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## 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;
endif
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' );
endif
## 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(:);
endif
##
## 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.' );
endif
## force COLUMN vector
y = varargin{1}(:);
arg2_is_y = 1;
break;
endif
endfor
##
## 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;
endif
elseif ( strcmp(arg, 'cross' ) )
if ( ~do_cross )
n_results = n_results+1;
do_cross = n_results;
endif
elseif ( strcmp(arg, 'trans' ) )
if ( ~do_trans )
n_results = n_results+1;
do_trans = n_results;
endif
elseif ( strcmp(arg, 'coher' ) )
if ( ~do_coher )
n_results = n_results+1;
do_coher = n_results;
endif
elseif ( strcmp(arg, 'ypower' ) )
if ( ~do_ypower )
n_results = n_results+1;
do_ypower = n_results;
endif
else
error( 'pwelch: string arg %d illegal value: %s', iarg+1, arg );
endif
## 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' );
endif
##
## 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;
endif
##
## -- "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;
endif
##
## 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(:);
endif
else
is_win = 0;
endif
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);
endif
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 );
endif
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 );
endif
Nfft = arg;
##
else
error( 'pwelch: arg %d must be string', iarg+1 );
endif
endfor
if ( conf>0 && (n_results && ~do_power ) )
error('pwelch: can give confidence interval for x power spectrum only' );
endif
##
## 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;
endif
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 );
endif
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;
endif
## make Hann window (don't depend on sigproc)
xx = seg_len - 1;
window = 0.5 - 0.5 * cos( (2*pi/xx)*[0:xx].' );
endif
##
## 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 );
endif
## make Hamming window (don't depend on sigproc)
xx = seg_len - 1;
window = 0.54 - 0.46 * cos( (2*pi/xx)*[0:xx].' );
endif
if ( isempty(Nfft) )
Nfft = max( 256, 2^ceil(log(seg_len)*nearly_one/log_two) );
endif
##
## 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;
endif
## make Hamming window (don't depend on sigproc)
xx = seg_len - 1;
window = 0.54 - 0.46 * cos( (2*pi/xx)*[0:xx].' );
endif
## Now we know segment length,
## so we can set default overlap as number of samples
if ( ~isempty(overlap) )
overlap = fix(seg_len * overlap / 100 );
endif
if ( isempty(Nfft) )
Nfft = max( 256, 2^ceil(log(seg_len)*nearly_one/log_two) );
endif
##
## 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;
endif
seg_len = 2 ^ ceil( log(sqrt(x_len/(1-overlap)))*nearly_one/log_two );
endif
## make Hamming window (don't depend on sigproc)
xx = seg_len - 1;
window = 0.54 - 0.46 * cos( (2*pi/xx)*[0:xx].' );
endif
## Now we know segment length,
## so we can set default overlap as number of samples
if ( ~isempty(overlap) )
overlap = fix(seg_len * overlap);
endif
##
## calculate FFT length
if ( isempty(Nfft) )
Nfft = seg_len;
endif
if ( is_sloppy )
Nfft = 2 ^ ceil( log(Nfft) * nearly_one / log_two );
endif
endif
## 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 );
endif
##
## 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)];
endif
x_len = seg_len;
endif
## 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;
endif
if ( arg2_is_y || need_Pxy)
y = y - sum( y(1:x_len) ) / x_len;
endif
endif
##
## 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 = [];
endif
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);
endif
fft_x = fft(xx);
endif
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);
endif
fft_y = fft(yy);
endif
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;
endif
endif
if ( need_Pxy )
## Pxy (cross power spectrum) is complex. Do not force to be real.
Pxy = Pxy + fft_y .* conj(fft_x);
endif
if ( need_Pyy )
## force Pyy to be real
Pyy = Pyy + real(fft_y .* conj(fft_y));
endif
n_ffts = n_ffts +1;
endfor
##
## 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);
endif
endif
##
## 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];
endif
endif
if ( need_Pxy )
Pxy = Pxy(1:psd_len) + conj([0; Pxy(Nfft:-1:psd_len+1); 0]);
endif
if ( need_Pyy )
Pyy = Pyy(1:psd_len) + [0; Pyy(Nfft:-1:psd_len+1); 0];
endif
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)];
endif
endif
if ( need_Pxy )
Pxy = Pxy(1:psd_len) + conj([0; Pxy(Nfft:-1:psd_len+1)]);
endif
if ( need_Pyy )
Pyy = Pyy(1:psd_len) + [0; Pyy(Nfft:-1:psd_len+1)];
endif
endif
else # two-sided (and shifted)
psd_len = Nfft;
endif
## 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;
endif
endif
if ( do_cross )
spectra(:,do_cross) = Pxy / scale;
spect_type(do_cross) = 2;
endif
if ( do_trans )
spectra(:,do_trans) = Pxy ./ Pxx;
spect_type(do_trans) = 3;
endif
if ( do_coher )
## force coherence to be real
spectra(:,do_coher) = real(Pxy .* conj(Pxy)) ./ Pxx ./ Pyy;
spect_type(do_coher) = 4;
endif
if ( do_ypower )
spectra(:,do_ypower) = Pyy / scale;
spect_type(do_ypower) = 5;
endif
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,:)];
endif
endif
##
## RETURN RESULTS or PLOT
if ( nargout>=2 && conf>0 )
varargout{2} = Vxx;
endif
if ( nargout>=(2+(conf>0)) )
## frequency is 2nd or 3rd return value,
## depends on if 2nd is confidence interval
varargout{2+(conf>0)} = freq;
endif
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 = [];
endif
if ( n_results > 1 )
figure();
endif
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))]);
endif
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)));
endif
title( char(plot_title(spect_type(ii),:)) );
ylabel( 'phase' );
endif
endfor
endif
endif
endfunction
%!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);
%! hold on
%! pwelch(skewed);
%! pwelch(signal,'shift','semilogy');
%! hold off
%! figure();
%! pwelch(skewed,[],[],[],Fs,'shift','semilogy');
%! pwelch(skewed,[],[],[],Fs,0.95,'shift','semilogy');
%! pwelch('R12+');
%! pwelch(signal,'squared');
%! figure();
%! pwelch([]);
%! pwelch(signal,3640,[],4096,2*pi,[],'no-strip');
%! figure();
%! pwelch(signal,[],[],[],2*pi,0.95,'no-strip');
%! pwelch(signal,64,[],[],2*pi,'no-strip');
%! hold on
%! pwelch(signal,64,[],256,2*pi,'no-strip');
%! figure();
%! pwelch('psd');
%! pwelch(signal,'squared');
%! hold off
%! pwelch({});
%! pwelch(white,signal,'trans','coher','short')

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