involute.m

linear time－frequency toolbox

function f=involute(f,dim);
%INVOLUTE  Involution 
%   Usage: finv=involute(f);
%          finv=involute(f,dim);
%
%   INVOLUTE(f) will return the involution of f.
%
%   INVOLUTE(f,dim) will return the involution of f along dimension dim.
%   This can for instance be used to calculate the 2D involution:
% 
%         f=involute(f,1);
%         f=involute(f,2);
% 
%   The involution finv of f is given by
% 
%         finv(l+1)=conj(f(mod(-l,L)+1));
% 
%   for l=0,...,L-1.
%
%   The relation between conjugation, Fourier transformation and involution
%   is expressed by
% 
%       conj(dft(f)) == dft(involute(f))
% 
%   for all signals f. The inverse discrete Fourier transform can be
%   expressed by
% 
%       idft(f) == conj(involute(dft(f)));
% 
%   SEE ALSO:  DFT, PCONV

% Assert correct input.

error(nargchk(1,2,nargin));

if nargin==1
  dim=1;
else
  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;
end;

if dim>1
  D=ndims(f);
  order=[dim, 1:dim-1,dim+1:D];
  f=permute(f,order);
end;

L=size(f,1);

% This is where the calculation is performed.
% The reshape(...,size(f) ensures that f will keep its
% original shape if it is multidimensional.
f=reshape(conj([f(1,:); ...
	  flipud(f(2:L,:))]),size(f));

if dim>1
  f=ipermute(f,order);
end;