence functions of time-
%%     series input data "x" and output data "y" by the Welch (1967)
%%     periodogram/FFT method.
%%       pwelch treats the second argument as "y" if there is a control-string
%%     argument "cross", "trans", "coher" or "ypower"; "power" does not force
%%     the 2nd argument to be treated as "y".  All other arguments are
%%     optional.  All spectra are returned in matrix "spectra". 
%%
%%  [spectra,Pxx_ci,freq] = pwelch(x,window,overlap,Nfft,Fs,conf,
%%                                 range,plot_type,detrend,sloppy)
%%  [spectra,Pxx_ci,freq] = pwelch(x,y,window,overlap,Nfft,Fs,conf,
%%                                 range,plot_type,detrend,sloppy,results)
%%     Estimates confidence intervals for the spectral density.
%%     See Hint (7) below for compatibility options.  Confidence level "conf"
%%     is the 6th or 7th numeric argument.  If "results" control-string 
%%     arguments are used, one of them must be "power" when the "conf"
%%     argument is present; pwelch can estimate confidence intervals only for
%%     the power spectrum of the "x" data.  It does not know how to estimate
%%     confidence intervals of the cross-power spectrum, transfer function or
%%     coherence; if you can suggest a good method, please send a bug report.
%%
%% ARGUMENTS
%% All but the first argument are optional and may be empty, except that
%% the "results" argument may require the second argument to be "y".
%%
%% x           %% [non-empty vector] system-input time-series data
%% y           %% [non-empty vector] system-output time-series data
%%
%% window      %% [real vector] of window-function values between 0 and 1; the
%%             %%       data segment has the same length as the window.
%%             %%       Default window shape is Hamming.
%%             %% [integer scalar] length of each data segment.  The default
%%             %%       value is window=sqrt(length(x)) rounded up to the
%%             %%       nearest integer power of 2; see 'sloppy' argument.
%%
%% overlap     %% [real scalar] segment overlap expressed as a multiple of
%%             %%       window or segment length.   0 <= overlap < 1,
%%             %%       The default is overlap=0.5 .
%%
%% Nfft        %% [integer scalar] Length of FFT.  The default is the length
%%             %%       of the "window" vector or has the same value as the
%%             %%       scalar "window" argument.  If Nfft is larger than the
%%             %%       segment length, "seg_len", the data segment is padded
%%             %%       with "Nfft-seg_len" zeros.  The default is no padding.
%%             %%       Nfft values smaller than the length of the data
%%             %%       segment (or window) are ignored silently.
%%
%% Fs          %% [real scalar] sampling frequency (Hertz); default=1.0
%%
%% conf        %% [real scalar] confidence level between 0 and 1.  Confidence
%%             %%       intervals of the spectral density are estimated from
%%             %%       scatter in the periodograms and are returned as Pxx_ci.
%%             %%       Pxx_ci(:,1) is the lower bound of the confidence
%%             %%       interval and Pxx_ci(:,2) is the upper bound.  If there
%%             %%       are three return values, or conf is an empty matrix,
%%             %%       confidence intervals are calculated for conf=0.95 .
%%             %%       If conf is zero or is not given, confidence intervals
%%             %%       are not calculated. Confidence intervals can be 
%%             %%       obtained only for the power spectral density of x;
%%             %%       nothing else.
%%
%% CONTROL-STRING ARGUMENTS -- each of these arguments is a character string.
%%   Control-string arguments must be after the other arguments but can be in
%%   any order.
%%  
%% range     %% 'half',  'onesided' : frequency range of the spectrum is
%%           %%       zero up to but not including Fs/2.  Power from
%%           %%       negative frequencies is added to the positive side of
%%           %%       the spectrum, but not at zero or Nyquist (Fs/2)
%%           %%       frequencies.  This keeps power equal in time and
%%           %%       spectral domains.  See reference [2].
%%           %% 'whole', 'twosided' : frequency range of the spectrum is
%%           %%       -Fs/2 to Fs/2, with negative frequencies
%%           %%       stored in "wrap around" order after the positive
%%           %%       frequencies; e.g. frequencies for a 10-point 'twosided'
%%           %%       spectrum are 0 0.1 0.2 0.3 0.4 0.5 -0.4 -0.3 -0.2 -0.1
%%           %% 'shift', 'centerdc' : same as 'whole' but with the first half
%%           %%       of the spectrum swapped with second half to put the
%%           %%       zero-frequency value in the middle. (See "help
%%           %%       fftshift".
%%           %% If data (x and y) are real, the default range is 'half',
%%           %% otherwise default range is 'whole'.
%%
%% plot_type %% 'plot', 'semilogx', 'semilogy', 'loglog', 'squared' or 'db':
%%           %% specifies the type of plot.  The default is 'plot', which
%%           %% means linear-linear axes. 'squared' is the same as 'plot'.
%%           %% 'dB' plots "10*log10(psd)".  This argument is ignored and a
%%           %% spectrum is not plotted if the caller requires a returned
%%           %% value.
%%
%% detrend   %% 'no-strip', 'none' -- do NOT remove mean value from the data
%%           %% 'short', 'mean' -- remove the mean value of each segment from
%%           %%                    each segment of the data.
%%           %% 'linear',       -- remove linear trend from each segment of
%%           %%                    the data.
%%           %% 'long-mean'     -- remove the mean value from the data before
%%           %%              splitting it into segments.  This is the default.
%%
%%   sloppy  %% 'sloppy': FFT length is rounded up to the nearest integer
%%           %%       power of 2 by zero padding.  FFT length is adjusted
%%           %%       after addition of padding by explicit Nfft argument.
%%           %%       The default is to use exactly the FFT and window/
%%           %%       segment lengths specified in argument list.
%%
%%   results %% specifies what results to return (in the order specified
%%           %%   and as many as desired).
%%           %% 'power' calculate power spectral density of "x"
%%           %% 'cross' calculate cross spectral density of "x" and "y"
%%           %% 'trans' calculate transfer function of a system with
%%           %%         input "x" and output "y"
%%           %% 'coher' calculate coherence function of "x" and "y"
%%           %% 'ypower' calculate power spectral density of "y"
%%           %%  The default is 'power', with argument "y" omitted. 
%%
%% RETURNED VALUES:
%%   If return values are not required by the caller, the results are
%%     plotted and nothing is returned.
%%
%%   spectra %% [real-or-complex matrix] columns of the matrix contain results
%%           %%        in the same order as specified by "results" arguments.
%%           %%        Each column contains one of the result vectors.
%%
%%   Pxx_ci  %% [real matrix] estimate of confidence interval for power
%%           %%        spectral density of x.  First column is the lower
%%           %%        bound.  Second column is the upper bound.
%%
%%   freq    %% [real column vector] frequency values 
%%
%% HINTS
%% 1) EMPTY ARGS:
%%    if you don't want to use an optional argument you can leave it empty
%%    by writing its value as [].
%% 2) FOR BEGINNERS:
%%    The profusion of arguments may make pwelch difficult to use, and an
%%    unskilled user can easily produce a meaningless result or can easily
%%    mis-interpret the result.  With real data "x" and sampling frequency
%%    "Fs", the easiest and best way for a beginner to use pwelch is
%%    probably "pwelch(x,[],[],[],Fs)".  Use the "window" argument to
%%    control the length of the spectrum vector.  For real data and integer
%%    scalar M, "pwelch(x,2*M,[],[],Fs)" gives an M+1 point spectrum.
%%    Run "demo pwelch" (octave only).
%% 3) WINDOWING FUNCTIONS:
%%    Without a window function, sharp spectral peaks can have strong
%%    sidelobes because the FFT of a data in a segment is in effect convolved
%%    with a rectangular window.  A window function which tapers off
%%    (gradually) at the ends produces much weaker sidelobes in the FFT.
%%    Hann (hanning), hamming, bartlett, blackman, flattopwin etc are
%%    available as separate Matlab/sigproc or Octave functions.  The sidelobes
%%    of the Hann window have a roll-off rate of 60dB/decade of frequency.
%%    The first sidelobe of the Hamming window is suppressed and is about 12dB
%%    lower than the first Hann sidelobe, but the roll-off rate is only
%%    20dB/decade.  You can inspect the FFT of a Hann window by plotting
%%    "abs(fft(postpad(hanning(256),4096,0)))".
%%    The default window is Hamming.
%% 4) ZERO PADDING:
%%    Zero-padding reduces the frequency step in the
%%    spectrum, and produces an artificially smoothed spectrum.  For example,
%%    "Nfft=2*length(window)" gives twice as many frequency values, but
%%    adjacent PSD (power spectral density) values are not independent;
%%    adjacent PSD values are independent if "Nfft=length(window)", which is
%%    the default value of Nfft.
%% 5) REMOVING MEAN FROM SIGNAL: 
%%    If the mean is not removed from the signal there is a large spectral
%%    peak at zero frequency and the sidelobes of this peak are likely to
%%    swamp the rest of the spectrum.  For this reason, the default behaviour
%%    is to remove the mean.  However, the matlab pwelch does not do this.
%% 6) WARNING ON CONFIDENCE INTERVALS
%%    Confidence intervals are obtained by measuring the sample variance of
%%    the periodograms and assuming that the periodograms have a Gaussian
%%    probability distribution.  This assumption is not accurate.  If, for
%%    example, the data (x) is Gaussian, the periodogram has a Rayleigh
%%    distribution.  However, the confidence intervals may still be  useful.
%%    
%% 7) COMPATIBILITY WITH Matlab R11, R12, etc
%%    When used without the second data (y) argument, arguments are compatible
%%    with the pwelch of Matlab R12, R13, R14, 2006a and 2006b except that
%%     1) overlap is expressed as a multiple of window length ---
%%        effect of overlap scales with window length
%%     2) default values of length(window), Nfft and Fs are more sensible, and
%%     3) Goertzel algorithm is not available so Nfft cannot be an array of
%%        frequencies as in Matlab 2006b.
%%    Pwelch has four persistent Matlab-compatibility levels.  Calling pwelch
%%    with an empty first argument sets the order of arguments and defaults
%%    specified above in the USAGE and ARGUMENTS section of this documentation.
%%          prev_compat=pwelch([]);
%%          [Pxx,f]=pwelch(x,window,overlap,Nfft,Fs,conf,...);
%%    Calling pwelch with a single string argument (as described below) gives
%%    compatibility with Matlab R11 or R12, or the R14 spectrum.welch
%%    defaults.  The returned value is the PREVIOUS compatibility string.
%%
%%    Matlab R11:  For compatibility with the Matlab R11 pwelch:
%%          prev_compat=pwelch('R11-');
%%          [Pxx,f]=pwelch(x,Nfft,Fs,window,overlap,conf,range,units);
%%          %% units of overlap are "number of samples"
%%          %% defaults: Nfft=min(length(x),256), Fs=2*pi, length(window)=Nfft,
%%          %%           window=Hanning, do not detrend,
%%          %% N.B.  "Sloppy" is not available.
%%
%%    Matlab R12:  For compatibility with Matlab R12 to 2006a pwelch:
%%          prev_compat=pwelch('R12+');
%%          [Pxx,f]=pwelch(x,window,overlap,nfft,Fs,...);
%%          %% units of overlap are "number of samples"
%%          %% defaults: length(window)==length(x)/8, window=Hamming,
%%          %%           Nfft=max(256,NextPow2), Fs=2*pi, do not detrend
%%          %% NextPow2 is the next power of 2 greater than or equal to the
%%          %% window length. "Sloppy", "conf" are not available.  Default
%%          %% window length gives very poor amplitude resolution.
%%
%%    To adopt defaults of the Matlab R14 "spectrum.welch" spectrum object
%%    associated "psd" method.
%%          prev_compat=pwelch('psd');
%%          [Pxx,f] = pwelch(x,window,overlap,Nfft,Fs,conf,...);
%%          %% overlap is expressed as a percentage of window length,
%%          %% defaults: length(window)==64, Nfft=max(256,NextPow2), Fs=2*pi
%%          %%           do not detrend
%%          %% NextPow2 is the next power of 2 greater than or equal to the
%%          %% window length. "Sloppy" is not available.
%%          %% Default window length gives coarse frequency resolution.
%%
%%
%% REFERENCES
%%  [1] Peter D. Welch (June 1967): 
%%   "The use of fast Fourier transform for the estimation of power spectra:
%%   a method based on time averaging over short, modified periodograms."
%%   IEEE Transactions on Audio Electroacoustics, Vol AU-15(6), pp 70-73
%%
%%  [2] William H. Press and Saul A. Teukolsky and William T. Vetterling and
%%               Brian P. Flannery",
%%   "Numerical recipes in C, The art of scientific computing", 2nd edition,
%%      Cambridge University Press, 2002 --- Section 13.7.
%%  [3] Paul Kienzle (1999-2001): "pwelch", http://octave.sourceforge.net/


function [varargout] = pwelch(x,varargin)
  %%
  %% COMPATIBILITY LEVEL
  %% Argument positions and defaults depend on compatibility level selected
  %% by calling pwelch without arguments or with a single string argument.
  %%   native:      compatib=1; prev_compat=pwelch(); prev_compat=pwelch([]);
  %%   matlab R11:  compatib=2; prev_compat=pwelch('R11-');
  %%   matlab R12:  compatib=3; prev_compat=pwelch('R12+');
  %%   spectrum.welch defaults:  compatib=4; prev_compat=pwelch('psd');
  %% In each case, the returned value is the PREVIOUS compatibility string.
  %%
compat_str = {; 'R11-'; 'R12+'; 'psd'};
persistent compatib;
if ( isempty(compatib) || compatib<=0 || compatib>4 )
  %% legal values are 1, 2, 3, 4
  compatib = 1;
end
if ( nargin <= 0 )
  error( 'pwelch: Need at least 1 arg. Use "help pwelch".' );
elseif ( nargin==1 && (ischar(x) || isempty(x)) )
  varargout{1} = compat_str{compatib};
  if ( isempty(x) ) % native
    compatib = 1;
  elseif ( strcmp(x,'R11-') )
    compatib = 2;
  elseif ( strcmp(x,'R12+') )
    compatib = 3;
  elseif ( strcmp(x,'psd') )
    compatib = 4;
  else
    error( 'pwelch: compatibility arg must be empty, R11-, R12+ or psd' );
  end
  %% return
%%
%% Check fixed argument
elseif ( isempty(x) || ~isvector(x) )
  error( 'pwelch: arg 1 (x) must be vector.' );
else
  %%  force x to be COLUMN vector
  if ( size(x,1)==1 )
    x=x(:);
  end
  %%
  %% Look through all args to check if  cross PSD, transfer function or
  %% coherence is required.  If yes, the second arg is data vector "y".
  arg2_is_y = 0;
  x_len = length(x);
  nvarargin = length(varargin);
  for iarg=1:nvarargin
    arg = varargin{iarg};
    if ( ~isempty(arg) && ischar(arg) && ...
         ( strcmp(arg,'cross') || strcmp(arg,'trans') || ...
           strcmp(arg,'coher') || strcmp(arg,'ypower') ))
      %% OK. Need "y". Grab it from 2nd arg.
      arg = varargin{1};
      if ( nargin<2 || isempty(arg) || ~isvector(arg) || length(arg)~=x_len )
        error( 'pwelch: arg 2 (y) must be vector, same length as x.' );
      end
      %% force  COLUMN vector
      y = varargin{1}(:);
      arg2_is_y = 1;
      break;
    end
  end
  %%
  %% COMPATIBILITY
  %% To select default argument values, "compatib" is used as an array index.
  %% Index values are   1=native,  2=R11,  3=R12,  4=spectrum.welch
  %%
  %%  argument positions:
  %%  arg_posn = varargin index of window, overlap, Nfft, Fs and conf
  %%             args respectively, a value of zero ==>> arg does not exist
  arg_posn = ; %% spectrum.welch defaults
  arg_posn  = arg_posn(compatib,:) + arg2_is_y;
  %%
  %%  SPECIFY SOME DEFAULT VALUES for (not all) optional arguments
  %%  Use compatib as array index.
  %%  Fs = sampling frequency
  Fs        = [ 1.0 2*pi 2*pi 2*pi ];
  Fs        = Fs(compatib);            
  %%  plot_type: 1='plot'|'squared'; 5='db'|'dB'
  plot_type = ;
  plot_type = plot_type(compatib);
  %%  rm_mean: 3='long-mean'; 0='no-strip'|'none'
  rm_mean   = ;
  rm_mean   = rm_mean(compatib);
  %% use max_overlap=x_len-1 because seg_len is not available yet
  %% units of overlap are different for each version:
  %%    fraction, samples, or percent
  max_overlap = [ 0.95 x_len-1 x_len-1 95];
  max_overlap = max_overlap(compatib);
  %% default confidence interval
  %%  if there are more than 2 return values and if there is a "conf" arg
  conf      = 0.95 * (nargout>2) * (arg_posn(5)>0);
  %%
  is_win    = 0;    % =0 means valid window arg is not provided yet
  Nfft      = ;   % default depends on segment length
  overlap   = ;   % WARNING: units can be #samples, fraction or percentage
  range     = ~isreal(x) || ( arg2_is_y && ~isreal(y) );
  is_sloppy = 0;
  n_results = 0;
  do_power  = 0;   
  do_cross  = 0;
  do_trans  = 0;
  do_coher  = 0;
  do_ypower = 0;
%%
%%  DECODE AND CHECK OPTIONAL ARGUMENTS
  end_numeric_args = 0;
  for iarg = 1+arg2_is_y:nvarargin
    arg = varargin{iarg};
    if ( ischar(arg) )
      %% first string arg ==> no more numeric args
      %% non-string args cannot follow a string arg
      end_numeric_args = 1;
      %%
      %% decode control-string arguments
      if ( strcmp(arg,'sloppy') )
        is_sloppy = ~is_win || is_win==1;
      elseif ( strcmp(arg,'plot') || strcmp(arg,'squared') )
        plot_type = 1;
      elseif ( strcmp(arg,'semilogx') )
        plot_type = 2;
      elseif ( strcmp(arg,'semilogy') )
        plot_type = 3;
      elseif ( strcmp(arg,'loglog') )
        plot_type = 4;
      elseif ( strcmp(arg,'db') || strcmp(arg,'dB') )
        plot_type = 5;
      elseif ( strcmp(arg,'half') || strcmp(arg,'onesided') )
        range = 0;
      elseif ( strcmp(arg,'whole') || strcmp(arg,'twosided') )
        range = 1;
      elseif ( strcmp(arg,'shift') || strcmp(arg,'centerdc') )
        range = 2;
      elseif ( strcmp(arg,'long-mean') )
        rm_mean = 3;
      elseif ( strcmp(arg,'linear') )
        rm_mean = 2;
      elseif ( strcmp(arg,'short') || strcmp(arg,'mean') )
        rm_mean = 1;
      elseif ( strcmp(arg,'no-strip') || strcmp(arg,'none') )
        rm_mean = 0;
      elseif ( strcmp(arg, 'power' ) )
        if ( ~do_power )
          n_results = n_results+1;
          do_power = n_results;
        end
      elseif ( strcmp(arg, 'cross' ) )
        if ( ~do_cross )
          n_results = n_results+1;
          do_cross = n_results;
        end
      elseif ( strcmp(arg, 'trans' ) )
        if ( ~do_trans )
          n_results = n_results+1;
          do_trans = n_results;
        end
      elseif ( strcmp(arg, 'coher' ) )
        if ( ~do_coher )
          n_results = n_results+1;
          do_coher = n_results;
        end
      elseif ( strcmp(arg, 'ypower' ) )
        if ( ~do_ypower )
          n_results = n_results+1;
          do_ypower = n_results;
        end
      else
        error( 'pwelch: string arg %d illegal value: %s', iarg+1, arg ); 
      end
      %% end of processing string args
      %%
    elseif ( end_numeric_args )
      if ( ~isempty(arg) )
        %% found non-string arg after a string arg ... oops
        error( 'pwelch: control arg must be string' );
      end
    %%
    %% first 4 optional arguments are numeric -- in fixed order
    %%
    %% deal with "Fs" and "conf" first because empty arg is a special default
    %% -- "Fs" arg -- sampling frequency
    elseif ( iarg == arg_posn(4) )
      if ( isempty(arg) )
        Fs = 1;
      elseif ( ~isscalar(arg) || ~isreal(arg) || arg<0 )
        error( 'pwelch: arg %d (Fs) must be real scalar >0', iarg+1 );
      else
        Fs = arg;
      end
    %%
    %%  -- "conf" arg -- confidence level
    %%    guard against the "it cannot happen" iarg==0
    elseif ( arg_posn(5) && iarg == arg_posn(5) )
      if ( isempty(arg) )
        conf = 0.95;
      elseif ( ~isscalar(arg) || ~isreal(arg) || arg < 0.0 || arg >= 1.0 )
        error( 'pwelch: arg %d (conf) must be real scalar, >=0, <1',iarg+1 );
      else
        conf = arg;
      end
    %%
    %% skip all empty args from this point onward
    elseif ( isempty(arg) )
      1;
    %%
    %%  -- "window" arg -- window function
    elseif ( iarg == arg_posn(1) )
      if ( isscalar(arg) )
        is_win = 1;
      elseif ( isvector(arg) )
        is_win = length(arg);
        if ( size(arg,2)>1 )  %% vector must be COLUMN vector
          arg = arg(:);
        end
      else 
        is_win = 0;
      end
      if ( ~is_win )
        error( 'pwelch: arg %d (window) must be scalar or vector', iarg+1 );
      elseif ( is_win==1 && ( ~isreal(arg) || fix(arg)~=arg || arg<=3 ) )
        error( 'pwelch: arg %d (window) must be integer >3', iarg+1 );
      elseif ( is_win>1 && ( ~isreal(arg) || any(arg<0) ) )
        error( 'pwelch: arg %d (window) vector must be real and >=0',iarg+1);
      end
      window = arg;
      is_sloppy = 0;
    %%
    %% -- "overlap" arg -- segment overlap
    elseif ( iarg == arg_posn(2) )
      if (~isscalar(arg) || ~isreal(arg) || arg<0 || arg>max_overlap )
        error( 'pwelch: arg %d (overlap) must be real from 0 to %f', ...
               iarg+1, max_overlap );
      end
      overlap = arg;
    %%
    %% -- "Nfft" arg -- FFT length
    elseif ( iarg == arg_posn(3) )
      if ( ~isscalar(arg) || ~isreal(arg) || fix(arg)~=arg || arg<0 )
        error( 'pwelch: arg %d (Nfft) must be integer >=0', iarg+1 );
      end
      Nfft = arg;
    %%
    else
      error( 'pwelch: arg %d  must be string', iarg+1 );
    end
  end
  if ( conf>0 && (n_results && ~do_power ) )
   error('pwelch: can give confidence interval for x power spectrum only' );
    end
  %%
  %% end DECODE AND CHECK OPTIONAL ARGUMENTS.
  %%
  %% SETUP REMAINING PARAMETERS
  %% default action is to calculate power spectrum only
  if ( ~n_results )
    n_results = 1;
    do_power = 1;
  end
  need_Pxx = do_power || do_trans || do_coher;
  need_Pxy = do_cross || do_trans || do_coher;
  need_Pyy = do_coher || do_ypower;
  log_two = log(2);
  nearly_one = 0.99999999999;
  %%
  %% compatibility-options
  %% provides exact compatibility with Matlab R11 or R12
  %%
  %% Matlab R11 compatibility
  if ( compatib==2 )
    if ( isempty(Nfft) )
      Nfft = min( 256, x_len );
    end
    if ( is_win > 1 )
      seg_len = min( length(window), Nfft );
      window = window(1:seg_len);
    else
      if ( is_win )
        %% window arg is scalar
        seg_len = window;
      else
        seg_len = Nfft;
      end
      %% make Hann window (don't depend on sigproc)
      xx = seg_len - 1;
      window = 0.5 - 0.5 * cos( (2*pi/xx)*[0:xx].' );
    end
  %%
  %% Matlab R12 compatibility
  elseif ( compatib==3 )
    if ( is_win > 1 )
      %% window arg provides window function
      seg_len = length(window);
    else
      %% window arg does not provide window function; use Hamming
      if ( is_win )
        %% window arg is scalar
        seg_len = window;
      else
        %% window arg not available; use R12 default, 8 windows
        %% ignore overlap arg; use overlap=50% -- only choice that makes sense
        %% this is the magic formula for 8 segments with 50% overlap
        seg_len = fix( (x_len-3)*2/9 );
      end
      %% make Hamming window (don't depend on sigproc)
      xx = seg_len - 1;
      window = 0.54 - 0.46 * cos( (2*pi/xx)*[0:xx].' );
      end
    if ( isempty(Nfft) )
      Nfft = max( 256, 2^ceil(log(seg_len)*nearly_one/log_two) );
    end
  %%
  %% Matlab R14 psd(spectrum.welch) defaults
  elseif ( compatib==4 )
    if ( is_win > 1 )
      %% window arg provides window function
      seg_len = length(window);
    else
      %% window arg does not provide window function; use Hamming
      if ( is_win )
        %% window arg is scalar
        seg_len = window;
      else
        %% window arg not available; use default seg_len = 64
        seg_len = 64;
      end
      %% make Hamming window (don't depend on sigproc)
      xx = seg_len - 1;
      window = 0.54 - 0.46 * cos( (2*pi/xx)*[0:xx].' );
      end
    %% Now we know segment length,
    %% so we can set default overlap as number of samples
    if ( ~isempty(overlap) )
      overlap = fix(seg_len * overlap / 100 );
      end
    if ( isempty(Nfft) )
      Nfft = max( 256, 2^ceil(log(seg_len)*nearly_one/log_two) );
    end
  %%
  %% default compatibility level
  else %if ( compatib==1 )
    %% calculate/adjust segment length, window function
    if ( is_win > 1 )
      %% window arg provides window function
      seg_len = length(window);
    else
      %% window arg does not provide window function; use Hamming
      if ( is_win )       %% window arg is scalar
        seg_len = window;
      else
        %% window arg not available; use default length:
        %% = sqrt(length(x)) rounded up to nearest integer power of 2
        if ( isempty(overlap) )
          overlap=0.5;
        end
        seg_len = 2 ^ ceil( log(sqrt(x_len/(1-overlap)))*nearly_one/log_two );
      end
      %% make Hamming window (don't depend on sigproc)
      xx = seg_len - 1;
      window = 0.54 - 0.46 * cos( (2*pi/xx)*[0:xx].' );
      end
    %% Now we know segment length,
    %% so we can set default overlap as number of samples
    if ( ~isempty(overlap) )
      overlap = fix(seg_len * overlap);
      end
    %%
    %% calculate FFT length
    if ( isempty(Nfft) )
      Nfft = seg_len;
    end
    if ( is_sloppy )
      Nfft = 2 ^ ceil( log(Nfft) * nearly_one / log_two );
    end
  end
  %% end of compatibility options
  %%
  %% minimum FFT length is seg_len
  Nfft = max( Nfft, seg_len );
  %% Mean square of window is required for normalising PSD amplitude.
  win_meansq = (window.' * window) / seg_len;
  %%
  %% Set default or check overlap.
  if ( isempty(overlap) )
    overlap = fix(seg_len /2);
  elseif ( overlap >= seg_len )
    error( 'pwelch: arg (overlap=%d) too big. Must be <length(window)=%d',...
           overlap, seg_len );
  end
  %%
  %% Pad data with zeros if shorter than segment. This should not happen.
  if ( x_len < seg_len )
    x = [x; zeros(seg_len-x_len,1)];
    if ( arg2_is_y )
      y = [y; zeros(seg_len-x_len,1)];
      end
    x_len = seg_len;
    end
  %% end SETUP REMAINING PARAMETERS
  %%
  %% 
  %% MAIN CALCULATIONS
  %% Remove mean from the data
  if ( rm_mean == 3 )
    n_ffts = max( 0, fix( (x_len-seg_len)/(seg_len-overlap) ) ) + 1;
    x_len  = min( x_len, (seg_len-overlap)*(n_ffts-1)+seg_len );
    if ( need_Pxx || need_Pxy )
      x = x - sum( x(1:x_len) ) / x_len;
    end
    if ( arg2_is_y || need_Pxy)
      y = y - sum( y(1:x_len) ) / x_len;
    end
  end
  %%
  %% Calculate and accumulate periodograms
  %%   xx and yy are padded data segments
  %%   Pxx, Pyy, Pyy are periodogram sums, Vxx is for confidence interval
  xx = zeros(Nfft,1);
  yy = xx;
  Pxx = xx;
  Pxy = xx;
  Pyy = xx;
  if ( conf>0 )
    Vxx = xx;
  else
    Vxx = [];
  end
  n_ffts = 0;
  for start_seg = [1:seg_len-overlap:x_len-seg_len+1]
    end_seg = start_seg+seg_len-1;
    %% Don't truncate/remove the zero padding in xx and yy
    if ( need_Pxx || need_Pxy )
      if ( rm_mean==1 ) % remove mean from segment
        xx(1:seg_len) = window .* ( ...
          x(start_seg:end_seg) - sum(x(start_seg:end_seg)) / seg_len);
      elseif ( rm_mean == 2 ) % remove linear trend from segment
        xx(1:seg_len) = window .* detrend( x(start_seg:end_seg) );
      else % rm_mean==0 or 3
        xx(1:seg_len) = window .* x(start_seg:end_seg);
      end
      fft_x = fft(xx);
    end
    if ( need_Pxy || need_Pyy )
      if ( rm_mean==1 ) % remove mean from segment
        yy(1:seg_len) = window .* ( ...
          y(start_seg:end_seg) - sum(y(start_seg:end_seg)) / seg_len);
      elseif ( rm_mean == 2 ) % remove linear trend from segment
        yy(1:seg_len) = window .* detrend( y(start_seg:end_seg) );
      else % rm_mean==0 or 3
        yy(1:seg_len) = window .* y(start_seg:end_seg);
      end
      fft_y = fft(yy);
    end
    if ( need_Pxx )
      %% force Pxx to be real; pgram = periodogram
      pgram = real(fft_x .* conj(fft_x));
      Pxx = Pxx + pgram;
      %% sum of squared periodograms is required for confidence interval
      if ( conf>0 )
        Vxx = Vxx + pgram .^2;
        end
    end
    if ( need_Pxy )
      %% Pxy (cross power spectrum) is complex. Do not force to be real.
      Pxy = Pxy + fft_y .* conj(fft_x);
    end
    if ( need_Pyy )
      %% force Pyy to be real
      Pyy = Pyy + real(fft_y .* conj(fft_y));
    end
    n_ffts = n_ffts +1;
  end
  %%
  %% Calculate confidence interval
  %%    -- incorrectly assumes that the periodogram has Gaussian probability
  %%       distribution (actually, it has a single-sided (e.g. exponential)
  %%       distribution.
  %% Sample variance of periodograms is (Vxx-Pxx.^2/n_ffts)/(n_ffts-1).
  %%    This method of calculating variance is more susceptible to round-off
  %%  error, but is quicker, and for double-precision arithmetic and the
  %%  inherently noisy periodogram (variance==mean^2), it should be OK.
  if ( conf>0 && need_Pxx )
    if ( n_ffts<2 )
      Vxx = zeros(Nfft,1);
    else
      %% Should use student distribution here (for unknown variance), but tinv
      %% is not a core Matlab function (is in statistics toolbox. Grrr)
      Vxx = (erfinv(conf)*sqrt(2*n_ffts/(n_ffts-1))) * sqrt(Vxx-Pxx.^2/n_ffts);
    end
  end
  %%
  %% Convert two-sided spectra to one-sided spectra (if range == 0).
  %% For one-sided spectra, contributions from negative frequencies are added 
  %% to the positive side of the spectrum -- but not at zero or Nyquist
  %% (half sampling) frequencies.  This keeps power equal in time and spectral
  %% domains, as required by Parseval theorem.
  %%
  if ( range == 0 )
    if ( ~ rem(Nfft,2) )    %% one-sided, Nfft is even
      psd_len = Nfft/2+1;
      if ( need_Pxx )
        Pxx = Pxx(1:psd_len) + [0; Pxx(Nfft:-1:psd_len+1); 0];
        if ( conf>0 )
          Vxx = Vxx(1:psd_len) + [0; Vxx(Nfft:-1:psd_len+1); 0];
        end
      end
      if ( need_Pxy )
        Pxy = Pxy(1:psd_len) + conj([0; Pxy(Nfft:-1:psd_len+1); 0]);
      end
      if ( need_Pyy )
        Pyy = Pyy(1:psd_len) + [0; Pyy(Nfft:-1:psd_len+1); 0];
      end
    else                    %% one-sided, Nfft is odd
      psd_len = (Nfft+1)/2;
      if ( need_Pxx )
        Pxx = Pxx(1:psd_len) + [0; Pxx(Nfft:-1:psd_len+1)];
        if ( conf>0 )
          Vxx = Vxx(1:psd_len) + [0; Vxx(Nfft:-1:psd_len+1)];
        end
      end
      if ( need_Pxy )
        Pxy = Pxy(1:psd_len) + conj([0; Pxy(Nfft:-1:psd_len+1)]);
      end
      if ( need_Pyy )
        Pyy = Pyy(1:psd_len) + [0; Pyy(Nfft:-1:psd_len+1)];
      end
    end
  else                      %% two-sided (and shifted)
    psd_len = Nfft;
  end
  %% end MAIN CALCULATIONS
  %%
  %% SCALING AND OUTPUT
  %% Put all results in matrix, one row per spectrum
  %%   Pxx, Pxy, Pyy are sums of periodograms, so "n_ffts"
  %%   in the scale factor converts them into averages
  spectra    = zeros(psd_len,n_results);
  spect_type = zeros(n_results,1);
  scale = n_ffts * seg_len * Fs * win_meansq;
  if ( do_power )
    spectra(:,do_power) = Pxx / scale;
    spect_type(do_power) = 1;
    if ( conf>0 )
      Vxx = [Pxx-Vxx Pxx+Vxx]/scale;
    end
  end
  if ( do_cross )
    spectra(:,do_cross) = Pxy / scale;
    spect_type(do_cross) = 2;
  end
  if ( do_trans )
    spectra(:,do_trans) = Pxy ./ Pxx;
    spect_type(do_trans) = 3;
  end
  if ( do_coher )
    %% force coherence to be real
    spectra(:,do_coher) = real(Pxy .* conj(Pxy)) ./ Pxx ./ Pyy;
    spect_type(do_coher) = 4;
  end
  if ( do_ypower )
    spectra(:,do_ypower) = Pyy / scale;
    spect_type(do_ypower) = 5;
  end
  freq = [0:psd_len-1].' * ( Fs / Nfft );
  %%
  %% range='shift': Shift zero-frequency to the middle
  if ( range == 2 )
    len2 = fix((Nfft+1)/2);
    spectra = [ spectra(len2+1:Nfft,:); spectra(1:len2,:)];
    freq    = [ freq(len2+1:Nfft)-Fs; freq(1:len2)];
    if ( conf>0 )
      Vxx = [ Vxx(len2+1:Nfft,:); Vxx(1:len2,:)];
    end
  end
  %%
  %%  RETURN RESULTS or PLOT
  if ( nargout>=2 && conf>0 )
    varargout{2} = Vxx;
  end
  if ( nargout>=(2+(conf>0)) )
    %% frequency is 2nd or 3rd return value,
    %% depends on if 2nd is confidence interval
    varargout{2+(conf>0)} = freq;
  end
  if ( nargout>=1 )
    varargout{1} = spectra;
  else
    %%
    %% Plot the spectra if there are no return variables.
    plot_title=;
    for ii = 1: n_results
      if ( conf>0 && spect_type(ii)==1 )
        Vxxxx = Vxx;
      else
        Vxxxx = ;
      end
      if ( n_results > 1 )
        figure();
        end
      if ( plot_type == 1 )
        plot(freq,[abs(spectra(:,ii)) Vxxxx]);
      elseif ( plot_type == 2 )
        semilogx(freq,[abs(spectra(:,ii)) Vxxxx]);
      elseif ( plot_type == 3 )
        semilogy(freq,[abs(spectra(:,ii)) Vxxxx]);
      elseif ( plot_type == 4 )
        loglog(freq,[abs(spectra(:,ii)) Vxxxx]);
      elseif ( plot_type == 5 )  % db
        ylabel( 'amplitude (dB)' );
        plot(freq,[10*log10(abs(spectra(:,ii))) 10*log10(abs(Vxxxx))]);
      end
      title( char(plot_title(spect_type(ii),:)) );
      ylabel( 'amplitude' );
      %% Plot phase of cross spectrum and transfer function
      if ( spect_type(ii)==2 || spect_type(ii)==3 )
        figure();
        if ( plot_type==2 || plot_type==4 )
          semilogx(freq,180/pi*angle(spectra(:,ii)));
        else
          plot(freq,180/pi*angle(spectra(:,ii)));
        end
        title( char(plot_title(spect_type(ii),:)) );
        ylabel( 'phase' );
      end
    end %for
  end 
end
end

%!demo
%! fflush(stdout);
%! rand('seed',2038014164);
%! a = [ 1.0 -1.6216505 1.1102795 -0.4621741 0.2075552 -0.018756746 ];
%! white = rand(1,16384);
%! signal = detrend(filter(0.70181,a,white));
%! % frequency shift by modulating with exp(j.omega.t) 
%! skewed = signal.*exp(2*pi*i*2/25*[1:16384]);
%! Fs = 25; % sampling frequency
%! hold off
%! pwelch([]);
%! pwelch(signal);
%! disp('Default settings: Fs=1Hz, overlap=0.5, no padding' )
%! input('Onesided power spectral density (real data). Press ENTER', 's' );
%! hold on
%! pwelch(skewed);
%! disp('Frequency-shifted complex data.  Twosided wrap-around spectrum.' );
%! input('Area is same as one-sided spectrum. Press ENTER', 's' );
%! pwelch(signal,'shift','semilogy');
%! input('Twosided, centred zero-frequency, lin-log plot. Press ENTER', 's' );
%! hold off
%! figure();
%! pwelch(skewed,[],[],[],Fs,'shift','semilogy');
%! input('Actual Fs=25 Hz. Note change of scales. Press ENTER', 's' );
%! pwelch(skewed,[],[],[],Fs,0.95,'shift','semilogy');
%! input('Spectral density with 95% confidence interval. Press ENTER', 's' );
%! pwelch('R12+');
%! pwelch(signal,'squared');
%! input('Spectral density with Matlab R12 defaults. Press ENTER', 's' );
%! figure();
%! pwelch([]);
%! pwelch(signal,3640,[],4096,2*pi,[],'no-strip');
%! input('Same spectrum with 95% confidence interval. Press ENTER', 's' );
%! figure();
%! pwelch(signal,[],[],[],2*pi,0.95,'no-strip');
%! input('95% confidence interval with native defaults. Press ENTER', 's' );
%! pwelch(signal,64,[],[],2*pi,'no-strip');
%! input('Only 32 frequency values in this spectrum. Press ENTER', 's' );
%! hold on
%! pwelch(signal,64,[],256,2*pi,'no-strip');
%! input('4:1 zero padding gives artificial smoothing. Press ENTER', 's' );
%! figure();
%! pwelch('psd');
%! pwelch(signal,'squared');
%! input('Just like Matlab spectrum.welch(...) defaults. Press ENTER', 's' );
%! hold off
%! pwelch({});
%! pwelch(white,signal,'trans','coher','short')
%! input('Transfer and coherence functions. Press ENTER', 's' );
%! disp('Use "close all" to remove plotting windows.' );


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