dcti.m

linear time－frequency toolbox

function c=dcti(f,N,dim)
%DCTI  Discrete Cosine Transform type I
%   Usage:  c=dcti(f);
%           c=dcti(f,N);
%           c=dcti(f,[],dim);
%           c=dcti(f,N,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
%   dimension.
%
%   DCTI(f,N) zero-pads or truncates f to length N before doing the
%   transformation.
%
%   DCTI(f,[],dim) applies the transformation along dimension dim. 
%   DCTI(f,N,dim) does the same, but pads or truncates to length N.
%   
%   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)]
%   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 
%
%   The implementation of this functions uses a simple algorithm that require
%   an FFT of length 2N-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:
%     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.

error(nargchk(1,3,nargin));

if nargin<3
  dim=1;
end;

if nargin<2
  N=[];
end;
    
D=ndims(f);
if (prod(size(dim))~=1 || ~isnumeric(dim))
  error('dim must be a scalar.');
end;
if rem(dim,1)~=0
  error('dim must be an integer.');
end;
if (dim<1) || (dim>D)
  error(sprintf('dim must be in the range from 1 to %d.',D));
end;

if (prod(size(N))>1 || ~isnumeric(dim))
  error('N must be a scalar or [].');
end;
if (~isempty(N) && rem(dim,1)~=0)
  error('N must be an integer.');
end;


if dim>1
  order=[dim, 1:dim-1,dim+1:D];

  % Put the desired dimension first.
  f=permute(f,order);

end;

% Remember the exact size for later.
permutedsize=size(f);
  
% Reshape f to a matrix.
f=reshape(f,size(f,1),prod(size(f))/size(f,1));

if ~isempty(N)
  f=postpad(f,N);

  if dim>1
    % Remember that we changed the length of the first dim.
    permutedsize(1)=N;
  end;
end;

L=size(f,1);
W=size(f,2);

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;

% Restore the original, permuted shape.
c=reshape(c,permutedsize);

if dim>1
  % Undo the permutation.
  c=ipermute(c,order);
end;