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filtfilt.m

## Copyright (C) 1999 Paul Kienzle
## Copyright (C) 2007 Francesco Potortì
## Copyright (C) 2008 Luca Citi
##
## 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 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: y = filtfilt(b, a, x)
##
## Forward and reverse filter the signal. This corrects for phase
## distortion introduced by a one-pass filter, though it does square the
## magnitude response in the process. That's the theory at least.  In
## practice the phase correction is not perfect, and magnitude response
## is distorted, particularly in the stop band.
####
## Example
##    [b, a]=butter(3, 0.1);                   % 10 Hz low-pass filter
##    t = 0:0.01:1.0;                         % 1 second sample
##    x=sin(2*pi*t*2.3)+0.25*randn(size(t));  % 2.3 Hz sinusoid+noise
##    y = filtfilt(b,a,x); z = filter(b,a,x); % apply filter
##    plot(t,x,';data;',t,y,';filtfilt;',t,z,';filter;')

## Changelog:
## 2000 02 pkienzle@kienzle.powernet.co.uk
##      - pad with zeros to load up the state vector on filter reverse.
##      - add example
## 2007 12 pot@gnu.org
##        - use filtic to compute initial and final states
##      - work for multiple columns as well
## 2008 12 lciti@essex.ac.uk
##      - fixed instability issues with IIR filters and noisy inputs
##        - initial states computed according to Likhterov & Kopeika, 2003
##      - use of a "reflection method" to reduce end effects
##      - added some basic tests

## TODO:  (pkienzle) My version seems to have similar quality to matlab,
##    but both are pretty bad.  They do remove gross lag errors, though.


function y = filtfilt(b, a, x)
  if (nargin != 3)
    usage("y=filtfilt(b,a,x)");
  end

  if ((rotate = (rows(x)==1)))      # a row vector
    x = x(:);                 # make it a column vector
  endif
  
  lx = size(x,1);
  a = a(:).';
  b = b(:).';
  lb = length(b);
  la = length(a);
  n = max(lb, la);
  lrefl = 3 * (n - 1);
  if la < n, a(n) = 0; end
  if lb < n, b(n) = 0; end

  ## Compute a the initial state taking inspiration from
  ## Likhterov & Kopeika, 2003. "Hardware-efficient technique for
  ##     minimizing startup transients in Direct Form II digital filters"
  kdc = sum(b) / sum(a);
  if (abs(kdc) < inf) # neither NaN nor +/- Inf
    si = fliplr(cumsum(fliplr(b - kdc * a)));
  else
    si = zeros(size(a)); # fall back to zero initialization
  end
  si(1) = ;

  for (c = 1:columns(x))      # filter all columns, one by one
    v = ; # a column vector

    ## Do forward and reverse filtering
    v = filter(b,a,v,si*v(1));                   # forward filter
    v = flipud(filter(b,a,flipud(v),si*v(end))); # reverse filter
    y(:,c) = v((lrefl+1):(lx+lrefl));
  endfor

  if (rotate)                 # x was a row vector
    y = rot90(y);       # rotate it back
  endif

endfunction

%!error filtfilt ();

%!error filtfilt (1, 2, 3, 4);

%!test
%! randn('state',0);
%! r = randn(1,200);
%! [b,a] = butter(10, [.2, .25]);
%! yfb = filtfilt(b, a, r);
%! assert (size(r), size(yfb));
%! assert (mean(abs(yfb)) < 1e3);
%! assert (mean(abs(yfb)) < mean(abs(r)));
%! ybf = fliplr(filtfilt(b, a, fliplr(r)));
%! assert (mean(abs(ybf)) < 1e3);
%! assert (mean(abs(ybf)) < mean(abs(r)));

%!test
%! randn('state',0);
%! r = randn(1,1000);
%! s = 10 * sin(pi * 4e-2 * (1:length(r)));
%! [b,a] = cheby1(2, .5, [4e-4 8e-2]);
%! y = filtfilt(b, a, r+s);
%! assert (size(r), size(y));
%! assert (mean(abs(y)) < 1e3);
%! assert (corrcoef(s(250:750), y(250:750)) > .95)
%! [b,a] = butter(2, [4e-4 8e-2]);
%! yb = filtfilt(b, a, r+s);
%! assert (mean(abs(yb)) < 1e3);
%! assert (corrcoef(y, yb) > .99)

%!test
%! randn('state',0);
%! r = randn(1,1000);
%! s = 10 * sin(pi * 4e-2 * (1:length(r)));
%! [b,a] = butter(2, [4e-4 8e-2]);
%! y = filtfilt(b, a, [r.' s.']);
%! yr = filtfilt(b, a, r);
%! ys = filtfilt(b, a, s);
%! assert (y, [yr.' ys.']);
%! y2 = filtfilt(b.', a.', [r.' s.']);
%! assert (y, y2);


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