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function f=ref_idwilt(c,g,a,M)
%-*- texinfo -*-
%@deftypefn {Function} ref_idwilt
%@verbatim
%REF_DWILT Reference Inverse Discrete Wilson Transform
% Usage: f=ref_idwilt(c,g,a,M);
%
%@end verbatim
%@strong{Url}: @url{http://ltfat.github.io/doc/reference/ref_idwilt.html}
%@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/>.
% Setup transformation matrix.
L=size(g,1);
W=size(c,2);
N=L/a;
F=zeros(L,M*N);
% Weight coefficients.
l=(0:L-1).';
pif=0;
if 1
% This version uses sines and cosines to express the basis functions.
for n=0:N/2-1
% Do the unmodulated coefficient.
F(:,2*M*n+1)=circshift(g,2*a*n);
% Setting this to -n*a should produce a time-invariant transform.
timeinv=0; %-n*a;
% m odd case
for m=1:2:M-1
F(:,m+2*M*n+1) = sqrt(2)*sin(pi*m/M*(l+timeinv)+pif).*circshift(g,2*n*a);
F(:,m+2*M*n+M+1) = sqrt(2)*cos(pi*m/M*(l+timeinv)+pif).*circshift(g,(2*n+1)*a);
end;
% m even case
for m=2:2:M-1
F(:,m+2*M*n+1) = sqrt(2)*cos(pi*m/M*(l+timeinv)+pif).*circshift(g,2*n*a);
F(:,m+2*M*n+M+1) = sqrt(2)*sin(pi*m/M*(l+timeinv)+pif).*circshift(g,(2*n+1)*a);
end;
% Most modulated coefficient, Nyquest frequency.
if mod(M,2)==0
F(:,M+2*M*n+1)=(-1).^(l+timeinv).*circshift(g,2*n*a);
else
F(:,M+2*M*n+1)=(-1).^(l+timeinv).*circshift(g,(2*n+1)*a);
end;
end;
else
% This version uses a cosine,
for n=0:N/2-1
% Do the unmodulated coefficient.
F(:,2*M*n+1)=circshift(g,2*a*n);
timeinv=-n*a;
% m odd case
for m=1:2:M-1
F(:,m+2*M*n+1) = sqrt(2)*cos(pi*m/M*(l+timeinv-M/2)+pif).*circshift(g,2*n*a);
F(:,m+2*M*n+M+1) = sqrt(2)*sin(pi*m/M*(l+timeinv-M/2-a)+pif).*circshift(g,(2*n+1)*a);
end;
% m even case
for m=2:2:M-1
F(:,m+2*M*n+1) = sqrt(2)*cos(pi*m/M*(l+timeinv-M/2)+pif).*circshift(g,2*n*a);
F(:,m+2*M*n+M+1) = sqrt(2)*sin(pi*m/M*(l+timeinv-M/2-a)+pif).*circshift(g,(2*n+1)*a);
end;
% Most modulated coefficient, Nyquest frequency.
if mod(M,2)==0
F(:,M+2*M*n+1)=(-1).^(l+timeinv).*circshift(g,2*n*a);
else
F(:,M+2*M*n+1)=(-1).^(l+timeinv-a).*circshift(g,(2*n+1)*a);
end;
end;
end;
f=F*c;
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