%% Copyright (C) 2006 Peter V. Lanspeary <pvl@mecheng.adelaide.edu.au>
%%
%% 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 3 of the License, 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 = [1 2 3 4 5;  %% native
                3 4 1 2 5;  %% Matlab R11- pwelch
                1 2 3 4 0;  %% Matlab R12+ pwelch
                1 2 3 4 5]; %% 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 = [ 1 5 5 5 ];
    plot_type = plot_type(compatib);
    %%  rm_mean: 3='long-mean'; 0='no-strip'|'none'
    rm_mean   = [ 3 0 0 0 ];
    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=['power spectrum x ';
                  'cross spectrum   ';
                  'transfer function';
                  'coherence        ';
                  'power spectrum y ' ];
      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.' );
