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function gd=nsgabdual(g,a,M,varargin)
%-*- texinfo -*-
%@deftypefn {Function} nsgabdual
%@verbatim
%NSGABDUAL Canonical dual window for non-stationary Gabor frames
% Usage: gd=nsgabdual(g,a,M);
% gd=nsgabdual(g,a,M,L);
%
% Input parameters:
% g : Cell array of windows.
% a : Vector of time shift.
% M : Vector of numbers of channels.
% L : Transform length.
% Output parameters:
% gd : Cell array of canonical dual windows
%
% NSGABDUAL(g,a,M,L) computes the canonical dual windows of the
% non-stationary discrete Gabor frame defined by windows given in g an
% time-shifts given by a.
%
% NSGABDUAL is designed to be used with the functions NSDGT and
% INSDGT. See the help on NSDGT for more details about the variables
% structure.
%
% The computed dual windows are only valid for the 'painless case', that
% is to say that they ensure perfect reconstruction only if for each
% window the number of frequency channels used for computation of NSDGT is
% greater than or equal to the window length. This correspond to cases
% for which the frame operator is diagonal.
%
%
% References:
% P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
% Theory, implementation and applications of nonstationary Gabor frames.
% J. Comput. Appl. Math., 236(6):1481--1496, 2011.
%
%@end verbatim
%@strong{Url}: @url{http://ltfat.github.io/doc/nonstatgab/nsgabdual.html}
%@seealso{nsgabtight, nsdgt, insdgt}
%@end deftypefn
% Copyright (C) 2005-2016 Peter L. Soendergaard <peter@sonderport.dk>.
% This file is part of LTFAT version 2.2.0
%
% 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/>.
% AUTHOR : Florent Jaillet
% TESTING: TEST_NSDGT
% REFERENCE:
if nargin<3
error('%s: Too few input parameters.',upper(mfilename));
end;
if ~isnumeric(a)
error('%s: a must be numeric.',upper(mfilename));
end;
if ~isnumeric(M)
error('%s: M must be numeric.',upper(mfilename));
end;
definput.keyvals.L=sum(a);
[flags,kv,L]=ltfatarghelper({'L'},definput,varargin);
timepos=cumsum(a)-a(1);
N=length(a);
[g,info]=nsgabwin(g,a,M);
a=info.a;
M=info.M;
if info.isfac
if info.ispainless
f=zeros(L,1); % Diagonal of the frame operator
% Compute the diagonal of the frame operator:
f=nsgabframediag(g,a,M);
% Initialize the result with g
gd=g;
% Correct each window to ensure perfect reconstrution
for ii=1:N
shift=floor(length(g{ii})/2);
tempind=mod((1:length(g{ii}))+timepos(ii)-shift-1,L)+1;
gd{ii}(:)=circshift(circshift(g{ii},shift)./f(tempind),-shift);
end
else
if 0
% Convert to freq. domain and run filterbankdual
% The code does not work
gf=cell(1,N);
gd=cell(1,N);
for ii=1:N
gf{ii}=circshift(fft(fir2long(g{ii},L)),timepos(ii));
end;
gfd=filterbankdual(gf,M);
for ii=1:N
gd{ii}=ifft(circshift(gfd{ii},-timepos(ii)));
end;
else
error('Not implemented yet.');
end;
end;
else
error(['%s: The canonical dual frame of this system is not a ' ...
'non-stationary Gabor frame. You must call an iterative ' ...
'method to perform the desired inverstion. Please see ' ...
'FRANAITER or FRSYNITER.'],upper(mfilename));
end;
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