dctiii.m

linear time－frequency toolbox

function c=dctiii(f)
%DCTIII  Discrete Consine Transform type III
%   Usage:  c=dctiii(f);
%           c=dctiii(f,N);
%           c=dctiii(f,[],dim);
%           c=dctiii(f,N,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,N) zero-pads or truncates f to length N before doing the
%   transformation.
%
%   DCTIII(f,[],dim) applies the transformation along dimension dim. 
%   DCTIII(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 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(n+1)*f(m+1)*cos(pi*(n+.5)*m/L) 
%                          m=0 
%
%   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.

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);
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;


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

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

% 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,:);