DCTIII - Discrete Consine Transform type III

function c=dctiii(f,L,dim)
%DCTIII  Discrete Consine Transform type III
%   Usage:  c=dctiii(f);
%           c=dctiii(f,L);
%           c=dctiii(f,[],dim);
%           c=dctiii(f,L,dim);
%
%   DCTIII(f) computes the discrete cosine transform of type III of the
%   input signal f. If f is multi-dimensional, the transformation is
%   applied along the first non-singleton dimension.
%
%   DCTIII(f,L) zero-pads or truncates f to length L before doing the
%   transformation.
%
%   DCTIII(f,[],dim) or DCTIII(f,L,dim) applies the transformation along
%   dimension dim.
%
%   The transform is real (output is real if input is real) and orthonormal.
%
%   This is the inverse of DCTII.
%
%   Let f be a signal of length L, let c=DCTIII(f) and define the vector
%   w of length L by  
%
%       w = [1/sqrt(2) 1 1 1 1 ...]
%
%   Then 
%
%                          L-1
%     c(n+1) = sqrt(2/L) * sum w(m+1)*f(m+1)*cos(pi*(n+.5)*m/L) 
%                          m=0 
%
%   Examples:
%   ---------
%
%   The following figures show the first 4 basis functions of the DCTIII of
%   length 20:
%
%     % The dctii is the adjoint of dctiii.
%     F=dctii(eye(20));
%
%     for ii=1:4
%       subplot(4,1,ii);
%       stem(F(:,ii));
%     end;
%
%   See also:  dctii, dctiv, dstii
%
%   References:
%     K. Rao and P. Yip. Discrete Cosine Transform, Algorithms, Advantages,
%     Applications. Academic Press, 1990.
%     
%     M. V. Wickerhauser. Adapted wavelet analysis from theory to software.
%     Wellesley-Cambridge Press, Wellesley, MA, 1994.
%     
%
%   Url: http://ltfat.github.io/doc/fourier/dctiii.html

% Copyright (C) 2005-2023 Peter L. Soendergaard <peter@sonderport.dk> and others.
% This file is part of LTFAT version 2.6.0
%
% 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: Peter L. Søndergaard
%   TESTING: TEST_PUREFREQ
%   REFERENCE: REF_DCTIII

complainif_argnonotinrange(nargin,1,3,mfilename);

if nargin<3
  dim=[];
end;

if nargin<2
  L=[];
end;

[f,L,Ls,W,dim,permutedsize,order]=assert_sigreshape_pre(f,L,dim,'DCTIII');

if ~isempty(L)
  f=postpad(f,L);
end;
c=comp_dct(f,3);
% c=zeros(2*L,W,assert_classname(f));
% 
% m1=1/sqrt(2)*exp(-(0:L-1)*pi*i/(2*L)).';
% m1(1)=1;
%   
% m2=1/sqrt(2)*exp((L-1:-1:1)*pi*i/(2*L)).';
% 
% for w=1:W
%   c(:,w)=[m1.*f(:,w);0;m2.*f(L:-1:2,w)];
% end;
% 
% c=fft(c)/sqrt(L);
% 
% c=c(1:L,:);
% 
% if isreal(f)
%   c=real(c);
% end;

c=assert_sigreshape_post(c,dim,permutedsize,order);

% This is a slow, but convenient way of expressing the above algorithm.
%R=1/sqrt(2)*[diag(exp(-(0:L-1)*pi*i/(2*L)));...
%	     zeros(1,L); ...
%	     [zeros(L-1,1),flipud(diag(exp((1:L-1)*pi*i/(2*L))))]];

%R(1,1)=1;

%c=fft(R*f)/sqrt(L);

%c=c(1:L,:);