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function c=dcti(f,L,dim)
%DCTI Discrete Cosine Transform type I
% Usage: c=dcti(f);
% c=dcti(f,L);
% c=dcti(f,[],dim);
% c=dcti(f,L,dim);
%
% `dcti(f)` computes the discrete cosine transform of type I 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
% non-singleton dimension.
%
% `dcti(f,L)` zero-pads or truncates *f* to length *L* before doing the
% transformation.
%
% `dcti(f,[],dim)` or `dcti(f,L,dim)` applies the transformation along
% dimension *dim*.
%
% The transform is real (output is real if input is real) and
% it is orthonormal.
%
% This transform is its own inverse.
%
% Let f be a signal of length *L*, let $c=dcti(f)$ and define the vector
% *w* of length *L* by
%
% .. w = [1/sqrt(2) 1 1 1 1 ...1/sqrt(2)]
%
%
% .. math:: w\left(n\right)=\begin{cases}\frac{1}{\sqrt{2}} & \text{if }n=0\text{ or }n=L-1\\1 & \text{otherwise}\end{cases}
%
% Then
%
% .. L-1
% c(n+1) = sqrt(2/(L-1)) * sum w(n+1)*w(m+1)*f(m+1)*cos(pi*n*m/(L-1))
% m=0
%
% .. math:: c\left(n+1\right)=\sqrt{\frac{2}{L-1}}\sum_{m=0}^{L-1}w\left(n\right)w\left(m\right)f\left(m+1\right)\cos\left(\frac{\pi nm}{L-1}\right)
%
% The implementation of this functions uses a simple algorithm that require
% an FFT of length *2L-2*, which might potentially be the product of a large
% prime number. This may cause the function to sometimes execute slowly.
% If guaranteed high speed is a concern, please consider using one of the
% other DCT transforms.
%
% See also: dctii, dctiv, dsti
%
% References: rayi90 wi94
% AUTHOR: Peter Soendergaard
% TESTING: TEST_PUREFREQ
% REFERENCE: REF_DCTI
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,'DCTI');
if ~isempty(L)
f=postpad(f,L);
end;
if L==1
c=f;
else
c=zeros(L,W);
f2=[f;flipud(f(2:L-1,:))]/sqrt(2);
f2(1,:)=f2(1,:)*sqrt(2);
f2(L,:)=f2(L,:)*sqrt(2);
% Do DFT.
s1=fft(f2)/sqrt(2*L-2);
% This could be done by a repmat instead.
for w=1:W
c(:,w)=s1(1:L,w)+[0;s1(2*L-2:-1:L+1,w);0];
end;
c(2:L-1,:)=c(2:L-1,:)/sqrt(2);
if isreal(f)
c=real(c);
end;
end;
c=assert_sigreshape_post(c,dim,permutedsize,order);