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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 consine transform of type III of the
% input signal f. If f is a matrix, then the transformation is applied to
% each column. For N-D arrays, the transformation is applied to the first
% dimension.
%
% DCTIII(f,L) zero-pads or truncates f to length L before doing the
% transformation.
%
% DCTIII(f,[],dim) applies the transformation along dimension dim.
% DCTIII(f,L,dim) does the same, but pads or truncates to length L.
%
% The transform is real (output is real if input is real) and
% it is 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
%N w = [1/sqrt(2) 1 1 1 1 ...]
%L \[w\left(n\right)=\begin{cases}\frac{1}{\sqrt{2}} & \text{if }n=0\\1 & \text{otherwise}\end{cases}\]
% Then
%M
%M L-1
%M c(n+1) = sqrt(2/L) * sum w(n+1)*f(m+1)*cos(pi*(n+.5)*m/L)
%M m=0
%F \[
%F c\left(n+1\right)=\sqrt{\frac{2}{L}}\sum_{m=0}^{L-1}w\left(n\right)f\left(m+1\right)\cos\left(\frac{\pi}{L}\left(n+\frac{1}{2}\right)m\right)
%F \]
%
% See also: dctii, dctiv, dstii
%
%R rayi90 wi94
% AUTHOR: Peter Soendergaard
% TESTING: TEST_PUREFREQ
% REFERENCE: REF_DCTIII
error(nargchk(1,3,nargin));
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=zeros(2*L,W);
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,:);