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## Copyright (C) 2012 Benjamin Lewis <benjf5@gmail.com>
##
## 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/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{t} =} lscomplex (@var{time}, @var{mag}, @var{maxfreq}, @var{numcoeff}, @var{numoctaves})
##
## Return the complex least-squares transform of the (@var{time},@var{mag})
## series, considering frequencies up to @var{maxfreq}, over @var{numoctaves}
## octaves and @var{numcoeff} coefficients.
##
## @seealso{lsreal}
## @end deftypefn
function transform = lscomplex (t, x, omegamax, ncoeff, noctave)
## VECTOR ONLY, and since t and x have the same number of
## entries, there's no problem.
n = length (t);
transform = zeros (1, ncoeff * noctave);
o = omegamax;
omul = 2 ^ (- 1 / ncoeff);
for iter = 1:ncoeff * noctave
ot = o .* t;
## See the paper for the expression below
transform(iter) = sum ((cos (ot) - (sin (ot) .* i)) .* x) / n;
## Advance the transform to the next coefficient in the octave
o *= omul;
endfor
endfunction
%!test
%! maxfreq = 4 / ( 2 * pi );
%! t = [0:0.008:8];
%! x = ( 2 .* sin (maxfreq .* t) +
%! 3 .* sin ( (3 / 4) * maxfreq .* t)-
%! 0.5 .* sin ((1/4) * maxfreq .* t) -
%! 0.2 .* cos (maxfreq .* t) +
%! cos ((1/4) * maxfreq .* t));
%! o = [ maxfreq , 3 / 4 * maxfreq , 1 / 4 * maxfreq ];
%! assert (lscomplex (t, x, maxfreq, 2, 2),
%! [(-0.400924546169395 - 2.371555305867469i), ...
%! (1.218065147708429 - 2.256125004156890i), ...
%! (1.935428592212907 - 1.539488163739336i), ...
%! (2.136692292751917 - 0.980532175174563i)], 5e-10);