hermbasis.m

linear time－frequency toolbox

function [V]=hermbasis(L)
%HERMBASIS  Orthonormal basis of discrete Hermite functions.
%   Usage:  V=hermbasis(L);
%
%   HERMBASIS(L) will compute an orthonormal basis of discrete Hermite
%   functions of length L. The vectors are returned as columns in the
%   output.
%
%   All the vectors in the output are eigenvectors of the discrete Fourier
%   transform, and resemble samplings of the continuous Hermite functions
%   to some degree.
%
%   SEE ALSO:  DFT, PHERM
%
%   REFERENCES:
%     H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay. The Fractional Fourier
%     Transform. John Wiley and Sons, 2001.

H=(spdiag(ones(L-1,1),-1)+spdiag(ones(L-1,1),1)+...
    spdiag(2*cos((0:L-1)*2*pi/L)-4));

H(1,L)=1;
H(L,1)=1;

H=H*pi/(i*2*pi)^2;

[V,D]=eig(H);

% If L is not a factor of 4, then all the eigenvalues of the tridiagonal
% matrix are distinct. If L IS a factor of 4, then one eigenvalue has
% multiplicity 2, and we must split the eigenspace belonging to this
% eigenvalue into a an even and an odd subspace.
if mod(L,4)==0
  x=V(:,L/2);
  x_e=(x+involute(x))/2;
  x_o=(x-involute(x))/2;

  x_e=x_e/norm(x_e);
  x_o=x_o/norm(x_o);

  V(:,L/2)=x_o;
  V(:,L/2+1)=x_e;

end;